adelson s tile and snake illusions: a helmholtzian type of ... · adelson s illusions. indeed, the...

Spatial Vision Vol 18 No 1 pp 25ndash72 (2005) VSP 2005Also available online - wwwvsppubcom

Adelsonrsquos tile and snake illusions A Helmholtzian typeof simultaneous lightness contrast

ALEXANDER D LOGVINENKO 1lowast and DEBORAH A ROSS 2lowastlowast1 Department of Vision Sciences Glasgow Caledonian University Glasgow G4 0BA UK2 School of Psychology The Queenrsquos University of Belfast Belfast BT7 1NN UK

Received 26 March 2004 accepted 28 April 2004

AbstractmdashAdelsonrsquos tile snake and some other lightness illusions of the same type were measuredwith the Munsell neutral scale for twenty observers It was shown that theories based on low-levelluminance contrast processing could hardly explain these illusions Neither can those based onluminance X-junctions On the other hand Helmholtzrsquos idea that simultaneous lightness contrastoriginates from an error in judgement of apparent illumination has been elaborated so as to accountfor the tile and snake illusions as well as other demonstrations presented in this report

Keywords Lightness perception anchoring effect apparent illumination apparent transparencyapparent illuminationlightness invariance lightness constancy

INTRODUCTION

A series of elegant lightness illusions produced by Adelson (1993) have had a far-reaching impact on lightness perception studies in the last decade (eg Albright1994 Blakeslee and McCourt 2003 Kingdom 1999 2002 Paradiso 2000Schirillo and Shevell 2002) Although a number of attempts have been made toexplain (Bressan 2001 Gilchrist et al 1999 Todorovic 2003 Wishart et al 1997)andor to model them (eg Blakeslee and McCourt 1999 Dakin and Bex 2002McCann 2001 Ross and Pessoa 2000) they still remain quite a challenge to visualscientists For instance we still do not know why the tile (Fig 1a) and snake (Fig 2)illusions are much stronger than the classical simultaneous lightness contrast effect(Fig 3) Does it follow that Adelsonrsquos illusions and classical simultaneous lightness

E-mail alogvinenkogcalacukE-mail drossqubacuk

26 A D Logvinenko and D A Ross

(a) (b)

Light strip Dark strip Light strip Dark strip

Reflectance Median Reflectance Median Reflectance Median Reflectance Median

079 925 048 775 079 900 048 725048 675 043 725 048 600 029 525043 475 029 550 043 450 026 425

Figure 1 (a) Tile pattern (after Adelson 1993) The original Adelson tile pattern is modified so that(i) there is a patch (with reflectance 048) that is included in both the lsquolightrsquo and lsquodarkrsquo strips (ii)the reflectances of the patches in the lsquolightrsquo and lsquodarkrsquo strips are in approximately same proportions048 029 and 079 048 The diamonds in both strips have the same reflectance (043) Howeverthey appear different in lightness (b) The modified tile pattern The diamonds in the lsquolightrsquo stripshave a reflectance of 043 while those in the lsquodarkrsquo strips have a reflectance of 026 yet they lookremarkably similar

contrast are different visual phenomena If not which features of Adelsonrsquospatterns strengthen the effect

There are at least four differing explanatory approaches to Adelsonrsquos illusions(i) the low-level explanation based on the local contrast around the target (diamond)borders (eg Cornsweet 1970 Whittle 1994a b) and its recent modifications(eg Blakeslee and McCourt 1999 2003 McCann 2001) (ii) a mid-levelexplanation emphasizing the role of the borders between strips or the type ofthe luminance junctions across the strips (Adelson 1993 2000 Anderson 1997Todorovic 1997) (iii) the anchoring theory (Gilchrist et al 1999) and (iv) a high-level explanation enunciated by H Helmholtz (1867) who thought that lightnessillusions such as simultaneous lightness contrast resulted from a lsquomisjudgementof illuminationrsquo (Adelson 1993 Adelson and Pentland 1996 Kingdom 2003Logvinenko 1999)

A Helmholtzian type of simultaneous lightness contrast 27

Light strip Dark strip

Reflectance Median Reflectance Median

073 875 052 875061 700 037 625052 550 031 550

Figure 2 Snake pattern (after Adelson 2000) The snake pattern is also modified so that make theratios of the abutting patches in the lsquolightrsquo and lsquodarkrsquo strips as close as possible (073 037 and 061 031) The diamonds in the lsquodarkrsquo and lsquolightrsquo strips have the same reflectance 052 however theylook different

Figure 3 The classical simultaneous lightness contrast display When presented against differentbackgrounds the square targets of the same reflectance look slightly different in lightness

It is easy to show that local luminance contrast plays a minor role if any inAdelsonrsquos illusions Indeed the tile illusion is known to almost disappear aftera slight rearrangement of the pattern (Adelson 1993) For instance the localcontrast around diamonds borders in Figs 4 and 5 is the same as in Fig 1a but




079 925 048 700048 700 043 650043 600 029 475

Figure 4 Ribbon pattern It is made from the tile pattern (Fig 1a) by shifting the lsquolightrsquo strips to theright so as to align the patches with the reflectance 048

the illusion is hardly observed in these pictures A similar dependence on thespatial rearrangement which does not affect the local luminance contrast aroundthe diamondsrsquo border was recently shown for the snake illusion too (Logvinenkoet al in press) On the other hand the snake illusion may be observed though inreduced form even when the target diamonds have the same local contrast (Fig 6)

Likewise the borders between strips and luminance junctions are not necessaryfor observing Adelsonrsquos tile and snake illusions Blurring the border between thestrips for instance does not reduce the snake illusion despite the distortion of someof the luminance junctions and borders (Fig 7) Furthermore the tile illusionmay even be enhanced by blurring the border between strips (Logvinenko 19992002b Logvinenko and Kane 2003) Figures 8 and 9 also show that the illusorylightness shift between the diamonds can be observed without borders betweenstrips Figure 8 is made up of only light strips of Fig 1a whereas Fig 9 consists ofonly dark strips of Fig 1a As one can see the difference in the diamondsrsquo lightnessin Figs 8 and 9 is quite large

A Helmholtzian type of explanation is essentially based on the assumption thatlightness and apparent illumination are not independent that is they are lockedin a certain relationship such as an apparent illuminationlightness invariance(Logvinenko 1997 1999) It implies that if a misjudgement of illumination occurs




079 925 048 700048 750 043 650043 625 029 500

Figure 5 Hex pattern (after Adelson 1993) It is made up from the patches of the same reflectanceas the tile pattern (Fig 1a)

it must affect lightness For example the tile pattern in Fig 1a is perceived as a 3Dwall of blocks viewed through a striped filter implying that apparent illuminationof the alternate strips is different This difference in apparent illumination betweenthe strips brings about the corresponding difference in lightness mdash the diamonds inthe strips which appear to have the lower illumination look lighter in accord withthe apparent illuminationlightness invariance

Since lightness constancy can be thought as a particular case of this invariance(Logvinenko 1997) Adelsonrsquos tile illusion and lightness constancy might have acommon explanation Fig 1b provides a strong support for this conjecture that isthat Adelsonrsquos tile illusion and lightness constancy are two sides of the same coinmdash the apparent illuminationlightness invariance It is the same pattern as Fig 1awhere the luminance ratio across the horizontal borders is 165 except that thediamonds in the lighter strips have reflectivity 165 times that of the diamonds in thedarker strips If the visual system interprets the luminance ratio across the horizontalborders in Figs 1a and 1b as that of the stripsrsquo illuminations then Fig 1b shouldproduce no illusion at all that is the diamonds in alternate rows in Fig 1b shouldlook the same Indeed being physically different the diamonds in the alternate rowsin Fig 1b look very similar thus exhibiting a nearly perfect lightness constancyphenomenon




043 500 043 725

Figure 6 Iso-contrast snake pattern While a striped structure seems to be quite distinctive it is anillusion So is an apparent difference in lightness between the diamonds Both diamonds are the same(reflectance 043) and they are surrounded by the surface of the same reflectance (050) Apparentstrips emerge from horizontal arrays of dark (reflectance 023) and light (reflectance 077) hoops

It is important for such an explanation to specify what sort of information thevisual system uses to infer that the alternating strips in Figs 1 and 2 are differentlyilluminated Adelson believes that it is luminance X-junctions that signal thedifference in apparent illumination between the strips (Adelson 1993 2000) Alsoit may be constancy of the luminance ratio across the borders (Logvinenko 2002d)Both may contribute to the illusions since removing the borders (Figs 8 and 9)reduces the illusion However neither luminance X-junctions nor constancy of theluminance ratio are the only cues for apparent illumination in Figs 1 and 2 since thewalls of blocks depicted in Figs 8 and 9 still look differently illuminated despite thefact that there is neither luminance X-junctions nor constancy of the luminance ratioin these pictures This implies that the global pictorial content of Figs 8 and 9 canin itself bring about the difference between the apparent (pictorial) illuminationsin these figures thus inducing (in line with a Helmholtzian type prediction) thecorresponding lightness shift

Still the illusion can be experienced when the global pictorial content (ie thewall of blocks) is absent Indeed the illusion emerges even for isolated strips(Fig 10) While there is neither 3D pictorial content nor luminance X-junctionsin Fig 10 the diamonds in the upper strip still look darker than those in the bottomstrip The fact that the isolation of the strips in Fig 1a only reduces the illusion




051 575 051 900

Figure 7 Blurred snake pattern The horizontal borders of the snake pattern (Fig 2) were blurred sothat the luminance varies sinusoidally along the vertical dimension

but does not completely eliminate it indicates that Helmholtzian misjudgement ofillumination is not the only cause of the illusion There should be some other localrather than global factors contributing to the effect The present report is devotedto studying the contribution of various factors local as well as global to Adelsonrsquostile and snake illusions

EXPERIMENT 1

The purpose of this experiment was (i) to measure the strength of the illusion forthe pictures presented above (ii) to study quantitatively the contribution of differentconfigurational elements by breaking the tile and snake patterns into their parts mdashstrips tiles and patches

The main experiment was preceded by a preliminary one during which theobservers had been trained to evaluate the lightness of simple grey patches on awhite background

Methods

Observers Twenty observers (8 males and 12 females age range 20ndash41) tookpart in the experiment All the observers were naiumlve as to the purpose of theexperiment All had normal or corrected to normal vision


Reflectance Median

079 925048 675043 500

Figure 8 lsquoLightrsquo wall of blocks pattern It comprises only lsquolightrsquo strips in Fig 1a

Stimuli and apparatus In the preliminary experiment we used eight greysquares on the white background the reflectances of which were as follows 079048 043 039 031 029 023 and 016 This choice was motivated by the factthat the patches constituting the tile and snake patterns had these reflectances

In the main experiment the observers were presented with the patterns (134 times134 cm) shown above (Figs 1ndash9) along with isolated strips for the tile (Fig 10) andsnake patterns (Figs 11 and 12) and isolated tiles for the tile pattern (Figs 13ndash15)Among these eight classical simultaneous lightness contrast displays (Fig 3) withdifferent target squares were used The reflectance of the target square was one ofthe eight values that were used in the preliminary experiment The rationale was tomeasure the classical simultaneous contrast effect for all the patches involved in thetile and snake patterns

Each pattern printed on an A4 sheet of white paper was mounted on the whitewall in front of an observer who sat at a distance of 1 m in an experimental roomwith ordinary illumination Two tungsten lamps were used to make an illuminationof the test pictures as homogeneous across space as possible Luminance measure-ments from eight different points across the display area showed that the illumina-tion variation was statistically insignificant (p = 036) The mean luminance forthe white background of the display area was 100 cdm2


Reflectance Median

048 750043 700029 525

Figure 9 lsquoDarkrsquo wall of blocks pattern It comprises only lsquodarkrsquo strips in Fig 1a

The 31-point Munsell neutral scale was used to evaluate the lightness of the testpatches The Munsell chips (2 times 5 cm each) were attached to the same white wallnext to the stimulus display

Procedure and experimental design Each stimulus display was presented oneat a time to an observer who was asked to select a Munsell chip that matchedthe test patch (Since the diamond patches (reflectance 043) and the patches withreflectance 048 were included in both light and dark strips of the tile pattern eachof them counted as two different test patches tested independently) Using a laserpointer the experimenter pointed out (in random order) which particular patch wasto be matched Observers also used a laser pointer to indicate their match Afteran observer completed the matches for all patches in the stimulus pattern it wasreplaced by another pattern on a random basis

The whole set of stimulus patterns was divided into four groups the picturesof just one group being presented in a random order during one experimentalsession lasting approximately half an hour Not more than one session a daywas conducted with each observer Each session was repeated ten times in thepreliminary experiment and five times in the main experiment with each observerso that in all two hundred matches were made for each test patch in the preliminaryand one hundred in the main experiment1




079 925 048 750048 650 043 700043 575 029 5375

Figure 10 Two separated strips of which the tile pattern (Fig 1a) is made up

Results

The results of the preliminary experiment are presented in Table 1 and in Fig 16 asa multiple boxplot graph (lsquoextractedrsquo histograms) Among other things the graphshows the median matches and the interquartile ranges for all eight reflectancesstudied in the preliminary experiment2

Table 2 and Fig 17 show the classical simultaneous lightness contrast effect forvarious target squares (Fig 3) To be more exact Fig 17 represents lsquoextractedrsquohistograms of differences between Munsell matches made for the same target squareon the white and black backgrounds While the Friedman rank test showed thereflectance of the target square in Fig 3 was significant (p = 004) as follows fromFig 17 the simultaneous lightness contrast shift was approximately the same for alltarget squares irrespective of their reflectance

It should be mentioned that while the Munsell neutral scale is generally believedto be of the interval type there is not sufficient evidence for this On the contraryit was argued that lightness matching was of the ordinal nature (Logvinenko2002d) So we chose to use non-parametric statistics in this study (with 5 levelof significance) Specifically we used the Wilcoxon signed-rank test to establishif there was a significant difference between lightness of the test objects in twodifferent surroundings (eg in lsquolightrsquo and lsquodarkrsquo strips in the tile and snake patterns)




073 875 052 875061 775 037 600052 625 031 475

Figure 11 Strips constituting the snake pattern (Fig 2)

If this difference was statistically significant we claimed that a lightness illusionwas observed To evaluate the magnitude of the illusion we used a non-parametricestimator of the shift between two distributions of the matches (ie obtained forlsquolightrsquo and lsquodarkrsquo strips in the tile and snake patterns) mdash the HodgesndashLehmannestimator3 associated with Wilcoxonrsquos signed rank statistic (Hollander and Wolfe1973 p 33) As seen in Table 2 the simultaneous lightness contrast effect in termsof the HodgesndashLehmann estimator varied from 0375 to 0625 Munsell units4

The median Munsell matches obtained in the main experiment for each patch arepresented beneath each pattern (Figs 1ndash15) Table 3 presents the median and meanMunsell matches obtained for the diamonds (reflectance 043 for Figs 1 2 4ndash10and 13ndash15 and 052 for Figs 11 and 12) in the lsquolightrsquo and lsquodarkrsquo surround Fig 18shows the lightness shift between the diamonds in the lsquolightrsquo and lsquodarkrsquo surroundThe HodgesndashLehmann estimator of the shift can be found in Table 3

As one can see the ribbon (Fig 4) and hex (Fig 5) patterns produced thesmallest though statistically significant lightness shifts (Wilcoxon signed-ranknormal statistic with correction Z = 518 and 522 respectively p lt 001) TheWilcoxon signed-rank test showed a significant difference between the simultaneouslightness contrast effect measured for the test patch of the same reflectance asthe diamonds (ie 043) and the lightness shift obtained for the ribbon pattern(Z = 233 p = 002) Therefore the illusion produced by the ribbon pattern




077 900 050 800050 675 043 675043 550 023 425

Figure 12 Strips constituting the iso-contrast snake pattern (Fig 6)

(Fig 4) is even weaker than the simultaneous lightness contrast effect (Fig 3)While the HodgesndashLehmann estimator for the hex pattern (Fig 5) was also foundto be smaller than that for the simultaneous lightness contrast display there wasno significant difference between these two distributions (Wilcoxon signed-ranknormal statistic with correction Z = 156 p = 012)

The lightness shift observed for the isolated tiles was approximately of the samemagnitude as the simultaneous lightness contrast effect The Friedman rank testshowed a non-significant difference between these patterns for both the diamond(p = 029) and the patch with reflectance 048 (p = 023)

The lightness shifts produced by the isolated strips were significantly strongerthan that produced by the isolated tiles Specifically the Friedman rank test showeda significant effect when the data registered for the isolated strips cut from the tilepattern were combined with those registered for isolated tiles (Friedman χ2 = 475df = 3 p lt 001)

In line with the previous studies a remarkably strong lightness shift was obtainedfor the tile and snake patterns the snake pattern producing the strongest illusion(Wilcoxon signed-rank normal statistic with correction Z = 670 p lt 001)Moreover the lightness shift observed for the isolated strips from the snake pattern(Fig 11) was of the same strength as that observed for the tile pattern (Fig 1a) therebeing no significant difference between them (Z = 034 p = 073) The blurred-


Light tile Dark tile


079 925 048 725048 650 043 650043 575 029 5125

Figure 13 Tiles from the Fig 1a

snake pattern (Fig 7) produced as strong an illusion as the original snake pattern(Fig 2) The Wilcoxon signed-rank test showed no significant differences betweenthe lightness shifts for these two patterns (Z = 049 p = 063) While the illusionproduced by the iso-contrast snake pattern (Fig 6) was significantly smaller thanthat measured for the tile-pattern in Fig 1a (Wilcoxon signed-rank normal statisticwith correction Z = 441 p lt 001) it was much higher than for the simultaneouslightness contrast effect (Fig 3)

The difference in lightness between the diamonds observed for the wall-of-blockpatterns (Figs 8 and 9) was significantly smaller than the lightness shift producedby the tile pattern in Fig 1a (Wilcoxon rank-sum5 normal statistic with correctionZ = 447 p lt 001) but larger than that produced by isolated tile strips in Fig 10(Wilcoxon rank-sum normal statistic with correction Z = 205 p = 004)

It should be pointed out that a significant lightness shift was observed not only forthe diamonds but also for the patches with reflectance 048 (Table 4 and Fig 19)While significantly less it was in the same direction as the lightness shift for thediamonds with one exception (Fig 20) mdash in Fig 5 it looked significantly darkerin the lsquodarkrsquo surround and lighter in the lsquolightrsquo (the Wilcoxon signed-rank testp lt 001)

The darkest patch in the tile pattern (reflectance 029) also changed its appearance(Table 5 and Fig 21) Specifically it became significantly lighter in the tile pattern




079 925 048 725048 650 043 650043 575 029 500

Figure 14 The same tiles as in Fig 13 except that the diamonds are separated from the other patches

in Fig 1a For example having considerably lower reflectance than the diamond itlooked lighter than the diamond in the lsquolightrsquo strip

Thus we observe that in the tile pattern (Fig 1a) all the patches in the lsquodarkrsquo stripsappeared lighter and those in the lsquolightrsquo strips darker except for the lightest patchwith reflectance 079 the median Munsell match for which was the same (925) forall of the patterns6

A similar lsquolightness shiftrsquo between alternating strips was observed in Fig 1b tooNote that the magnitude of this shift was approximately as much as to make thediamonds in the alternating strips in Fig 1b look nearly the same Indeed themedian difference between the Munsell matches (as well as the HodgesndashLehmannestimator) for the diamonds in the light and dark strips for Fig 1b was 025 Whilebeing statistically significant (the signed rank Wilcoxon test p lt 001) the illusionin the modified tile pattern (Fig 1b) was reduced by a factor of 10 as compared tothat in Fig 1a

Discussion

These results provide strong evidence against any low-level explanation of the tileand snake illusions based on the local luminance contrast between the diamondsand their immediate surround Indeed the diamonds in the tile (Fig 1a) ribbon




079 925 048 7375048 675 043 675043 575 029 500

Figure 15 Another set of tiles from Fig 1a

(Fig 4) and hex (Fig 5) patterns as well as in the isolated strips (Fig 10) and tiles(Figs 13ndash15) patterns have the same local contrast However the illusion observedfor these patterns varies in strength across a rather wide range mdash from 025 Munsellunits (the ribbon pattern) to 2375 Munsell units (the tile pattern) There should besome other factor which reduces the tile and snake illusions by nearly a factor of 10Furthermore as shown recently the tile illusion completely disappears when the tilepattern is implemented as a real 3D wall of blocks with the same diamondsurroundlocal contrast (Logvinenko et al 2002)

On the other hand the iso-contrast snake pattern (Fig 6) produces the illusionwhich is much stronger than the ribbon (Fig 4) and hex (Fig 5) patterns Whilethe diamondsurround local contrast is equal for all the strips in this pattern ityields almost as strong an illusion as that produced by the tile pattern Hencethe difference in local contrast is neither necessary nor sufficient to experience theillusion

The mid-level explanation based on the luminance junctions and constancy ofthe luminance ratio only has not been supported by the data either Reallyremoving the borders between the strips in the snake pattern (Fig 7) was notshown to affect the illusion Also quite large differences between the correspondingdiamondsrsquo lightness was found in Figs 8ndash10 where there was no striped structure


Figure 16 The results of the preliminary experiment Reflectance of the target is on the horizontalaxis Munsell match is along the vertical axis The ends of the boxes are the first and third quartilesHence the height of the boxes is the interquartile range A horizontal line in the box is drawn at themedian An upper whisker is drawn at the largest match that is less than or equal to the third quartileplus 15 times the interquartile range Likewise a bottom whisker is drawn at the smallest match thatis greater than or equal to the first quartile plus 15 times the interquartile range All the matcheswhich fall outsides of the range marked by the whiskers are indicated by individual lines

Table 1Median and mean Munsell matches obtained in the preliminary experiment

Target reflectance Median Mean

016 400 415023 500 504029 575 568031 575 583039 650 649043 675 676048 700 701079 900 886

Therefore the luminance junctions and sharp luminance borders are not necessaryfor observing the illusion

Still the illusion produced by the plain walls (Figs 8 and 9) as well as the isolatedstrips (Fig 10) is significantly smaller than for the tile pattern (Fig 1a) The obvious


Figure 17 Classical simultaneous lightness contrast effect The horizontal axis is reflectance of thetarget square in Fig 3 The difference between Munsell matches for the black and white backgroundsis on the vertical axis

Table 2Median and mean Munsell matches and the HodgesndashLehmann estimator obtained for the classicalsimultaneous contrast display (Fig 3)

Target Light surround Dark surround HodgesndashLehmannreflectance estimatorMedian Mean Median Mean

016 400 403 475 466 050023 500 510 575 571 050029 575 578 625 637 050031 600 597 650 663 0625039 650 651 700 712 0625043 700 690 725 740 050048 725 721 775 776 0625079 900 886 925 925 0375

difference between these patterns is that Fig 1a contains the luminance borderwith a constant luminance ratio across it (and the X-luminance junctions) whereasFigs 8ndash10 do not Hence the luminance junctions and constancy of the luminanceratio may have an enhancing effect on the illusion This issue will be looked at inmore detail in the next section (Experiment 2)


Figure 18 Lightness illusory shift observed for the diamonds (reflectance 043) in various displays

Table 3Median and mean Munsell matches and the HodgesndashLehmann estimator obtained for the diamondsin Figs 1a 1b 2 and 4ndash15

Figure Reflectance Light strip Dark strip HodgesndashLehmannnumber estimatorMedian Mean Median Mean

1a 043 475 487 725 735 23751b 043026 450 436 425 429 0252 052 550 527 875 870 33754 043 600 596 650 640 0255 043 625 611 650 650 0256 043 500 516 725 719 21257 051 575 552 900 886 337589 043 500 513 700 675 17510 043 575 545 700 685 137511 052 625 605 875 853 25012 043 550 532 675 663 12513 043 575 559 650 654 087514 043 575 553 650 649 10015 043 575 554 675 659 0875


Figure 19 Lightness illusory shift observed for the patch with reflectance 048 in various displays

Table 4Median and mean Munsell matches and the HodgesndashLehmann estimator for the patch with re-flectance 048

Figure Light strip Dark strip HodgesndashLehmannnumber estimatorMedian Mean Median Mean

1a 675 656 775 767 1001b 600 591 725 728 1254 700 698 700 690 0005 750 747 700 702 minus02589 675 668 750 739 07510 650 653 750 743 07513 650 650 725 709 037514 650 645 725 711 05015 675 663 7375 731 050

At the same time the data testify unequivocally in favour of the Helmholtziantype of explanation based on the idea of misjudgement of illumination Accordingto this idea the black half of the background in the classical simultaneous lightnesscontrast display might be perceived as if it is less illuminated than the white half(Fig 3) If this is the case then the luminance edge dividing the backgroundinto the black and white halves gives rise to not only a lightness edge but to anapparent illumination edge as well However it remains unclear in Helmholtzian


Figure 20 The HodgesndashLehmann estimator of the illusory lightness shift for the diamonds(reflectance 043) and the patch with reflectance 048 in various displays

writings why such a lsquomisjudgementrsquo of the illumination of the black half of thebackground should affect the lightness We believe that this is because the apparentillumination and lightness are interlocked into the apparent illuminationlightnessinvariance (Logvinenko 1997 1999) Furthermore a luminance edge determinesa reciprocal pair of lightness and apparent illumination edges As a result givena particular contrast of the luminance border if the apparent illumination of theblack background is underestimated it entails a corresponding overestimation ofthe lightness of the target on this background and of the background itself Whileit is not clear whether such an explanation is valid for the classical simultaneouslightness contrast it certainly works for the tile and snake illusions

Consider for instance the original and modified tile patterns (Figs 1a and 1b) Atfirst glance we seem to have obtained a paradoxical result When the diamonds inthe alternated rows in Fig 1a are physically the same they appear very different butwhen they are different (Fig 1b) they look quite similar in lightness However thisis exactly what would be expected if the tile illusion and lightness constancy have acommon root (the apparent illuminationlightness invariance) If the visual systeminterprets the alternative strips in Fig 1b as being differently illuminated and takesinto account this difference when assigning the same lightness to the diamonds indifferent rows then it is more than likely that the same taking-into-account willoccur for Fig 1a as well

It should be pointed out however that the idea of lsquomisjudgement of illuminationrsquois not specific enough to be a genuine explanation It requires further elaborationFirst of all one has to specify what illumination is supposed to be subject tolsquomisjudgementrsquo In the present context it is worth distinguishing between anabsolute (ambient) and relative illuminations (Kingdom 2002 Logvinenko 1997)An increase of the intensity of the only light source in the scene results in a change



Table 5Median and mean Munsell matches for the patch with reflectance 029

Figure number Median Mean

1a 550 5411b 525 5184 475 4895 500 5179 525 49910 5375 53413 5125 51614 500 52015 500 524

in only the absolute not relative illumination A difference in relative illuminationcan be observed between shadowed and non-shadowed (highlighted) areas7 Theluminance ratio between the shadowed and non-shadowed areas remains constantwhen the ambient illumination changes (Logvinenko 2002d Marr 1982 p 90)As known there are two types of shadows namely cast and attached ones Theformer are caused by the spatial layout of the scene The latter arise due to thespatial relief of a particular object Accordingly we shall distinguish between the


relative illumination of the cast-shadow type and the relative illumination of theattached shadow type

The difference in illumination of all the three types can be observed in ourpictures8 For instance the difference in the apparent ambient illumination is seenbetween Figs 8 and 9 The difference in the apparent relative illumination of thecast-shadow type is clearly observed between the horizontal strips in Figs 1 and 2The lateral sides of the cubes in Fig 5 differ in the apparent relative illumination ofthe attached-shadow type

It is easy to see that every picture presented above is readily segmented intoareas of equal apparent illumination We shall call them equi-illuminated frames9According to the three types of apparent illumination there are three levels of equi-illuminated frames These levels are hierarchally subordinated More specificallya pictorial fragment can belong to only one equi-illuminated frame of the samelevel but it can belong to different equi-illuminated frames of different levels Forinstance in Fig 5 there is just one equi-illuminated frame at the level of ambientillumination and at the level of cast shadow (ie the pattern as a whole) and thereare three equi-illuminated frames at the level of attached shadow (the sides of theblocks) Likewise Fig 1a contains the same three equi-illuminated frames at thelevel of attached shadow and one equi-illuminated frame at the level of ambientillumination but in this picture there are two different equi-illuminated frames atthe level of cast shadow (ie the horizontal strips) In Fig 10 there are two differentequi-illuminated frames at the level of ambient illumination (the strips) one equi-illuminated frame at the level of cast shadow and three equi-illuminated frames atthe level of attached shadow

The apparent illuminationlightness invariance predicts that two equiluminant (ieof the same luminance) patches belonging to different equi-illuminated frames willbe perceived as being of a different lightness More specifically the equiluminantpatch belonging to the darker equi-illuminated frame will appear lighter and theequiluminant patch belonging to the brighter equi-illuminated frame will lookdarker It accounts for why the diamonds in the dark strips of the tile pattern appearlighter than the same diamonds in the light strips mdash these alternating strips belongto the different equi-illuminated frames at the level of cast shadow Furthermoreit also explains why the patch with the reflectance 048 in the hex pattern (Fig 5)appeared darker in the dark strip contrary to what is observed in the tile pattern(Fig 1a) where it appeared lighter in the dark strip In Fig 5 this patch belongsto different equi-illuminated frames only at one level (attached shadow) On thecontrary in Fig 1a this patch belongs to different equi-illuminated frames at twolevels (attached and cast shadow) At the level of attached shadow it belongs to themore illuminated frame This explains why in Fig 5 it looks darker10 However atthe level of cast shadow it belongs to the less illuminated frame thus it has to looklighter As we can see in Fig 1a this apparent perceptual conflict is resolved infavour of the equi-illuminated frame at the level of cast shadow that is the patch inquestion looks lighter Nevertheless the lightness shift observed for the patch with


reflectance 048 is generally lower as compared to that for the diamonds (Fig 20)Such a reduction of the illusory shift is a consequence of the perceptual conflict inwhich this patch is involved

A further problem is how the visual system carries out the segmentation of thewhole scene into equi-illuminated frames In other words what cues does thevisual system use to infer differences in illumination It is clear that such cuesmight be different at different levels of illumination For example a distributionof luminances in the whole scene may be an important source of informationabout the ambient illumination (Adelson 2000) If it is shifted towards the darker(respectively lighter) end in one scene as compared to another it may indicate thatthe ambient illumination in this scene is lower (respectively higher) than in theother Perhaps this is why Fig 8 looks more illuminated than Fig 9

As mentioned above the type of luminance junctions and the constancy of theluminance ratio across the luminance border may play an important role in thesegmentation into equi-illuminated frames at the level of cast shadow Indeedsplitting the tile pattern into separate strips where there are neither luminancejunctions nor luminance borders considerably reduces the illusion

As the segmentation into equi-illuminated frames at the level of attached shadowis intimately connected with the perception of 3D shape the classical depth cuesmay contribute to it thus affecting lightness perception While the role of depthcues in lightness perception is well-known (Bloj and Hurlbert 2002 Freeman etal 1993 Gilchrist 1977 Knill and Kersten 1991 Logvinenko and Menshikova1994 Mach 1959) it has not always been realised that their effect on lightness ismediated by that they first of all affect the apparent illumination and as a result ofthis mdash lightness

This explains why the tile illusion is so sensitive to spatial rearrangements ofthe pictorial content For example the ribbon pattern (Fig 4) differs from theoriginal tile pattern (Fig 1a) only by a small horizontal shift of the alternating strips(the patches with reflectance 048 are abutting in Fig 4 whereas they are shiftedrelative to each other in Fig 1a) However the illusion in Fig 4 nearly disappearsIt happens because the 3D pictorial content in Fig 4 is rather different (a ribbonagainst the black-white striped background) A new pictorial content invokes a newsegmentation into equi-illuminated frames In contrast with Fig 1a where there aretwo different equi-illuminated frames at the level of cast shadow Fig 4 containsonly one equi-illuminated frame at the level of cast shadow As all the diamondsbelong to the same equi-illuminated frame at the level of cast shadow they looknearly the same

The segmentation into equi-illuminated frames must be followed by evaluation ofhow frames differ from each other in terms of the illumination magnitude Havingclaimed this we do not necessarily mean that such evaluation takes place in termsof ratio or interval scale It might be the case that the visual system only decideswhich frame is lighter and which is darker In other words the segmentation maytake place only in ordinal terms


If the apparent illuminationlightness holds true then assignment of a particularillumination to different frames has to be accompanied by assigning a correspondinglightness to any luminance in a frame In other words we suggest that the apparentillumination of a frame plays the role of the lightness anchor within the frame

In the anchoring theory of lightness perception the maximal luminance in a frameis claimed to serve as an anchor (Gilchrist 2003 Gilchrist et al 1999) To be moreexact the region of the maximal luminance in a frame is supposed to be assignedwhite in this frame Such anchoring is equivalent to the suggestion that apparentillumination is assigned to equi-illuminated frames in the same proportion as thatof maximal illuminations in these frames It is easy to show that this predicts 100lightness constancy and huge simultaneous lightness contrast effect (Gilchrist1988) both predictions being obviously wrong11 The authors of the anchoringtheory resort to weighting the lightness values assigned to a given luminance indifferent frames so as to reconcile their predictions with the experimental dataHowever the lack of a strict definition of frame and weighting process itself makesthe anchoring theory unclear on this subject

The results suggest that the assigned apparent illuminations are not in the samerelation as the maximal luminances in the frames In other words the range of theassigned apparent illuminations is a great deal narrower than that of the maximalluminances in the equi-illuminated frames Such a compression of this range canbe accounted for if one assumes that it is maximal brightness rather than maximalluminance that underlies assigning the apparent illuminations12 Specifically if theapparent illuminations are assigned in direct proportion to the maximal brightnessesin the frames then the range of the assigned apparent illuminations will undergothe same compressive transformation as that relating brightness to luminance Forexample both WeberndashFechner and Stevens laws would predict such a compressionof the apparent illumination range

While we have not measured the apparent illumination in the pictures it is easy tosee that it is in line with the Helmholtzian account of the illusion presented aboveThe impression of the apparent illumination in the pictures generally correlateswith the strength of the illusion that is the greater the difference in the apparentillumination the greater the difference in the lightness Really the difference in theapparent illumination between alternating strips in Fig 1a is bigger than that of theisolated strips in Fig 10 This is in line with the fact that the illusion as measuredfor Fig 1a is stronger than that for Fig 10 On the other hand the difference inthe apparent illumination between walls in Figs 8 and 9 is clearly larger than thatbetween the isolated strips in Fig 10 which is in line with the reduction of theillusion in Fig 10 as compared to that in Figs 8 and 9

However the statistically significant difference in lightness between the diamondswas also found for isolated tiles (Figs 13ndash15) where a difference in apparentillumination can hardly be seen Therefore the Helmholtzian account is unlikely tobe appropriate here Moreover as shown elsewhere the patches may be separatedfrom the diamonds for quite a distance with the same result mdash the diamond


accompanied by the patch with reflectance 029 looks lighter than that accompaniedby the patch with reflectance 079 (Logvinenko 2002c) It was argued that such anillusion might be a result of the so-called anchoring effect well established in thepsychophysical literature (Luce and Galanter 1965)

EXPERIMENT 2

As pointed out above when the isolated strips (Fig 10) are combined so as tomake up the tile pattern (Fig 1a) the illusion considerably increases It occursbecause such combining gives rise to at least two cues which may signal that theluminance border between the strips is produced by an illumination edge One ofthese cues is the X-luminance junction (Adelson 1993) the other is luminance-ratioinvariance across the border (Marr 1982 p 90) Specifically the patches with thereflectance 079 and 048 in the light strips make the same luminance ratio (165)with the patches with the reflectance 048 and 029 in the dark strip respectivelyIt was shown recently that although luminance-ratio invariance is not necessary toproduce the illusion the illusion might disappear when it is significantly violated(Logvinenko 2002d) In this experiment we studied more systematically theeffect of violation of the luminance-ratio invariance on the tile illusion using moreobservers than in previous study (Logvinenko 2002d)

Methods

Altering the patch with reflectance 048 in the light strip of the tile pattern (Fig 1a)the luminance ratio (between this and the abutting patch with reflectance 029) wasvaried at the following levels 134 106 079 and 066 (Figs 22ndash25) As shownbefore the illusion could be reduced even by reversing the contrast polarity in justone column (Logvinenko 2002d) To study this effect in more detail we preparedthree more patterns where the contrast polarity was reversed in the same way asfor Fig 25 but only for one (Fig 26) two (Fig 27) or three (Fig 28) of the fivecolumns

The modified tile patterns (Figs 22ndash28) were presented to the same twentyobservers who took part in Experiment 1 The procedure and experimental designwere also the same as in Experiment 1 Observers were asked to match thelightness of the indicated patches with a chip from the 31-point Munsell neutralscale Lightness of all patches with different reflectance in each strip was measuredfive times for each observer Measurements in altered and unaltered columns weredone separately13

Results

As above the table beneath each figure presents the median Munsell matches for allthe patches measured in experiment Figure 29 and Table 6 show how the strength




079 925 048 750043 575 043 725039 500 029 475

Figure 22 An altered version of Fig 1a The patches with reflectance 048 in the light strips inFig 1a are replaced with those having reflectance 039 thus decreasing the luminance ratio from 160to 134 for these tiles

of the illusions varies with the luminance ratio at the border between the stripsWhen luminance ratio becomes less than 1 the contrast polarity at the strip borderfor this patch become reversed as compared to an unaltered pair As can be seenin Fig 29 the illusion decreases gradually when the luminance ratio decreasesbeing completely reduced for the lowest ratio 066 (Fig 25) The Wilcoxon signed-rank test showed no significant difference between the diamondsrsquo lightness in thealternating strips in Fig 25 (Z = 186 p = 006) this difference being significantin all the other pictures (Figs 22ndash24) While the effect of the luminance ratio onthe diamondsrsquo lightness in the lsquodarkrsquo strips was found to be significant (Friedmanχ2 = 969 df = 4 p lt 001) it was relatively small As seen in Fig 29 it isthe diamondsrsquo lightness in the lsquolightrsquo strips (where the alternation was made) thatmainly changed with the luminance ratio

Figure 30 and Table 7 show how the median lightness match measured in thealtered as well as unaltered columns depends on the number of columns alteredThe Munsell matches for the diamonds in the lsquodarkrsquo strips and unaltered columnswas the same for all Figs 26ndash28 (725 Munsell units) being equal to that for theoriginal tile pattern (Friedman χ2 = 290 df = 3 p = 041) While the matchfor the diamonds in the lsquodarkrsquo strips and altered columns significantly decreased




079 925 048 700043 600 043 675031 475 029 475


with the number of altered columns (Friedman χ2 = 4676 df = 3 p lt 001)the decrease was relatively small Once again the illusory change of lightness wasmainly observed in the lsquolightrsquo strips

Discussion

Decreasing the luminance ratio at the strip borders for one of two tiles in Adelsontile pattern was found to drastically reduce the illusion Such a gradual decreaseof the illusion strength in Figs 22ndash25 as compared to Fig 1a challenges both theaccount based on the luminance X-junctions and that based on the luminance ratioinvariance

Indeed the luminance X-junctions of the same sort are present in both Figs 1aand 23 However being rather large in Fig 1a (2375 Munsell units) the illusionin Fig 23 is reduced to the level observed for the classical simultaneous lightnesscontrast effect (0625 Munsell units) Such a reduction cannot be accounted forby a theory based on the luminance X-junctions since being ordinal in natureluminance junctions can only indicate a difference in the stripsrsquo illumination butnot the magnitude of this difference Moreover as pointed out by Kersten (1992Fig 155 p 215) one luminance X-junction is not enough to induce a difference




079 925 048 725043 650 043 6875023 425 029 500


in apparent illumination between the regions making the junction A series ofthe luminance X-junctions is necessary for an apparent illumination difference toemerge However when the luminances X-junctions of different types are arrangedtogether as in Figs 26ndash28 they become ambiguous

For instance in Figs 26ndash28 the type of luminance X-junctions is different forthe unaltered and altered columns It is double-contrast preserving in unalteredcolumns and single-contrast preserving in altered columns Single-contrast pre-serving X-junctions were found to produce a very small illusion (03125 Munsellunits) for Fig 24 and no illusion for Fig 25 So the theory based on X-junctionswould predict no illusion in altered columns and a full illusion in unaltered columnsin Figs 26ndash28 Quite the contrary the diamonds in the unaltered and altered columnsin Figs 26ndash28 look much the same (Fig 30) It means that the visual system showsa tendency towards global interpretation of the luminance distribution across stripsborders Being local by definition the concept of a luminance X-junction does notseem to be relevant to this tendency

The notion of the luminance ratio invariance fails to account for the resultseither Indeed Figs 22 and 23 induce a rather strong illusion despite that thereis no such invariance in these figures Strictly speaking none of the Figs 22ndash25




073 925 048 7375043 700 043 700016 3875 029 475


should produce an illusion since the luminance ratio is not constant across the stripsborders However it is only Fig 25 that induces no illusion the rest of thesefigures exhibiting a significant difference between the lightness of the diamondsin the lsquolightrsquo and lsquodarkrsquo strips contrary to this prediction Moreover we observe agradual decrease in the illusion strength from Fig 22 to Fig 25 What causes thisgradual deterioration of the illusion

It is unlikely that the illusion in Figs 22ndash25 is reduced because of the differencein the local contrast around the diamonds Truly the altered patch in Figs 22ndash25 becomes physically darker as the luminance ratio gets smaller Hence onemight argue that it is a result of this progressive reduction that leads to the apparentdarkening of the diamonds in the lsquolightrsquo strips (Fig 29) However as follows fromFig 30 a similar (only slightly less) reduction was observed for the diamonds inthe unaltered columns in Figs 26ndash28 where the local contrast around the diamondsremained unchanged Therefore it is hardly the local contrast that gradually reducesthe illusion in Figs 22ndash25

Still the notion of luminance ratio invariance can be further elaborated so as tobe applicable to Figs 22ndash25 too So far it was implicitly assumed that the opticalmedia represented in the pictures is perfect that is completely transparent When


Light strip mdash Dark strip mdashunaltered column unaltered column


079 925 048 775048 700 043 725043 6375 029 575

Light strip mdash Dark strip mdashaltered column altered column


079 925 048 775043 675 043 7375016 375 029 475

Figure 26 An altered version of Fig 1a The patches with reflectance 048 in the light strips inFig 1a are replaced with those having reflectance 016 but only in one column

it is not perfect (eg as in the presence of fog haze etc) the light coming throughto the eyes can be at least in the first approximation represented as a sum of twocomponents

l = I r + a

where l is the intensity of the light entering the eyes r is reflectance I and a aremultiplicative and additive constants which depend on the intensity of the incidentlight and the transmittance of the optical media respectively the latter emergingdue to scattering light

Such a relationship between the luminance and reflectance was called the at-mosphere transfer function by Adelson (2000 pp 346ndash348) When the opticalmedia (atmosphere in Adelsonrsquos terms) is completely clean the additive compo-




079 925 048 7875048 725 043 725043 650 029 600



079 925 048 7875043 700 043 725016 375 029 475

Figure 27 Same as Fig 26 except that an alteration was made in two columns

nent is zero and we have a perfect transmittance when the intensity of light comingto the eyes is multiplicatively related to the intensity of incident light and the sur-face reflectance In this case the luminance ratio across the illumination border isconstant (Logvinenko 2002d)

Figure 1a as well as Fig 2 simulates a situation when the atmosphere transferfunctions for the alternating strips differ only in multiplicative component bothhaving zero additive components On the other hand Fig 6 simulates the oppositecase when the atmosphere transfer functions for the alternating strips differ only inadditive component the multiplicative component being the same Indeed in Fig 6the reflectance of the strips the lighter and darker hoops were 050 077 and 023respectively Given a homogeneous illumination of the whole pattern say with theintensity I we have the following four intensities of the reflected light (Fig 31)




079 925 048 7875048 725 043 725043 675 029 600



079 925 048 800043 700 043 725016 375 029 475

Figure 28 Same as Fig 26 except that an alteration was made in three columns

l1 = 050I l2 = 023I lprime1 = 077I lprime2 = 050I It is easy to see that lprime1 = l1 + aprimeand lprime2 = l2 + aprime where aprime = 017I It follows first that the atmospheric functionfor the strips with the darker hoops can be represented as l = I r Second theatmospheric function for the strips with the lighter hoops differs only in an additiveconstant lprime = I r + aprime Such a difference between the strips in additive componentsimulates a haze and indeed a sort of an apparent haze14 over the strips with thelighter hoops is experienced in Fig 6 Note that an additive component brings aboutnot only an apparent haze but it induces quite a strong (225 Munsell units) lightnessillusion15 Adelson recently demonstrated a similar haze illusion (Adelson 2000Fig 24ndash17 p 348)

The haze illusion can be well understood within the same explanatory frameworkas the tile illusion Indeed if one of two regions of the same luminance looks hazywhereas the other clear it would mean that the surface viewed through the murky


Figure 29 Match data as a function of luminance ratio across the horizontal border in Figs 1 and 22ndash25

Table 6Median and mean Munsell matches and the HodgesndashLehmann estimator obtained for the diamondsin Figs 22ndash25


22 575 543 725 717 162523 600 596 675 674 062524 650 650 6875 681 0312525 700 689 700 697 0125

atmosphere should have less reflectivity to reflect the same amount of light Thusapparent haze reduces lightness16

When the optical transmittance is reduced thus the additive component is notzero the luminance ratio across the illumination border is not constant any longerIndeed let us consider a Mondrian-like pattern with reflectances r1 rn twoparts of which is viewed through two different translucent filters (Fig 33) Letlprime = I primer + aprime and lprimeprime = I primeprimer + aprimeprime be the atmosphere transfer functions ofthese filters Then lprimei lprimeprimei = (I primeri + aprime)(I primeprimeri + aprimeprime) is obviously not equal tolprimej lprimeprimej = (I primerj + aprime)(I primeprimerj + aprimeprime) Nevertheless the ratio of luminance differences(lprimei minus lprimej )(l

primeprimei minus lprimeprimej ) is invariant of reflectance Really

(lprimei minus lprimej )(lprimeprimei minus lprimeprimej ) = ((I prime + aprime) minus (I primerj + aprime))((I primeprimeri + aprimeprime) minus (I primeprimerj + aprimeprime))

= I prime(ri minus rj )Iprimeprime(ri minus rj ) = I primeI primeprime


Figure 30 Match data for Figs 1 and 25ndash28

Such luminance-difference ratio invariance can be used as a cue to segment thevisual scene into frames with the same atmosphere transfer function In otherwords this ratio can be used to distinguish reflectance edges from atmosphereborders Moreover since the ratio of luminance differences is equal to the ratioof the multiplicative constants in the atmosphere transfer functions this ratio can beused to infer reflectance from luminance

The haze illusion (Fig 6) shows however that when assigning lightness toluminance the visual system takes into account not only luminance-difference ratiobut the additive constant too Indeed direct calculation shows that the luminance-difference ratio across the horizontal stripsrsquo border in Fig 6 is equal to 1 It meansthat there is no shadow difference between the strips If lightness were assignedonly according to the luminance-difference ratio in this case the diamonds wouldappear equal in lightness thus no illusion being predicted


Tabl

e7

Med

ian

and

mea

nM

unse

llm

atch

esa

ndth

eH

odge

sndashL

ehm

ann

esti

mat

oro

btai

ned

for

the

diam

onds

inF

igs

26ndash2

8

Fig

ure

Alt

ered

Hod

gesndash

Una

lter

edH

odge

sndashnu

mbe

rL

ehm

ann

Leh

man

nes

tim

ator

esti

mat

orL

ight

stri

pD

ark

stri

pL

ight

stri

pD

ark

stri

p

Med

ian

Mea

nM

edia

nM

ean

Med

ian

Mea

nM

edia

nM

ean

266

756

697

375

735

062

56

375

625

725

736

112

527

700

689

725

727

037

56

506

497

257

260

7528

700

690

725

722

037

56

756

677

257

310

625


Figure 31 Luminancies constituting the iso-contrast snake pattern (Fig 6) evaluated under assump-tion that the illumination border is linear ie that between the strips (see explanation in the text)

Figure 32 Luminancies constituting the iso-contrast snake pattern (Fig 6) evaluated under assump-tion that the illumination border is snake-shaped (see explanation in the text)

Figure 33 A Mondrian-like pattern viewed through two different atmospheres The atmosphereborder is a curvilinear line dividing the pattern into the upper and bottom halves (see explanation inthe text)


Table 8The luminance difference ratio as evaluated for Figs 1a and 22ndash25

Fig 1a Fig 22 Fig 23 Fig 24 Fig 25

163 211 253 295 332

Table 9The additive component as evaluated for Figs 1a and 22ndash25


000 009 015 020 023

As distinct from figures 1a and 6 the atmosphere transfer functions for the alter-nating strips in Figs 22ndash25 differ in both multiplicative and additive componentsThe difference in the multiplicative component is experienced as a difference inapparent illumination (or shadow) The difference in the additive component is ex-perienced as an apparent haze

The luminance-difference ratio in Figs 22ndash25 can be evaluated as follows (079minusx)(048 minus 029) where x is the reflectance of the altered patch in the lsquolightrsquo stripThis ratio varies from 165 for Fig 1a to 33 for Fig 25 (Table 8) In other wordsthe multiplicative component for the lsquolightrsquo strips becomes more than three times asmuch as that for the lsquodarkrsquo strips for Fig 25 Such an increase of the luminance-difference ratio signals an even larger difference in apparent lightness between thediamonds in the alternating strips Hence if the luminance-difference ratio were theonly determinant of the illusion it should have become stronger in Figs 22ndash25 ascompared to Fig 1a

To evaluate the additive component let us assume for the sake of simplicitythat the additive component aprime is zero This amounts to assuming no haze forthe lsquolightrsquo strips (where the alteration was made) which is in line with what wesee in Figs 22ndash25 In this case one can use the episcotister luminance model17

(Gerbino 1994 pp 240ndash245) to evaluate the additive component aprimeprime for the lsquodarkrsquostrips in Figs 22ndash25 Direct calculation shows that the additive component aprimeprimerapidly increases when the reflectance of the altered patch in Figs 22ndash25 decreases(Table 9) which is in agreement with the amount of apparent haze in Figs 22ndash25Increasing the luminance-difference ratio and increasing the additive componentare in apparent conflict since the former contributes towards enhancing whereas thelatter towards diminishing the illusion Since the illusion gradually reduces fromFig 22 to Fig 25 the additive component proves to be more effective than themultiplicative component

It should be noted however that while a conflict between the additive andmultiplicative components explains the apparent difference in lightness between thediamonds in the different strips it remains unclear why the diamonds in the lsquolightrsquostrips come to look lighter and the diamonds in the lsquodarkrsquo strips undergo only little


change becoming slightly darker Note however that the same takes place also inthe original Adelson tile pattern (Fig 1a) that is the diamonds in the lsquolightrsquo stripsundergo the main change in appearance Indeed the diamonds in the lsquodarkrsquo stripsin Fig 1a become only slightly lighter as compared to how the patch with the samereflectance (043) looks in Figs 3 and 4

GENERAL DISCUSSION

A distinctive feature of psychology is that it is full of long-standing controversieswhich look as if they will never be solved (eg nativism vs empirism naturevs nurture) One such controversy is personified by the debate between H vonHelmholtz and E Hering on lightness simultaneous contrast (Kingdom 1997Turner 1994) Since these theories have generally been considered as mutuallyexclusive the history of the issue is often thought of as an oscillation between thesetwo extremes (eg Kingdom 1997 1999) with a shift towards the Helmholtzianstance for the last two decades (Adelson 1993 Adelson and Pentland 1996Arend 1994 Bergstrom 1977 Gilchrist 1977 Logvinenko 1999 Williams etal 1998a b)

That the HelmholtzndashHering controversy has not been resolved for more than ahundred years suggests that there might have been strong evidence in favour of boththeories Indeed since then each side has developed their own class of lightnessphenomena that the other side are hard pressed to account for These may be calledHelmholtzian and Hering types of simultaneous lightness contrast (Logvinenko2002c Logvinenko and Kane in press)

Adelsonrsquos tile and snake illusions certainly belong to the Helmholtzian typeIndeed local luminance contrast was not shown to play a significant role in theirproduction thus challenging numerous low-level models of lightness perceptionbased on local luminance contrast (Cornsweet 1970 Hering 1874 trans 1964Hurvich and Jameson 1966 Ratliff 1972 Wallach 1963 Whittle 1994a b)Furthermore even recent versions of low-level models based on a series of linearspatial filters (eg Blakeslee and McCourt 1999 Kingdom and Moulden 1992)cannot explain for example why the tile illusion which is quite strong in Fig 1adisappears completely for a 3D implementation of Fig 1a so that the retinal imagesof this 3D wall of blocks made of the same paper as Fig 1a and Fig 1a itself arepractically the same (Logvinenko et al 2002) Indeed as pointed out elsewhere(Logvinenko 2002a) all these models no matter how complicated processing takesplace in them after the filter stage predict the same output in response to the sameinput Since the input (retinal pattern) is nearly the same they should predict nearlythe same effect for both Fig 1a and its 3D implementation which is in disagreementwith what was observed in experiment (Logvinenko et al 2002) On the otherhand as shown above Adelsonrsquos tile and snake illusions lend themselves readily toan account developing further Helmholtzian idea of misjudged illumination


Still there are some lightness illusions of Heringrsquos type mdash Mach bands Hermanngrid grating induction to mention a few mdash which are hard to account for in terms ofHelmholtzian ideas We believe that these two types of illusions are rather differentin their nature Indeed the tile illusion was shown to disappear when the samewall of blocks was made as a 3D object (Logvinenko et al 2002) whereas gratinginduction can easily be observed from 3D cylinders (Logvinenko and Kane inpress) Such robustness of grating induction to the perceptual context (2D vs 3Dshape) shows that it is probably underlain by some mechanisms based on low-levelluminance contrast processing It is no surprise that low-level models have provedto be successful in accounting for grating induction (eg Blakeslee and McCourt1999 2003 McCourt and Foley 1985 McCourt and Blakeslee 1994 McCourt andKingdom 1996 Moulden and Kingdom 1991)

Each of the two different types of simultaneous lightness contrast obviously re-quires its own account Although the idea of a monistic explanation of simultane-ous lightness contrast seems to be very attractive (eg Gilchrist 2003) all attemptsto extend either account onto the whole variety of lightness illusions have provedunsuccessful For example there is no doubt that luminance contrast (or lumi-nance ratio) is more appropriate than absolute luminance as an input variable sinceit is well-established that due to light adaptation brightness (subjective intensity) oflight correlates with luminance contrast rather than absolute luminance (eg Shap-ley and Enroth-Cugel 1984 Whittle 1994a) However as pointed out by Gilchristluminance contrast (as well as luminance ratio) is ambiguous as a determinant oflightness thus it should be anchored (Gilchrist 1994 Gilchrist et al 1999) Webelieve that it is apparent illumination or to be more exact apparent atmosphere(ie apparent transmittance and apparent haze) that plays the role of such an an-chor For example both diamonds in the iso-contrast snake pattern (Fig 6) have thesame luminance contrast with their immediate surrounds Moreover their immedi-ate surrounds are the same However they look rather different As claimed abovethe difference in lightness in this figure is likely to be caused by the correspondingdifference in apparent haze over the alternating strips

Admittedly encoding contrast is an efficient sensory mechanism for discountingthe ambient illumination that makes the sensory input independent of any possiblechange in the illuminant intensity across a range However being ecologically asvalid as material changes (ie in reflectance) the illumination changes such asshadows are worth being perceived to the same extent as lightness And we are asgood at perceiving shadows as at perceiving lightness Therefore the question isnot how the visual system discounts the illumination changes but how it encodesthem and takes them into account when calculating lightness

One might argue however that taking apparent illumination into account whencalculating lightness implies an evaluation of illumination that is a problem onits own Hering was probably the first to point out this problem However firsthe seemed to mean the physical illumination whereas we believe that it is anapparent rather than physical illumination that affects lightness Second we do


not suggest that computing apparent illumination (andor apparent transmittance)precedes computing lightness Neither do we suggest the opposite (ie lightnessprecedes apparent illumination) They both appear together as a pair of perceptualdimensions of an object In other words the luminance ratio determines not a singledimension (lightness or apparent illumination) but a pair mdash apparent illuminationand lightness Of course a particular luminance ratio can give rise to a numberof apparent illuminationlightness pairs However not just any possible apparentilluminationlightness pair can emerge in response to a particular luminance ratioApparent illumination and lightness in a pair are bound by certain constraintsIn other words there is a sort of percept-percept coupling (Epstein 1973 1982Hochberg 1974) between these two perceptual dimensions mdash apparent illuminationand lightness

The albedo hypothesis was probably the first quantitative formulation of therelationship between apparent illumination and lightness (Beck 1972 Koffka1935 Kozaki and Noguchi 1976 Noguchi and Kozaki 1985) However the albedohypothesis in its original form was criticised from both the theoretical (Logvinenko1997) and experimental point of view (Beck 1972) The experimental criticismwas based on the fact that subjective estimates of the ambient illumination donot follow the relation with lightness that is predicted by the albedo hypothesis(Beck 1961 Kozaki 1973 MacLeod 1940 Oyama 1968 Rutherford andBrainard 2002) However as pointed out elsewhere (Logvinenko 1997) theapparent illuminationlightness invariance implies that the apparent illumination isa perceptual dimension of an object rather than a characteristic of the illuminantIndeed the apparent illumination of a surface depends on the spatial orientationof this surface the general spatial layout of the scene etc Altering the spatialorientation of the surface for example may change its apparent illumination evenwhen the illuminant and thus the ambient illumination remains unchanged Thereis abundant evidence that a change in an apparent slant may cause a change inapparent illumination and this in turn may cause a corresponding change inlightness while the retinal stimulation remains the same (eg Bloj and Hurlbert2002 Gilchrist 1977 Knill and Kersten 1991 Logvinenko and Menshikova 1994Mach 1959 Wishart et al 1997) It is also in line with our results For instance theapparent ambient illumination of the wall of blocks in Fig 5 is perceived as beinghomogeneous The patches with reflectance 048 have the same luminance Whythen do they look as though they have different lightness The most likely reason forthis is that they have different apparent slant Surfaces having different slants cannothave the same apparent illumination under even ambient light falling onto thesesurfaces Such a constraint is built in the apparent illuminationlightness invariance

Such apparent illuminationlightness invariance is a form in which the prior expe-rience exists in our perception To put it another way this and other invariances ofthis sort (Epstein 1982) are a way in which the prior experience affects perceiving

It should be kept in mind that the apparent illuminationlightness invariancediscussed so far implies apparent illumination and lightness of real objects whereas


as mentioned above our explanation of the tile and snake illusion involves pictorialapparent illumination However it does not mean that we imply that pictorialapparent illumination is involved in the same sort of relationship with lightnessas is apparent illumination in real scenes Nor do we suggest that the tile andsnake illusion require a conscious experience of an lsquoerroneousrsquo illumination in thepictures We believe that the lightness illusions in question result from an indirecteffect of pictorial cues for illumination available in the pictures (Logvinenko1999) Although all these cues are effective in producing a pictorial impressionof illumination it is unlikely that they contribute to an apparent illumination of thepicture as a real object embedded in the natural environment since there are alsoabundant cues for real illumination of the picture (at the level of natural vision)which provide veridical information about real illumination So we are usually ableto distinguish between real and pictorial illuminations having no doubts in what thereal illumination of the picture is

Nevertheless we believe that the pictorial cues of illumination may trigger the hy-pothetical perceptual mechanism securing the invariant relationship between light-ness and apparent illumination (and transmittance) Although veridical cues forreal illumination of the picture (which is usually homogeneous) override a directeffect of these pictorial cues an indirect effect (through the apparent illumina-tionlightness invariance mechanism) on the picture lightness may take place

Most displays presented in this report contain pictorial illumination cues whichare in apparent conflict with real illumination cues For example pictorial illumina-tion cues in Fig 7 testify to an illumination gradient whereas real illumination cuestell us that the picture is homogeneously lit The way such a conflict is resolved bythe visual system probably depends on the relative strength of the illumination cuesavailable Also it could be hypothesised that some process of weighting these cuesmay play a role in producing the resultant apparent illumination However what-ever the perceptual mechanism of producing apparent illumination is we suggestthat original information concerning pictorial illumination cues is driven in paralleland independently into both mechanisms mdash the apparent illumination mechanismand the lightness mechanism This information may be overridden in the former(thus producing no effect on apparent illumination) but it could be taken into ac-count in the latter (producing the lightness induction effect) This line of reasoningis similar to that involved in Gregoryrsquos lsquoinappropriate constancy scalingrsquo theoryof geometrical illusions resting on the apparent size-distance invariance (Gregory1974 Logvinenko 1999) and it is very close to the stance of Gilchrist (1979) andGilchrist Delman and Jacobsen (1983)

Such an account predicts that lightness illusions might depend on the relativestrength of the available pictorial illumination cues and if these cues are contradic-tory on the result of some compromise resolving this contradiction (eg weightingpooling etc) In other words it is flexible enough to account qualitatively for thedifferences in the magnitude of the lightness induction effect in the different figuresabove


As to the classical simultaneous lightness contrast display it is not clear what typeof illusion it produces Perhaps both mechanisms partly contribute to it and thisexplains why such a long dispute concerning the nature of simultaneous lightnesscontrast has not been resolved yet One might argue however that the Helmholtzianexplanation is hardly applicable to the classical simultaneous lightness contrasteffect because of the difference in the magnitude between simultaneous lightnesscontrast which is usually rather weak and lightness constancy which is muchstronger (eg Gilchrist 1988) Indeed if the black background in the classicalsimultaneous lightness contrast display is mistakenly perceived as an area ofreduced illumination then the simultaneous lightness contrast effect should bemuch stronger than that which is usually observed It should be kept in mindhowever that this would be true only if the trade-off between lightness and apparentillumination were complete In other words one can expect a much strongercontrast effect only if the black background is perceived as white paper (same as thewhite background) which is dimly illuminated But this never happens We alwayssee black and white backgrounds However the question is what shade of black isseen in the black part of the classical simultaneous lightness contrast display

As a matter of fact a complete trade-off between lightness and apparent illumi-nation did not take place even in the most favourable condition of classical Gelbrsquosdisplay (Gelb 1929) Actually it is Gelbrsquos effect that Gilchrist (1988) measuredin his first experiment under what he called lsquocontrast conditionrsquo To be more ex-act he found that in spite of his observers seeing the shadowed part of the back-ground as if it was made of a dark paper the observersrsquo matches showed that thetrade-off between shadow and lightness was not complete Specifically his ob-servers were found to match the apparent lightness of the shadowed part of thebackground as being lighter than what was predicted by a complete trade-off be-tween shadow and lightness This result might have been in line with the apparentilluminationlightness invariance provided that his observers saw the shadowed partof the background as being slightly less illuminated than the highlighted part Un-fortunately the apparent illumination was not measured in Gilchristrsquos experiment

The Helmholtzian account of simultaneous lightness contrast is therefore equiv-alent to the claim that a luminance edge produced by the border between the blackand white backgrounds is never perceived as a pure reflectance (lightness) edge Itis always accompanied by a coinciding apparent illumination edge though of lowcontrast which might cause the lightness shift in a classical simultaneous lightnesscontrast display This is in line with a growing body of evidence that a border be-tween shadowed and highlighted areas is also never perceived as a pure apparentillumination edge Specifically Gilchrist and his collaborators have recently shownthat lightness matches for the same grey paper on two sides of a shadow borderare never evaluated as having the same lightness (Gilchrist and Zdravkovic 2000Zdravkovic and Gilchrist 2000) The paper in the shadowed area is always un-derestimated (looks darker) Unfortunately these authors have not reported on anymeasurements of apparent illumination so it only remains to speculate if the ap-


parent illumination of the shadowed area was also underestimated (as follows fromthe shadowlightness invariance) or not However when shadow along with light-ness was measured it was found that they went hand in hand as predicted by theshadowlightness invariance (Logvinenko and Menshikova 1994)

Thus in spite of Gilchristrsquos criticism Helmholtzrsquos lsquomisjudgement of illuminationrsquomay prove to contribute to the classical simultaneous lightness contrast effectMoreover we believe that the anchoring effect also plays an important role inproducing this effect (Logvinenko 2002c) Therefore there are generally speakingthree types of mechanisms which might contribute into the classical simultaneouslightness contrast effect However it remains for the future investigation to ascertainin which proportions these contributions are made

CONCLUSION

Adelsonrsquos tile and snake illusions constitute a special Helmholtzian type ofsimultaneous lightness contrast that lends itself to an explanation based on theapparent illuminationlightness invariance Such an explanation implies an abilityto classify luminance edges into reflectance and illumination ones We suggest thatthe visual system uses the luminance-differences ratio invariance across a luminanceborder to distinguish between the reflectance and illumination edges

Acknowledgement

This work was supported by a research grant from the BBSRC (Ref 81S13175) toA Logvinenko

NOTES

1 For technical reasons the data for Fig 9 were collected from only 15 of 20observers So only 75 matches are available for this picture

2 Note that for some technical reasons the reflectances in Fig 16 are equallyspaced so this graph does not give the right idea of how the median match varieswith reflectance

3 The HodgesndashLehmann estimator gives a shift value such that if we shift all theobservations of one distribution by this value there will be no significant differencebetween the two distributions

4 Note that the HodgesndashLehmann estimator proved to be equal to neither thedifference of the medians nor the median difference of the matches

5 Since the number of matches was unequal (75 for the Fig 9 and 100 for Fig 1)we used the Wilcoxon rank-sum rather than signed-rank test

6 Still there were significant differences in its appearance in different patterns(Friedman χ2 = 191 df = 7 p lt 001) Although it might be against the


intuition that a significant difference can exist between two distributions with thesame medians there is no theoretical reason why it cannot happen Moreoverthe opposite may happen too More specifically two distributions with differentmedians can manifest no significant difference For instance the difference betweenthe median matches for the patch with reflectance 029 in Figs 4 and 9 is 05 Munsellunits though it was found to be not significant (Wilcoxon rank-sum normal statisticwith correction Z = 155 p = 012) In fact the Wilcoxon signed-rank testtests the difference between the distributions as the whole rather than between themedians

7 Apart from shadows the difference in relative illumination can be produced bya transparent media (eg transparent filter)

8 It should be mentioned that once we deal with pictures a distinction betweenan apparent and pictorial apparent illumination of a picture should also be madeIt reflects a dual nature of a picture as an object to be perceived As pointed outby Gibson (1979) on the one hand a picture is just a surface contaminated withpaint on the other there is a pictorial content which is rendered by this surfaceAccordingly there is an apparent illumination of a contaminated surface and anapparent illumination of a rendered (pictorial) object which might be different Forexample an apparent illumination of the paper on which the patterns were depictedwas always perceived as even and equal for all parts of the displays by all ourobservers On the contrary a pictorial apparent illumination was judged differentlyfor different parts of the displays In this discussion apparent illumination will meanpictorial apparent illumination unless it says otherwise

9 Although frame is a key notion in the anchoring theory of lightness perception(Gilchrist et al 1999) unfortunately it was never clarified by its authors whetherframe is supposed to be an area of equal illumination

10 While the wall of blocks looks equally illuminated in Fig 5 the lateral sides ofthe blocks have different slant Because of this difference in the apparent slant onelateral side looks be more shadowed than the other thus creating another differencein apparent illumination namely that of the lateral sides of the blocks As a resulta patch with the reflectance 048 belonging to the side in shadow looks lighter thanthat belonging to the highlighted side being in line with the prediction based on theapparent illuminationlightness invariance In a sense here we have an example ofslant-independent lightness constancy that has recently been documented (Boyaciet al 2003 Ripamonti et al submitted)

11 If the maximal luminance in Figs 8 and 9 were the anchor the diamonds inthese figures would look similar since in both figures the same white background isthe area of maximum luminance but they look rather different

12 We understand brightness as a subjective intensity of light It should bedistinguished from lightness and apparent illumination which are both perceptualdimensions of object rather than light For instance a change in apparent depthmay affect lightness but not brightness (Schirillo et al 1990)


13 In fact both experiments were run simultaneously with the same observersThey were divided into two only for the convenience of presentation

14 In the visual literature apparent haze is often called apparent transparency(eg Gerbino 1994 Kersten 1992 Mettelli 1974) We find such terminology inthe present context rather misleading since apparent haze implies that the opticalmedia is not transparent For example it would be strange to apply the termtransparency to translucency that is the limiting case of haze when the additivecomponent dominates

15 It should be noted that Fig 6 is ambiguous in the sense that the snake-shapedluminance border can also be interpreted as an atmospheric boundary with the sameatmospheric functions In order to show this one has simply to swap the designationsof identical patches with reflectance 050 (Fig 32)

16 Note that if the snake-shaped luminance border is interpreted as an at-mospheric boundary then the lightness shift should be in the opposite direction In-deed in this case the lower diamond in Fig 6 would belong to a hazy snake Hencethe lower diamond should look darker than the upper one when the atmosphericboundary is snake-shaped The fact that all our observers saw the upper ratherthan lower diamond darker indicates that the visual system is reluctant to treat thesnake-shaped luminance border as an atmospheric boundary Perhaps as claimedelsewhere (Adelson and Somers 2000 Logvinenko et al in press) it is a gen-eral rule built in somewhere in the visual system that for a luminance border to beinterpreted as an atmospheric boundary it should be straight rather than curved

17 The filter model (Gerbino 1994) yields similar results

REFERENCES

Adelson E H (1993) Perceptual organization and the judgment of brightness Science 262 2042ndash2044

Adelson E H (2000) Lightness perception and lightness illusions in The New Cognitive Neuro-sciences 2nd edn M Gazzaniga (Ed) pp 339ndash351 MIT Press Cambridge MA USA

Adelson E H and Pentland A P (1996) The perception of shading and reflectance in Perceptionas Bayesian Inference D Knill and W Richards (Eds) pp 409ndash423 Cambridge University PressNew York

Adelson E H and Somers D (2000) Shadows are fuzzy and straight paint is sharp and crookedPerception 29 (Suppl) 46

Albright D (1994) Why do things look as they do Trends In Neuroscience 17 175ndash177Anderson B L (1997) A theory of illusory lightness and transparency in monocular and binocular

images The role of contour junctions Perception 26 419ndash453Arend L E (1994) Surface colours illumination and surface geometry Intrinsic-image models

of human colour perception in Lightness Brightness and Transparency A Gilchrist (Ed) pp159ndash213 Lawrence Erlbaum Associates Hillsdale NJ USA

Beck J (1961) Judgments of surface illumination and lightness J Exper Psychol 61 368ndash375Beck J (1972) Surface Colour Perception Cornell University Press Ithaca NewYorkBergstrom S S (1977) Common and relative components of reflected light as information about the

illumination colour and three-dimensional form of objects Scand J Psychol 18 180ndash186


Blakeslee B and McCourt M E (1999) A multiscale spatial filtering account of the White effectsimultaneous brightness contrast and grating induction Vision Research 39 4361ndash4377

Blakeslee B and McCourt M E (2003) A multiscale spatial filtering account of brightnessphenomena in Levels of Perception L Harris and M Jenkin (Eds) Springer-Verlag Berlin

Bloj M G and Hurlbert A C (2002) An empirical study of the traditional Mach card effectPerception 31 233ndash246

Boyaci H Maloney L T and Hersh S (2003) The effect of perceived surface orientation onperceived surface albedo in three-dimensional scenes J Vision 3 541ndash553

Brainard D H and Maloney S I (2002) The effect of object shape and pose on perceived lightnessin Vision Sciences Society Sarasota Florida (Proc pp 191)

Bressan P (2001) Explaining lightness illusions Perception 30 1031ndash1046Cornsweet T (1970) Visual Perception Academic Press New YorkDakin S C and Bex P J (2002) 1f channel reweighting predicts many aspects of lightness

perception Perception 31 (Suppl) 149Epstein W (1982) Perceptndashpercept couplings Perception 11 75ndash83Gelb A (1929) Die farbenkonstanzrsquo der sehdinge in Handbuch der Normalen und Patologischen

Physiologie A Bethe (Ed) pp 594ndash678 Springer BerlinGerbino W (1994) Achromatic transparency in Lightness Brightness and Transparency

A L Gilchrist (Ed) pp 215ndash255 Lawrence Erlbaum Associates Hillsdale NJ USAGibson J J (1979) The Ecological Approach to Visual Perception Houghton Mifflin Boston MA

USAGilchrist A L (1977) Perceived lightness depends on perceived spatial arrangement Science 195

185ndash187Gilchrist A L (1979) The perception of surface blacks and whites Scientific American 240 112ndash

123Gilchrist A (1988) Lightness contrast and failures of lightness constancy a common explanation

Perception and Psychophysics 43 415ndash424Gilchrist A L (1994) Absolute versus relative theories of lightness perception in Lightness

Brightness and Transparency Gilchrist A L (Ed) pp 1ndash31 Lawrence Erlbaum AssociatesHillsdale NJ USA

Gilchrist A (2003) Dualistic versus monistic accounts of lightness perception in Levels ofPerception L Harris and M Jenkin (Eds) Springer-Verlag Berlin

Gilchrist A L and Zdravkovic S (2000) Mining lightness errors Investigative Ophthalmology andVisual Science Supplement 41 226

Gilchrist A L Delman S and Jacobsen A (1983) The classification and integration of edges ascritical to the perception of reflectance and illumination Perception and Psychophysics 33 425ndash436

Gilchrist A Kossyfidis C Bonato F Agostini T Cataliotti J Li X Spehar B Annan V andEconomou E (1999) An anchoring theory of lightness perception Psychol Rev 106 795ndash834

Gregory R L (1974) Distortion of visual space as inappropriate constancy scaling in Concepts andMechanisms of Perception pp 342ndash349 Duckworth London

Helmholtz H von (1867) Handbuch der Physiologischen Optik Voss LeipzigHering E (1964) Outlines of a Theory of the Light Sense (original work published in 1874)

L M Hurvich and D Jameson (trans) Harvard University Press Cambridge MA USAHochberg J (1974) Higher-order stimuli and inter-response coupling in the perception of the visual

world in Perception Essays in Honour of James J Gibson R B McLeod and H L Pick Jr(Eds) pp 17ndash39 Cornell University Press Ithaca New York

Hollander M and Wolfe D A (1973) Nonparametric Statistical Methods John Wiley New YorkKersten D (1992) Transparency and the co-operative computation of scene attributes in Compu-

tational Models of Visual Processing M S Landy and J A Movshon (Eds) pp 209ndash228 MITPress Cambridge MA USA


Kingdom F A A (1997) Simultaneous contrast the legacies of Hering and Helmholtz Perception26 673ndash677

Kingdom F A A (1999) Old wine in new bottles Some thoughts on Logvinenkorsquos ldquoLightnessinduction revisitedrdquo Perception 28 929ndash934

Kingdom F A A (2003) Levels of brightness perception in Levels of Perception L Harris andM Jenkin (Eds) Springer-Verlag Berlin

Kingdom F and Moulden B (1992) A multi-channel approach to brightness coding Vision Research32 1565ndash1582

Knill D C and Kersten D (1991) Apparent surface curvature affects lightness perception Nature351 228ndash230

Koffka K (1935) Principles of Gestalt Psychology Harcourt Brace New YorkKozaki A (1973) Perception of lightness and brightness of achromatic surface color and impression

of illumination Japanese Psychol Res 15 194ndash203Kozaki A and Noguchi K (1976) The relationship between perceived surface-lightness and

perceived illumination Psychol Res 39 1ndash16Logvinenko A D (1997) Invariant relationship between achromatic colour apparent illumination

and shape of surface Implications for the colour perception theories in John Daltonrsquos ColourVision Legacy C M Dickinson I J Murray and D Carden (Eds) Taylor and Francis London

Logvinenko A D (1999) Lightness induction revisited Perception 28 803ndash816Logvinenko A D (2002a) Does bandpass linear filter response predict gradient induction A reply

to Fred Kingdom Perception 32 (5) 621ndash626Logvinenko A D (2002b) A fair test of the effect of a shadow-incompatible luminance gradient on

the simultaneous lightness contrast Perception 32 717ndash720Logvinenko A D (2002c) The anchoring effect in lightness perception Neuroscience Letters 334

5ndash8Logvinenko A D (2002d) Articulation in the context of edge classification Perception 31 201ndash207Logvinenko A D and Kane J (2003) Luminance gradient can break background-independent

lightness constancy Perception 32 263ndash268Logvinenko A D and Kane J (2004) Heringrsquos and Helmholtzian types of simultaneous lightness

contrast Journal of Vision (in press)Logvinenko A D and Menshikova G (1994) Trade-off between achromatic colour and perceived

illumination as revealed by the use of pseudoscopic inversion of apparent depth Perception 231007ndash1023

Logvinenko A D Kane J and Ross D A (2002) Is lightness induction a pictorial illusionPerception 31 73ndash82

Logvinenko A D Adelson E Ross D A and Somers D (2004) Linearity as a cue for luminanceedge classification Perception and Psychophysics (in press)

Luce R D and Galanter E (1965) Psychophysical scaling in Handbook of MathematicalPsychology Vol I R D Luce R R Bush and E Galanter (Eds) pp 245ndash307 John WileyNew York

MacLeod R B (1940) Brightness constancy in unrecognised shadows J Exper Psychol 27 1ndash22Mach E (1959) The Analysis of Sensations translated from the 5th German edition by S Waterlow

Dover Publications New YorkMaloney L T Boyaci H and Hersh S (2002) Human observers do not correct perceived lightness

for perceived orientation in ProcVision Sciences Society Sarasota Florida p 192Marr D (1982) Vision W H Freeman San FranciscoMcCann J (2001) Calculating lightnesses in a single depth plane J Electronic Imaging 10 110ndash122McCourt M E (1982) A spatial frequency dependent grating-induction effect Vision Research 22

119ndash134McCourt M E and Blakeslee B (1994) Contrast-matching analysis of grating induction and

suprathreshold contrast perception J Opt Soc Amer A 11 14ndash24


McCourt M E and Foley J M (1985) Spatial frequency interference on grating-induction VisionResearch 25 1507ndash1515

McCourt M E and Kingdom F A A (1996) Facilitation of luminance grating detection by inducedgratings Vision Research 36 2563ndash2575

Mettelli F (1974) Achromatic colour conditions in the perception of transparency in PerceptionEssays in Honour of James J Gibson R B McLeod and H L Pick Jr (Eds) pp 95ndash116 CornellUniversity Press Ithaca New York

Moulden B and Kingdom F (1991) The local border mechanism in grating induction VisionResearch 31 1999ndash2008

Noguchi K and Kozaki A (1985) Perceptual scission of surface-lightness and illumination Anexamination of the Gelb effect Psychol Res 47 19ndash25

Oyama T (1968) Stimulus determination of brightness constancy and the perception of illuminationJapanese Psychol Res 10 146ndash155

Paradiso M A (2000) Visual neuroscience Illuminating the dark corners Current Biology 10 R15ndashR18

Ratliff F (1972) Contour and contrast Scientific American 226 91ndash101Ripamonti C Bloj M Hauck R Mitha K Greenwald S Maloney S I and Brainard

D H Measurements of the effect of surface slant on perceived lightness Journal of Vision 4735ndash746

Ross W D and Pessoa L (2000) Lightness from contrast A selective integration model Perceptionand Psychophysics 62 1160ndash1181

Rutherford M D and Brainard D H (2002) Lightness constancy a direct test of the illuminationestimation hypothesis Psychological Science 13 142ndash149

Schirillo J A and Shevell S K (2002) Articulation brightness apparent illumination and contrastratios Perception 31 161ndash169

Schirillo J Reeves A and Arend L (1990) Perceived lightness but not brightness of achromaticsurfaces depends on perceived depth information Perception and Psychophysics 48 82ndash90

Shapely R M and Enroth-Cugell C (1984)Visual adaptation and retinal gain controls in Progressin Retinal Research Vol 3 N N Osborne and G J Chader (Eds) pp 263ndash343 Pergamon PressOxford

Todorovic D (1997) Lightness and junctions Perception 26 379ndash394Todorovic D (2003) Lightness and illumination comments on Kingdom Blakeslee and McCourt

in Levels of Perception L Harris and M Jenkin (Eds) Springer-Verlag BerlinTurner R S (1994) In the Eyersquos Mind Vision and the Helmholtz-Hering Controversy Princeton

University Press Princeton NJ USAWallach H (1963) The perception of neutral colours Scientific American 208 107ndash116Whittle P (1994a) The psychophysics of contrast brightness in Lightness Brightness and

Transparency A L Gilchrist (Ed) pp 35ndash110 Lawrence Erlbaum Associates Hillsdale NJUSA

Whittle P (1994b) Contrast brightness and ordinary seeing in Lightness Brightness and Trans-parency A L Gilchrist (Ed) pp 111ndash157 Lawrence Erlbaum Associates Hillsdale NJ USA

Williams S M McCoy A N and Purves D (1998a)The influence of depicted illumination onbrightness Proc Natl Acad Sci USA 95 13296ndash13300

Williams S M McCoy A N and Purves D (1998b) An empirical explanation of brightness ProcNatl Acad Sci USA 95 13301ndash13306

Wishart K A Frisby J P and Buckley D (1997) The role of 3-D surface slope in a light-nessbrightness effect Vision Research 37 467ndash473

Zdravkovic S and Gilchrist A L (2000) Lightness determination for an object under twoilluminations Perception 29 (Suppl) 73


(a) (b)

Light strip Dark strip Light strip Dark strip

Reflectance Median Reflectance Median Reflectance Median Reflectance Median

079 925 048 775 079 900 048 725048 675 043 725 048 600 029 525043 475 029 550 043 450 026 425

Figure 1 (a) Tile pattern (after Adelson 1993) The original Adelson tile pattern is modified so that(i) there is a patch (with reflectance 048) that is included in both the lsquolightrsquo and lsquodarkrsquo strips (ii)the reflectances of the patches in the lsquolightrsquo and lsquodarkrsquo strips are in approximately same proportions048 029 and 079 048 The diamonds in both strips have the same reflectance (043) Howeverthey appear different in lightness (b) The modified tile pattern The diamonds in the lsquolightrsquo stripshave a reflectance of 043 while those in the lsquodarkrsquo strips have a reflectance of 026 yet they lookremarkably similar

contrast are different visual phenomena If not which features of Adelsonrsquospatterns strengthen the effect

There are at least four differing explanatory approaches to Adelsonrsquos illusions(i) the low-level explanation based on the local contrast around the target (diamond)borders (eg Cornsweet 1970 Whittle 1994a b) and its recent modifications(eg Blakeslee and McCourt 1999 2003 McCann 2001) (ii) a mid-levelexplanation emphasizing the role of the borders between strips or the type ofthe luminance junctions across the strips (Adelson 1993 2000 Anderson 1997Todorovic 1997) (iii) the anchoring theory (Gilchrist et al 1999) and (iv) a high-level explanation enunciated by H Helmholtz (1867) who thought that lightnessillusions such as simultaneous lightness contrast resulted from a lsquomisjudgementof illuminationrsquo (Adelson 1993 Adelson and Pentland 1996 Kingdom 2003Logvinenko 1999)




073 875 052 875061 700 037 625052 550 031 550







079 925 048 700048 700 043 650043 600 029 475








079 925 048 700048 750 043 650043 625 029 500







043 500 043 725







051 575 051 900



EXPERIMENT 1



Methods



Reflectance Median

079 925048 675043 500






Reflectance Median

048 750043 700029 525








079 925 048 750048 650 043 700043 575 029 5375


Results







073 875 052 875061 775 037 600052 625 031 475








077 900 050 800050 675 043 675043 550 023 425









079 925 048 725048 650 043 650043 575 029 5125









079 925 048 725048 650 043 650043 575 029 500





Discussion





079 925 048 7375048 675 043 675043 575 029 500









016 400 415023 500 504029 575 568031 575 583039 650 649043 675 676048 700 701079 900 886







016 400 403 475 466 050023 500 510 575 571 050029 575 578 625 637 050031 600 597 650 663 0625039 650 651 700 712 0625043 700 690 725 740 050048 725 721 775 776 0625079 900 886 925 925 0375






1a 043 475 487 725 735 23751b 043026 450 436 425 429 0252 052 550 527 875 870 33754 043 600 596 650 640 0255 043 625 611 650 650 0256 043 500 516 725 719 21257 051 575 552 900 886 337589 043 500 513 700 675 17510 043 575 545 700 685 137511 052 625 605 875 853 25012 043 550 532 675 663 12513 043 575 559 650 654 087514 043 575 553 650 649 10015 043 575 554 675 659 0875





1a 675 656 775 767 1001b 600 591 725 728 1254 700 698 700 690 0005 750 747 700 702 minus02589 675 668 750 739 07510 650 653 750 743 07513 650 650 725 709 037514 650 645 725 711 05015 675 663 7375 731 050











1a 550 5411b 525 5184 475 4895 500 5179 525 49910 5375 53413 5125 51614 500 52015 500 524






















EXPERIMENT 2


Methods



Results





079 925 048 750043 575 043 725039 500 029 475







079 925 048 700043 600 043 675031 475 029 475



Discussion






079 925 048 725043 650 043 6875023 425 029 500








073 925 048 7375043 700 043 700016 3875 029 475








079 925 048 775048 700 043 725043 6375 029 575



079 925 048 775043 675 043 7375016 375 029 475



l = I r + a






079 925 048 7875048 725 043 725043 650 029 600



079 925 048 7875043 700 043 725016 375 029 475







079 925 048 7875048 725 043 725043 675 029 600



079 925 048 800043 700 043 725016 375 029 475








22 575 543 725 717 162523 600 596 675 674 062524 650 650 6875 681 0312525 700 689 700 697 0125











Tabl

e7

Med

ian

and

mea

nM

unse

llm

atch

esa

ndth

eH

odge

sndashL

ehm

ann

esti

mat

oro

btai

ned

for

the

diam

onds

inF

igs

26ndash2

8

Fig

ure

Alt

ered

Hod

gesndash

Una

lter

edH

odge

sndashnu

mbe

rL

ehm

ann

Leh

man

nes

tim

ator

esti

mat

orL

ight

stri

pD

ark

stri

pL

ight

stri

pD

ark

stri

p

Med

ian

Mea

nM

edia

nM

ean

Med

ian

Mea

nM

edia

nM

ean

266

756

697

375

735

062

56

375

625

725

736

112

527

700

689

725

727

037

56

506

497

257

260

7528

700

690

725

722

037

56

756

677

257

310

625








163 211 253 295 332



000 009 015 020 023








GENERAL DISCUSSION


























CONCLUSION


Acknowledgement


NOTES





















REFERENCES























































































073 875 052 875061 700 037 625052 550 031 550







079 925 048 700048 700 043 650043 600 029 475








079 925 048 700048 750 043 650043 625 029 500







043 500 043 725







051 575 051 900



EXPERIMENT 1



Methods



Reflectance Median

079 925048 675043 500






Reflectance Median

048 750043 700029 525








079 925 048 750048 650 043 700043 575 029 5375


Results







073 875 052 875061 775 037 600052 625 031 475








077 900 050 800050 675 043 675043 550 023 425









079 925 048 725048 650 043 650043 575 029 5125









079 925 048 725048 650 043 650043 575 029 500





Discussion





079 925 048 7375048 675 043 675043 575 029 500









016 400 415023 500 504029 575 568031 575 583039 650 649043 675 676048 700 701079 900 886







016 400 403 475 466 050023 500 510 575 571 050029 575 578 625 637 050031 600 597 650 663 0625039 650 651 700 712 0625043 700 690 725 740 050048 725 721 775 776 0625079 900 886 925 925 0375






1a 043 475 487 725 735 23751b 043026 450 436 425 429 0252 052 550 527 875 870 33754 043 600 596 650 640 0255 043 625 611 650 650 0256 043 500 516 725 719 21257 051 575 552 900 886 337589 043 500 513 700 675 17510 043 575 545 700 685 137511 052 625 605 875 853 25012 043 550 532 675 663 12513 043 575 559 650 654 087514 043 575 553 650 649 10015 043 575 554 675 659 0875





1a 675 656 775 767 1001b 600 591 725 728 1254 700 698 700 690 0005 750 747 700 702 minus02589 675 668 750 739 07510 650 653 750 743 07513 650 650 725 709 037514 650 645 725 711 05015 675 663 7375 731 050











1a 550 5411b 525 5184 475 4895 500 5179 525 49910 5375 53413 5125 51614 500 52015 500 524






















EXPERIMENT 2


Methods



Results





079 925 048 750043 575 043 725039 500 029 475







079 925 048 700043 600 043 675031 475 029 475



Discussion






079 925 048 725043 650 043 6875023 425 029 500








073 925 048 7375043 700 043 700016 3875 029 475








079 925 048 775048 700 043 725043 6375 029 575



079 925 048 775043 675 043 7375016 375 029 475



l = I r + a






079 925 048 7875048 725 043 725043 650 029 600



079 925 048 7875043 700 043 725016 375 029 475







079 925 048 7875048 725 043 725043 675 029 600



079 925 048 800043 700 043 725016 375 029 475








22 575 543 725 717 162523 600 596 675 674 062524 650 650 6875 681 0312525 700 689 700 697 0125











Tabl

e7

Med

ian

and

mea

nM

unse

llm

atch

esa

ndth

eH

odge

sndashL

ehm

ann

esti

mat

oro

btai

ned

for

the

diam

onds

inF

igs

26ndash2

8

Fig

ure

Alt

ered

Hod

gesndash

Una

lter

edH

odge

sndashnu

mbe

rL

ehm

ann

Leh

man

nes

tim

ator

esti

mat

orL

ight

stri

pD

ark

stri

pL

ight

stri

pD

ark

stri

p

Med

ian

Mea

nM

edia

nM

ean

Med

ian

Mea

nM

edia

nM

ean

266

756

697

375

735

062

56

375

625

725

736

112

527

700

689

725

727

037

56

506

497

257

260

7528

700

690

725

722

037

56

756

677

257

310

625








163 211 253 295 332



000 009 015 020 023








GENERAL DISCUSSION


























CONCLUSION


Acknowledgement


NOTES





















REFERENCES























































































079 925 048 700048 700 043 650043 600 029 475








079 925 048 700048 750 043 650043 625 029 500







043 500 043 725







051 575 051 900



EXPERIMENT 1



Methods



Reflectance Median

079 925048 675043 500






Reflectance Median

048 750043 700029 525








079 925 048 750048 650 043 700043 575 029 5375


Results







073 875 052 875061 775 037 600052 625 031 475








077 900 050 800050 675 043 675043 550 023 425









079 925 048 725048 650 043 650043 575 029 5125









079 925 048 725048 650 043 650043 575 029 500





Discussion





079 925 048 7375048 675 043 675043 575 029 500









016 400 415023 500 504029 575 568031 575 583039 650 649043 675 676048 700 701079 900 886







016 400 403 475 466 050023 500 510 575 571 050029 575 578 625 637 050031 600 597 650 663 0625039 650 651 700 712 0625043 700 690 725 740 050048 725 721 775 776 0625079 900 886 925 925 0375






1a 043 475 487 725 735 23751b 043026 450 436 425 429 0252 052 550 527 875 870 33754 043 600 596 650 640 0255 043 625 611 650 650 0256 043 500 516 725 719 21257 051 575 552 900 886 337589 043 500 513 700 675 17510 043 575 545 700 685 137511 052 625 605 875 853 25012 043 550 532 675 663 12513 043 575 559 650 654 087514 043 575 553 650 649 10015 043 575 554 675 659 0875





1a 675 656 775 767 1001b 600 591 725 728 1254 700 698 700 690 0005 750 747 700 702 minus02589 675 668 750 739 07510 650 653 750 743 07513 650 650 725 709 037514 650 645 725 711 05015 675 663 7375 731 050











1a 550 5411b 525 5184 475 4895 500 5179 525 49910 5375 53413 5125 51614 500 52015 500 524






















EXPERIMENT 2


Methods



Results





079 925 048 750043 575 043 725039 500 029 475







079 925 048 700043 600 043 675031 475 029 475



Discussion






079 925 048 725043 650 043 6875023 425 029 500








073 925 048 7375043 700 043 700016 3875 029 475








079 925 048 775048 700 043 725043 6375 029 575



079 925 048 775043 675 043 7375016 375 029 475



l = I r + a






079 925 048 7875048 725 043 725043 650 029 600



079 925 048 7875043 700 043 725016 375 029 475







079 925 048 7875048 725 043 725043 675 029 600



079 925 048 800043 700 043 725016 375 029 475








22 575 543 725 717 162523 600 596 675 674 062524 650 650 6875 681 0312525 700 689 700 697 0125











Tabl

e7

Med

ian

and

mea

nM

unse

llm

atch

esa

ndth

eH

odge

sndashL

ehm

ann

esti

mat

oro

btai

ned

for

the

diam

onds

inF

igs

26ndash2

8

Fig

ure

Alt

ered

Hod

gesndash

Una

lter

edH

odge

sndashnu

mbe

rL

ehm

ann

Leh

man

nes

tim

ator

esti

mat

orL

ight

stri

pD

ark

stri

pL

ight

stri

pD

ark

stri

p

Med

ian

Mea

nM

edia

nM

ean

Med

ian

Mea

nM

edia

nM

ean

266

756

697

375

735

062

56

375

625

725

736

112

527

700

689

725

727

037

56

506

497

257

260

7528

700

690

725

722

037

56

756

677

257

310

625








163 211 253 295 332



000 009 015 020 023








GENERAL DISCUSSION


























CONCLUSION


Acknowledgement


NOTES





















REFERENCES























































































079 925 048 700048 750 043 650043 625 029 500







043 500 043 725







051 575 051 900



EXPERIMENT 1



Methods



Reflectance Median

079 925048 675043 500






Reflectance Median

048 750043 700029 525








079 925 048 750048 650 043 700043 575 029 5375


Results







073 875 052 875061 775 037 600052 625 031 475








077 900 050 800050 675 043 675043 550 023 425









079 925 048 725048 650 043 650043 575 029 5125









079 925 048 725048 650 043 650043 575 029 500





Discussion





079 925 048 7375048 675 043 675043 575 029 500









016 400 415023 500 504029 575 568031 575 583039 650 649043 675 676048 700 701079 900 886







016 400 403 475 466 050023 500 510 575 571 050029 575 578 625 637 050031 600 597 650 663 0625039 650 651 700 712 0625043 700 690 725 740 050048 725 721 775 776 0625079 900 886 925 925 0375






1a 043 475 487 725 735 23751b 043026 450 436 425 429 0252 052 550 527 875 870 33754 043 600 596 650 640 0255 043 625 611 650 650 0256 043 500 516 725 719 21257 051 575 552 900 886 337589 043 500 513 700 675 17510 043 575 545 700 685 137511 052 625 605 875 853 25012 043 550 532 675 663 12513 043 575 559 650 654 087514 043 575 553 650 649 10015 043 575 554 675 659 0875





1a 675 656 775 767 1001b 600 591 725 728 1254 700 698 700 690 0005 750 747 700 702 minus02589 675 668 750 739 07510 650 653 750 743 07513 650 650 725 709 037514 650 645 725 711 05015 675 663 7375 731 050











1a 550 5411b 525 5184 475 4895 500 5179 525 49910 5375 53413 5125 51614 500 52015 500 524






















EXPERIMENT 2


Methods



Results





079 925 048 750043 575 043 725039 500 029 475







079 925 048 700043 600 043 675031 475 029 475



Discussion






079 925 048 725043 650 043 6875023 425 029 500








073 925 048 7375043 700 043 700016 3875 029 475








079 925 048 775048 700 043 725043 6375 029 575



079 925 048 775043 675 043 7375016 375 029 475



l = I r + a






079 925 048 7875048 725 043 725043 650 029 600



079 925 048 7875043 700 043 725016 375 029 475







079 925 048 7875048 725 043 725043 675 029 600



079 925 048 800043 700 043 725016 375 029 475








22 575 543 725 717 162523 600 596 675 674 062524 650 650 6875 681 0312525 700 689 700 697 0125











Tabl

e7

Med

ian

and

mea

nM

unse

llm

atch

esa

ndth

eH

odge

sndashL

ehm

ann

esti

mat

oro

btai

ned

for

the

diam

onds

inF

igs

26ndash2

8

Fig

ure

Alt

ered

Hod

gesndash

Una

lter

edH

odge

sndashnu

mbe

rL

ehm

ann

Leh

man

nes

tim

ator

esti

mat

orL

ight

stri

pD

ark

stri

pL

ight

stri

pD

ark

stri

p

Med

ian

Mea

nM

edia

nM

ean

Med

ian

Mea

nM

edia

nM

ean

266

756

697

375

735

062

56

375

625

725

736

112

527

700

689

725

727

037

56

506

497

257

260

7528

700

690

725

722

037

56

756

677

257

310

625








163 211 253 295 332



000 009 015 020 023








GENERAL DISCUSSION


























CONCLUSION


Acknowledgement


NOTES





















REFERENCES























































































043 500 043 725







051 575 051 900



EXPERIMENT 1



Methods



Reflectance Median

079 925048 675043 500






Reflectance Median

048 750043 700029 525








079 925 048 750048 650 043 700043 575 029 5375


Results







073 875 052 875061 775 037 600052 625 031 475








077 900 050 800050 675 043 675043 550 023 425









079 925 048 725048 650 043 650043 575 029 5125









079 925 048 725048 650 043 650043 575 029 500





Discussion





079 925 048 7375048 675 043 675043 575 029 500









016 400 415023 500 504029 575 568031 575 583039 650 649043 675 676048 700 701079 900 886







016 400 403 475 466 050023 500 510 575 571 050029 575 578 625 637 050031 600 597 650 663 0625039 650 651 700 712 0625043 700 690 725 740 050048 725 721 775 776 0625079 900 886 925 925 0375






1a 043 475 487 725 735 23751b 043026 450 436 425 429 0252 052 550 527 875 870 33754 043 600 596 650 640 0255 043 625 611 650 650 0256 043 500 516 725 719 21257 051 575 552 900 886 337589 043 500 513 700 675 17510 043 575 545 700 685 137511 052 625 605 875 853 25012 043 550 532 675 663 12513 043 575 559 650 654 087514 043 575 553 650 649 10015 043 575 554 675 659 0875





1a 675 656 775 767 1001b 600 591 725 728 1254 700 698 700 690 0005 750 747 700 702 minus02589 675 668 750 739 07510 650 653 750 743 07513 650 650 725 709 037514 650 645 725 711 05015 675 663 7375 731 050











1a 550 5411b 525 5184 475 4895 500 5179 525 49910 5375 53413 5125 51614 500 52015 500 524






















EXPERIMENT 2


Methods



Results





079 925 048 750043 575 043 725039 500 029 475







079 925 048 700043 600 043 675031 475 029 475



Discussion






079 925 048 725043 650 043 6875023 425 029 500








073 925 048 7375043 700 043 700016 3875 029 475








079 925 048 775048 700 043 725043 6375 029 575



079 925 048 775043 675 043 7375016 375 029 475



l = I r + a






079 925 048 7875048 725 043 725043 650 029 600



079 925 048 7875043 700 043 725016 375 029 475







079 925 048 7875048 725 043 725043 675 029 600



079 925 048 800043 700 043 725016 375 029 475








22 575 543 725 717 162523 600 596 675 674 062524 650 650 6875 681 0312525 700 689 700 697 0125











Tabl

e7

Med

ian

and

mea

nM

unse

llm

atch

esa

ndth

eH

odge

sndashL

ehm

ann

esti

mat

oro

btai

ned

for

the

diam

onds

inF

igs

26ndash2

8

Fig

ure

Alt

ered

Hod

gesndash

Una

lter

edH

odge

sndashnu

mbe

rL

ehm

ann

Leh

man

nes

tim

ator

esti

mat

orL

ight

stri

pD

ark

stri

pL

ight

stri

pD

ark

stri

p

Med

ian

Mea

nM

edia

nM

ean

Med

ian

Mea

nM

edia

nM

ean

266

756

697

375

735

062

56

375

625

725

736

112

527

700

689

725

727

037

56

506

497

257

260

7528

700

690

725

722

037

56

756

677

257

310

625








163 211 253 295 332



000 009 015 020 023








GENERAL DISCUSSION


























CONCLUSION


Acknowledgement


NOTES





















REFERENCES























































































051 575 051 900



EXPERIMENT 1



Methods



Reflectance Median

079 925048 675043 500






Reflectance Median

048 750043 700029 525








079 925 048 750048 650 043 700043 575 029 5375


Results







073 875 052 875061 775 037 600052 625 031 475








077 900 050 800050 675 043 675043 550 023 425









079 925 048 725048 650 043 650043 575 029 5125









079 925 048 725048 650 043 650043 575 029 500





Discussion





079 925 048 7375048 675 043 675043 575 029 500









016 400 415023 500 504029 575 568031 575 583039 650 649043 675 676048 700 701079 900 886







016 400 403 475 466 050023 500 510 575 571 050029 575 578 625 637 050031 600 597 650 663 0625039 650 651 700 712 0625043 700 690 725 740 050048 725 721 775 776 0625079 900 886 925 925 0375






1a 043 475 487 725 735 23751b 043026 450 436 425 429 0252 052 550 527 875 870 33754 043 600 596 650 640 0255 043 625 611 650 650 0256 043 500 516 725 719 21257 051 575 552 900 886 337589 043 500 513 700 675 17510 043 575 545 700 685 137511 052 625 605 875 853 25012 043 550 532 675 663 12513 043 575 559 650 654 087514 043 575 553 650 649 10015 043 575 554 675 659 0875





1a 675 656 775 767 1001b 600 591 725 728 1254 700 698 700 690 0005 750 747 700 702 minus02589 675 668 750 739 07510 650 653 750 743 07513 650 650 725 709 037514 650 645 725 711 05015 675 663 7375 731 050











1a 550 5411b 525 5184 475 4895 500 5179 525 49910 5375 53413 5125 51614 500 52015 500 524






















EXPERIMENT 2


Methods



Results





079 925 048 750043 575 043 725039 500 029 475







079 925 048 700043 600 043 675031 475 029 475



Discussion






079 925 048 725043 650 043 6875023 425 029 500








073 925 048 7375043 700 043 700016 3875 029 475








079 925 048 775048 700 043 725043 6375 029 575



079 925 048 775043 675 043 7375016 375 029 475



l = I r + a






079 925 048 7875048 725 043 725043 650 029 600



079 925 048 7875043 700 043 725016 375 029 475







079 925 048 7875048 725 043 725043 675 029 600



079 925 048 800043 700 043 725016 375 029 475








22 575 543 725 717 162523 600 596 675 674 062524 650 650 6875 681 0312525 700 689 700 697 0125











Tabl

e7

Med

ian

and

mea

nM

unse

llm

atch

esa

ndth

eH

odge

sndashL

ehm

ann

esti

mat

oro

btai

ned

for

the

diam

onds

inF

igs

26ndash2

8

Fig

ure

Alt

ered

Hod

gesndash

Una

lter

edH

odge

sndashnu

mbe

rL

ehm

ann

Leh

man

nes

tim

ator

esti

mat

orL

ight

stri

pD

ark

stri

pL

ight

stri

pD

ark

stri

p

Med

ian

Mea

nM

edia

nM

ean

Med

ian

Mea

nM

edia

nM

ean

266

756

697

375

735

062

56

375

625

725

736

112

527

700

689

725

727

037

56

506

497

257

260

7528

700

690

725

722

037

56

756

677

257

310

625








163 211 253 295 332



000 009 015 020 023








GENERAL DISCUSSION


























CONCLUSION


Acknowledgement


NOTES





















REFERENCES





















































































Reflectance Median

079 925048 675043 500






Reflectance Median

048 750043 700029 525








079 925 048 750048 650 043 700043 575 029 5375


Results







073 875 052 875061 775 037 600052 625 031 475








077 900 050 800050 675 043 675043 550 023 425









079 925 048 725048 650 043 650043 575 029 5125









079 925 048 725048 650 043 650043 575 029 500





Discussion





079 925 048 7375048 675 043 675043 575 029 500









016 400 415023 500 504029 575 568031 575 583039 650 649043 675 676048 700 701079 900 886







016 400 403 475 466 050023 500 510 575 571 050029 575 578 625 637 050031 600 597 650 663 0625039 650 651 700 712 0625043 700 690 725 740 050048 725 721 775 776 0625079 900 886 925 925 0375






1a 043 475 487 725 735 23751b 043026 450 436 425 429 0252 052 550 527 875 870 33754 043 600 596 650 640 0255 043 625 611 650 650 0256 043 500 516 725 719 21257 051 575 552 900 886 337589 043 500 513 700 675 17510 043 575 545 700 685 137511 052 625 605 875 853 25012 043 550 532 675 663 12513 043 575 559 650 654 087514 043 575 553 650 649 10015 043 575 554 675 659 0875





1a 675 656 775 767 1001b 600 591 725 728 1254 700 698 700 690 0005 750 747 700 702 minus02589 675 668 750 739 07510 650 653 750 743 07513 650 650 725 709 037514 650 645 725 711 05015 675 663 7375 731 050











1a 550 5411b 525 5184 475 4895 500 5179 525 49910 5375 53413 5125 51614 500 52015 500 524






















EXPERIMENT 2


Methods



Results





079 925 048 750043 575 043 725039 500 029 475







079 925 048 700043 600 043 675031 475 029 475



Discussion






079 925 048 725043 650 043 6875023 425 029 500








073 925 048 7375043 700 043 700016 3875 029 475








079 925 048 775048 700 043 725043 6375 029 575



079 925 048 775043 675 043 7375016 375 029 475



l = I r + a






079 925 048 7875048 725 043 725043 650 029 600



079 925 048 7875043 700 043 725016 375 029 475







079 925 048 7875048 725 043 725043 675 029 600



079 925 048 800043 700 043 725016 375 029 475








22 575 543 725 717 162523 600 596 675 674 062524 650 650 6875 681 0312525 700 689 700 697 0125











Tabl

e7

Med

ian

and

mea

nM

unse

llm

atch

esa

ndth

eH

odge

sndashL

ehm

ann

esti

mat

oro

btai

ned

for

the

diam

onds

inF

igs

26ndash2

8

Fig

ure

Alt

ered

Hod

gesndash

Una

lter

edH

odge

sndashnu

mbe

rL

ehm

ann

Leh

man

nes

tim

ator

esti

mat

orL

ight

stri

pD

ark

stri

pL

ight

stri

pD

ark

stri

p

Med

ian

Mea

nM

edia

nM

ean

Med

ian

Mea

nM

edia

nM

ean

266

756

697

375

735

062

56

375

625

725

736

112

527

700

689

725

727

037

56

506

497

257

260

7528

700

690

725

722

037

56

756

677

257

310

625








163 211 253 295 332



000 009 015 020 023








GENERAL DISCUSSION


























CONCLUSION


Acknowledgement


NOTES





















REFERENCES





















































































Reflectance Median

048 750043 700029 525








079 925 048 750048 650 043 700043 575 029 5375


Results







073 875 052 875061 775 037 600052 625 031 475








077 900 050 800050 675 043 675043 550 023 425









079 925 048 725048 650 043 650043 575 029 5125









079 925 048 725048 650 043 650043 575 029 500





Discussion





079 925 048 7375048 675 043 675043 575 029 500









016 400 415023 500 504029 575 568031 575 583039 650 649043 675 676048 700 701079 900 886







016 400 403 475 466 050023 500 510 575 571 050029 575 578 625 637 050031 600 597 650 663 0625039 650 651 700 712 0625043 700 690 725 740 050048 725 721 775 776 0625079 900 886 925 925 0375






1a 043 475 487 725 735 23751b 043026 450 436 425 429 0252 052 550 527 875 870 33754 043 600 596 650 640 0255 043 625 611 650 650 0256 043 500 516 725 719 21257 051 575 552 900 886 337589 043 500 513 700 675 17510 043 575 545 700 685 137511 052 625 605 875 853 25012 043 550 532 675 663 12513 043 575 559 650 654 087514 043 575 553 650 649 10015 043 575 554 675 659 0875





1a 675 656 775 767 1001b 600 591 725 728 1254 700 698 700 690 0005 750 747 700 702 minus02589 675 668 750 739 07510 650 653 750 743 07513 650 650 725 709 037514 650 645 725 711 05015 675 663 7375 731 050











1a 550 5411b 525 5184 475 4895 500 5179 525 49910 5375 53413 5125 51614 500 52015 500 524






















EXPERIMENT 2


Methods



Results





079 925 048 750043 575 043 725039 500 029 475







079 925 048 700043 600 043 675031 475 029 475



Discussion






079 925 048 725043 650 043 6875023 425 029 500








073 925 048 7375043 700 043 700016 3875 029 475








079 925 048 775048 700 043 725043 6375 029 575



079 925 048 775043 675 043 7375016 375 029 475



l = I r + a






079 925 048 7875048 725 043 725043 650 029 600



079 925 048 7875043 700 043 725016 375 029 475







079 925 048 7875048 725 043 725043 675 029 600



079 925 048 800043 700 043 725016 375 029 475








22 575 543 725 717 162523 600 596 675 674 062524 650 650 6875 681 0312525 700 689 700 697 0125











Tabl

e7

Med

ian

and

mea

nM

unse

llm

atch

esa

ndth

eH

odge

sndashL

ehm

ann

esti

mat

oro

btai

ned

for

the

diam

onds

inF

igs

26ndash2

8

Fig

ure

Alt

ered

Hod

gesndash

Una

lter

edH

odge

sndashnu

mbe

rL

ehm

ann

Leh

man

nes

tim

ator

esti

mat

orL

ight

stri

pD

ark

stri

pL

ight

stri

pD

ark

stri

p

Med

ian

Mea

nM

edia

nM

ean

Med

ian

Mea

nM

edia

nM

ean

266

756

697

375

735

062

56

375

625

725

736

112

527

700

689

725

727

037

56

506

497

257

260

7528

700

690

725

722

037

56

756

677

257

310

625








163 211 253 295 332



000 009 015 020 023








GENERAL DISCUSSION


























CONCLUSION


Acknowledgement


NOTES





















REFERENCES























































































079 925 048 750048 650 043 700043 575 029 5375


Results







073 875 052 875061 775 037 600052 625 031 475








077 900 050 800050 675 043 675043 550 023 425









079 925 048 725048 650 043 650043 575 029 5125









079 925 048 725048 650 043 650043 575 029 500





Discussion





079 925 048 7375048 675 043 675043 575 029 500









016 400 415023 500 504029 575 568031 575 583039 650 649043 675 676048 700 701079 900 886







016 400 403 475 466 050023 500 510 575 571 050029 575 578 625 637 050031 600 597 650 663 0625039 650 651 700 712 0625043 700 690 725 740 050048 725 721 775 776 0625079 900 886 925 925 0375






1a 043 475 487 725 735 23751b 043026 450 436 425 429 0252 052 550 527 875 870 33754 043 600 596 650 640 0255 043 625 611 650 650 0256 043 500 516 725 719 21257 051 575 552 900 886 337589 043 500 513 700 675 17510 043 575 545 700 685 137511 052 625 605 875 853 25012 043 550 532 675 663 12513 043 575 559 650 654 087514 043 575 553 650 649 10015 043 575 554 675 659 0875





1a 675 656 775 767 1001b 600 591 725 728 1254 700 698 700 690 0005 750 747 700 702 minus02589 675 668 750 739 07510 650 653 750 743 07513 650 650 725 709 037514 650 645 725 711 05015 675 663 7375 731 050











1a 550 5411b 525 5184 475 4895 500 5179 525 49910 5375 53413 5125 51614 500 52015 500 524






















EXPERIMENT 2


Methods



Results





079 925 048 750043 575 043 725039 500 029 475







079 925 048 700043 600 043 675031 475 029 475



Discussion






079 925 048 725043 650 043 6875023 425 029 500








073 925 048 7375043 700 043 700016 3875 029 475








079 925 048 775048 700 043 725043 6375 029 575



079 925 048 775043 675 043 7375016 375 029 475



l = I r + a






079 925 048 7875048 725 043 725043 650 029 600



079 925 048 7875043 700 043 725016 375 029 475







079 925 048 7875048 725 043 725043 675 029 600



079 925 048 800043 700 043 725016 375 029 475








22 575 543 725 717 162523 600 596 675 674 062524 650 650 6875 681 0312525 700 689 700 697 0125











Tabl

e7

Med

ian

and

mea

nM

unse

llm

atch

esa

ndth

eH

odge

sndashL

ehm

ann

esti

mat

oro

btai

ned

for

the

diam

onds

inF

igs

26ndash2

8

Fig

ure

Alt

ered

Hod

gesndash

Una

lter

edH

odge

sndashnu

mbe

rL

ehm

ann

Leh

man

nes

tim

ator

esti

mat

orL

ight

stri

pD

ark

stri

pL

ight

stri

pD

ark

stri

p

Med

ian

Mea

nM

edia

nM

ean

Med

ian

Mea

nM

edia

nM

ean

266

756

697

375

735

062

56

375

625

725

736

112

527

700

689

725

727

037

56

506

497

257

260

7528

700

690

725

722

037

56

756

677

257

310

625








163 211 253 295 332



000 009 015 020 023








GENERAL DISCUSSION


























CONCLUSION


Acknowledgement


NOTES





















REFERENCES























































































073 875 052 875061 775 037 600052 625 031 475








077 900 050 800050 675 043 675043 550 023 425









079 925 048 725048 650 043 650043 575 029 5125









079 925 048 725048 650 043 650043 575 029 500





Discussion





079 925 048 7375048 675 043 675043 575 029 500









016 400 415023 500 504029 575 568031 575 583039 650 649043 675 676048 700 701079 900 886







016 400 403 475 466 050023 500 510 575 571 050029 575 578 625 637 050031 600 597 650 663 0625039 650 651 700 712 0625043 700 690 725 740 050048 725 721 775 776 0625079 900 886 925 925 0375






1a 043 475 487 725 735 23751b 043026 450 436 425 429 0252 052 550 527 875 870 33754 043 600 596 650 640 0255 043 625 611 650 650 0256 043 500 516 725 719 21257 051 575 552 900 886 337589 043 500 513 700 675 17510 043 575 545 700 685 137511 052 625 605 875 853 25012 043 550 532 675 663 12513 043 575 559 650 654 087514 043 575 553 650 649 10015 043 575 554 675 659 0875





1a 675 656 775 767 1001b 600 591 725 728 1254 700 698 700 690 0005 750 747 700 702 minus02589 675 668 750 739 07510 650 653 750 743 07513 650 650 725 709 037514 650 645 725 711 05015 675 663 7375 731 050











1a 550 5411b 525 5184 475 4895 500 5179 525 49910 5375 53413 5125 51614 500 52015 500 524






















EXPERIMENT 2


Methods



Results





079 925 048 750043 575 043 725039 500 029 475







079 925 048 700043 600 043 675031 475 029 475



Discussion






079 925 048 725043 650 043 6875023 425 029 500








073 925 048 7375043 700 043 700016 3875 029 475








079 925 048 775048 700 043 725043 6375 029 575



079 925 048 775043 675 043 7375016 375 029 475



l = I r + a






079 925 048 7875048 725 043 725043 650 029 600



079 925 048 7875043 700 043 725016 375 029 475







079 925 048 7875048 725 043 725043 675 029 600



079 925 048 800043 700 043 725016 375 029 475








22 575 543 725 717 162523 600 596 675 674 062524 650 650 6875 681 0312525 700 689 700 697 0125











Tabl

e7

Med

ian

and

mea

nM

unse

llm

atch

esa

ndth

eH

odge

sndashL

ehm

ann

esti

mat

oro

btai

ned

for

the

diam

onds

inF

igs

26ndash2

8

Fig

ure

Alt

ered

Hod

gesndash

Una

lter

edH

odge

sndashnu

mbe

rL

ehm

ann

Leh

man

nes

tim

ator

esti

mat

orL

ight

stri

pD

ark

stri

pL

ight

stri

pD

ark

stri

p

Med

ian

Mea

nM

edia

nM

ean

Med

ian

Mea

nM

edia

nM

ean

266

756

697

375

735

062

56

375

625

725

736

112

527

700

689

725

727

037

56

506

497

257

260

7528

700

690

725

722

037

56

756

677

257

310

625








163 211 253 295 332



000 009 015 020 023








GENERAL DISCUSSION


























CONCLUSION


Acknowledgement


NOTES





















REFERENCES























































































077 900 050 800050 675 043 675043 550 023 425









079 925 048 725048 650 043 650043 575 029 5125









079 925 048 725048 650 043 650043 575 029 500





Discussion





079 925 048 7375048 675 043 675043 575 029 500









016 400 415023 500 504029 575 568031 575 583039 650 649043 675 676048 700 701079 900 886







016 400 403 475 466 050023 500 510 575 571 050029 575 578 625 637 050031 600 597 650 663 0625039 650 651 700 712 0625043 700 690 725 740 050048 725 721 775 776 0625079 900 886 925 925 0375






1a 043 475 487 725 735 23751b 043026 450 436 425 429 0252 052 550 527 875 870 33754 043 600 596 650 640 0255 043 625 611 650 650 0256 043 500 516 725 719 21257 051 575 552 900 886 337589 043 500 513 700 675 17510 043 575 545 700 685 137511 052 625 605 875 853 25012 043 550 532 675 663 12513 043 575 559 650 654 087514 043 575 553 650 649 10015 043 575 554 675 659 0875





1a 675 656 775 767 1001b 600 591 725 728 1254 700 698 700 690 0005 750 747 700 702 minus02589 675 668 750 739 07510 650 653 750 743 07513 650 650 725 709 037514 650 645 725 711 05015 675 663 7375 731 050











1a 550 5411b 525 5184 475 4895 500 5179 525 49910 5375 53413 5125 51614 500 52015 500 524






















EXPERIMENT 2


Methods



Results





079 925 048 750043 575 043 725039 500 029 475







079 925 048 700043 600 043 675031 475 029 475



Discussion






079 925 048 725043 650 043 6875023 425 029 500








073 925 048 7375043 700 043 700016 3875 029 475








079 925 048 775048 700 043 725043 6375 029 575



079 925 048 775043 675 043 7375016 375 029 475



l = I r + a






079 925 048 7875048 725 043 725043 650 029 600



079 925 048 7875043 700 043 725016 375 029 475







079 925 048 7875048 725 043 725043 675 029 600



079 925 048 800043 700 043 725016 375 029 475








22 575 543 725 717 162523 600 596 675 674 062524 650 650 6875 681 0312525 700 689 700 697 0125











Tabl

e7

Med

ian

and

mea

nM

unse

llm

atch

esa

ndth

eH

odge

sndashL

ehm

ann

esti

mat

oro

btai

ned

for

the

diam

onds

inF

igs

26ndash2

8

Fig

ure

Alt

ered

Hod

gesndash

Una

lter

edH

odge

sndashnu

mbe

rL

ehm

ann

Leh

man

nes

tim

ator

esti

mat

orL

ight

stri

pD

ark

stri

pL

ight

stri

pD

ark

stri

p

Med

ian

Mea

nM

edia

nM

ean

Med

ian

Mea

nM

edia

nM

ean

266

756

697

375

735

062

56

375

625

725

736

112

527

700

689

725

727

037

56

506

497

257

260

7528

700

690

725

722

037

56

756

677

257

310

625








163 211 253 295 332



000 009 015 020 023








GENERAL DISCUSSION


























CONCLUSION


Acknowledgement


NOTES





















REFERENCES























































































079 925 048 725048 650 043 650043 575 029 5125









079 925 048 725048 650 043 650043 575 029 500





Discussion





079 925 048 7375048 675 043 675043 575 029 500









016 400 415023 500 504029 575 568031 575 583039 650 649043 675 676048 700 701079 900 886







016 400 403 475 466 050023 500 510 575 571 050029 575 578 625 637 050031 600 597 650 663 0625039 650 651 700 712 0625043 700 690 725 740 050048 725 721 775 776 0625079 900 886 925 925 0375






1a 043 475 487 725 735 23751b 043026 450 436 425 429 0252 052 550 527 875 870 33754 043 600 596 650 640 0255 043 625 611 650 650 0256 043 500 516 725 719 21257 051 575 552 900 886 337589 043 500 513 700 675 17510 043 575 545 700 685 137511 052 625 605 875 853 25012 043 550 532 675 663 12513 043 575 559 650 654 087514 043 575 553 650 649 10015 043 575 554 675 659 0875





1a 675 656 775 767 1001b 600 591 725 728 1254 700 698 700 690 0005 750 747 700 702 minus02589 675 668 750 739 07510 650 653 750 743 07513 650 650 725 709 037514 650 645 725 711 05015 675 663 7375 731 050











1a 550 5411b 525 5184 475 4895 500 5179 525 49910 5375 53413 5125 51614 500 52015 500 524






















EXPERIMENT 2


Methods



Results





079 925 048 750043 575 043 725039 500 029 475







079 925 048 700043 600 043 675031 475 029 475



Discussion






079 925 048 725043 650 043 6875023 425 029 500








073 925 048 7375043 700 043 700016 3875 029 475








079 925 048 775048 700 043 725043 6375 029 575



079 925 048 775043 675 043 7375016 375 029 475



l = I r + a






079 925 048 7875048 725 043 725043 650 029 600



079 925 048 7875043 700 043 725016 375 029 475







079 925 048 7875048 725 043 725043 675 029 600



079 925 048 800043 700 043 725016 375 029 475








22 575 543 725 717 162523 600 596 675 674 062524 650 650 6875 681 0312525 700 689 700 697 0125











Tabl

e7

Med

ian

and

mea

nM

unse

llm

atch

esa

ndth

eH

odge

sndashL

ehm

ann

esti

mat

oro

btai

ned

for

the

diam

onds

inF

igs

26ndash2

8

Fig

ure

Alt

ered

Hod

gesndash

Una

lter

edH

odge

sndashnu

mbe

rL

ehm

ann

Leh

man

nes

tim

ator

esti

mat

orL

ight

stri

pD

ark

stri

pL

ight

stri

pD

ark

stri

p

Med

ian

Mea

nM

edia

nM

ean

Med

ian

Mea

nM

edia

nM

ean

266

756

697

375

735

062

56

375

625

725

736

112

527

700

689

725

727

037

56

506

497

257

260

7528

700

690

725

722

037

56

756

677

257

310

625








163 211 253 295 332



000 009 015 020 023








GENERAL DISCUSSION


























CONCLUSION


Acknowledgement


NOTES





















REFERENCES























































































079 925 048 725048 650 043 650043 575 029 500





Discussion





079 925 048 7375048 675 043 675043 575 029 500









016 400 415023 500 504029 575 568031 575 583039 650 649043 675 676048 700 701079 900 886







016 400 403 475 466 050023 500 510 575 571 050029 575 578 625 637 050031 600 597 650 663 0625039 650 651 700 712 0625043 700 690 725 740 050048 725 721 775 776 0625079 900 886 925 925 0375






1a 043 475 487 725 735 23751b 043026 450 436 425 429 0252 052 550 527 875 870 33754 043 600 596 650 640 0255 043 625 611 650 650 0256 043 500 516 725 719 21257 051 575 552 900 886 337589 043 500 513 700 675 17510 043 575 545 700 685 137511 052 625 605 875 853 25012 043 550 532 675 663 12513 043 575 559 650 654 087514 043 575 553 650 649 10015 043 575 554 675 659 0875





1a 675 656 775 767 1001b 600 591 725 728 1254 700 698 700 690 0005 750 747 700 702 minus02589 675 668 750 739 07510 650 653 750 743 07513 650 650 725 709 037514 650 645 725 711 05015 675 663 7375 731 050











1a 550 5411b 525 5184 475 4895 500 5179 525 49910 5375 53413 5125 51614 500 52015 500 524






















EXPERIMENT 2


Methods



Results





079 925 048 750043 575 043 725039 500 029 475







079 925 048 700043 600 043 675031 475 029 475



Discussion






079 925 048 725043 650 043 6875023 425 029 500








073 925 048 7375043 700 043 700016 3875 029 475








079 925 048 775048 700 043 725043 6375 029 575



079 925 048 775043 675 043 7375016 375 029 475



l = I r + a






079 925 048 7875048 725 043 725043 650 029 600



079 925 048 7875043 700 043 725016 375 029 475







079 925 048 7875048 725 043 725043 675 029 600



079 925 048 800043 700 043 725016 375 029 475








22 575 543 725 717 162523 600 596 675 674 062524 650 650 6875 681 0312525 700 689 700 697 0125











Tabl

e7

Med

ian

and

mea

nM

unse

llm

atch

esa

ndth

eH

odge

sndashL

ehm

ann

esti

mat

oro

btai

ned

for

the

diam

onds

inF

igs

26ndash2

8

Fig

ure

Alt

ered

Hod

gesndash

Una

lter

edH

odge

sndashnu

mbe

rL

ehm

ann

Leh

man

nes

tim

ator

esti

mat

orL

ight

stri

pD

ark

stri

pL

ight

stri

pD

ark

stri

p

Med

ian

Mea

nM

edia

nM

ean

Med

ian

Mea

nM

edia

nM

ean

266

756

697

375

735

062

56

375

625

725

736

112

527

700

689

725

727

037

56

506

497

257

260

7528

700

690

725

722

037

56

756

677

257

310

625








163 211 253 295 332



000 009 015 020 023








GENERAL DISCUSSION


























CONCLUSION


Acknowledgement


NOTES





















REFERENCES























































































079 925 048 7375048 675 043 675043 575 029 500









016 400 415023 500 504029 575 568031 575 583039 650 649043 675 676048 700 701079 900 886







016 400 403 475 466 050023 500 510 575 571 050029 575 578 625 637 050031 600 597 650 663 0625039 650 651 700 712 0625043 700 690 725 740 050048 725 721 775 776 0625079 900 886 925 925 0375






1a 043 475 487 725 735 23751b 043026 450 436 425 429 0252 052 550 527 875 870 33754 043 600 596 650 640 0255 043 625 611 650 650 0256 043 500 516 725 719 21257 051 575 552 900 886 337589 043 500 513 700 675 17510 043 575 545 700 685 137511 052 625 605 875 853 25012 043 550 532 675 663 12513 043 575 559 650 654 087514 043 575 553 650 649 10015 043 575 554 675 659 0875





1a 675 656 775 767 1001b 600 591 725 728 1254 700 698 700 690 0005 750 747 700 702 minus02589 675 668 750 739 07510 650 653 750 743 07513 650 650 725 709 037514 650 645 725 711 05015 675 663 7375 731 050











1a 550 5411b 525 5184 475 4895 500 5179 525 49910 5375 53413 5125 51614 500 52015 500 524






















EXPERIMENT 2


Methods



Results





079 925 048 750043 575 043 725039 500 029 475







079 925 048 700043 600 043 675031 475 029 475



Discussion






079 925 048 725043 650 043 6875023 425 029 500








073 925 048 7375043 700 043 700016 3875 029 475








079 925 048 775048 700 043 725043 6375 029 575



079 925 048 775043 675 043 7375016 375 029 475



l = I r + a






079 925 048 7875048 725 043 725043 650 029 600



079 925 048 7875043 700 043 725016 375 029 475







079 925 048 7875048 725 043 725043 675 029 600



079 925 048 800043 700 043 725016 375 029 475








22 575 543 725 717 162523 600 596 675 674 062524 650 650 6875 681 0312525 700 689 700 697 0125











Tabl

e7

Med

ian

and

mea

nM

unse

llm

atch

esa

ndth

eH

odge

sndashL

ehm

ann

esti

mat

oro

btai

ned

for

the

diam

onds

inF

igs

26ndash2

8

Fig

ure

Alt

ered

Hod

gesndash

Una

lter

edH

odge

sndashnu

mbe

rL

ehm

ann

Leh

man

nes

tim

ator

esti

mat

orL

ight

stri

pD

ark

stri

pL

ight

stri

pD

ark

stri

p

Med

ian

Mea

nM

edia

nM

ean

Med

ian

Mea

nM

edia

nM

ean

266

756

697

375

735

062

56

375

625

725

736

112

527

700

689

725

727

037

56

506

497

257

260

7528

700

690

725

722

037

56

756

677

257

310

625








163 211 253 295 332



000 009 015 020 023








GENERAL DISCUSSION


























CONCLUSION


Acknowledgement


NOTES





















REFERENCES
























































































016 400 415023 500 504029 575 568031 575 583039 650 649043 675 676048 700 701079 900 886







016 400 403 475 466 050023 500 510 575 571 050029 575 578 625 637 050031 600 597 650 663 0625039 650 651 700 712 0625043 700 690 725 740 050048 725 721 775 776 0625079 900 886 925 925 0375






1a 043 475 487 725 735 23751b 043026 450 436 425 429 0252 052 550 527 875 870 33754 043 600 596 650 640 0255 043 625 611 650 650 0256 043 500 516 725 719 21257 051 575 552 900 886 337589 043 500 513 700 675 17510 043 575 545 700 685 137511 052 625 605 875 853 25012 043 550 532 675 663 12513 043 575 559 650 654 087514 043 575 553 650 649 10015 043 575 554 675 659 0875





1a 675 656 775 767 1001b 600 591 725 728 1254 700 698 700 690 0005 750 747 700 702 minus02589 675 668 750 739 07510 650 653 750 743 07513 650 650 725 709 037514 650 645 725 711 05015 675 663 7375 731 050











1a 550 5411b 525 5184 475 4895 500 5179 525 49910 5375 53413 5125 51614 500 52015 500 524






















EXPERIMENT 2


Methods



Results





079 925 048 750043 575 043 725039 500 029 475







079 925 048 700043 600 043 675031 475 029 475



Discussion






079 925 048 725043 650 043 6875023 425 029 500








073 925 048 7375043 700 043 700016 3875 029 475








079 925 048 775048 700 043 725043 6375 029 575



079 925 048 775043 675 043 7375016 375 029 475



l = I r + a






079 925 048 7875048 725 043 725043 650 029 600



079 925 048 7875043 700 043 725016 375 029 475







079 925 048 7875048 725 043 725043 675 029 600



079 925 048 800043 700 043 725016 375 029 475








22 575 543 725 717 162523 600 596 675 674 062524 650 650 6875 681 0312525 700 689 700 697 0125











Tabl

e7

Med

ian

and

mea

nM

unse

llm

atch

esa

ndth

eH

odge

sndashL

ehm

ann

esti

mat

oro

btai

ned

for

the

diam

onds

inF

igs

26ndash2

8

Fig

ure

Alt

ered

Hod

gesndash

Una

lter

edH

odge

sndashnu

mbe

rL

ehm

ann

Leh

man

nes

tim

ator

esti

mat

orL

ight

stri

pD

ark

stri

pL

ight

stri

pD

ark

stri

p

Med

ian

Mea

nM

edia

nM

ean

Med

ian

Mea

nM

edia

nM

ean

266

756

697

375

735

062

56

375

625

725

736

112

527

700

689

725

727

037

56

506

497

257

260

7528

700

690

725

722

037

56

756

677

257

310

625








163 211 253 295 332



000 009 015 020 023








GENERAL DISCUSSION


























CONCLUSION


Acknowledgement


NOTES





















REFERENCES
























































































016 400 403 475 466 050023 500 510 575 571 050029 575 578 625 637 050031 600 597 650 663 0625039 650 651 700 712 0625043 700 690 725 740 050048 725 721 775 776 0625079 900 886 925 925 0375






1a 043 475 487 725 735 23751b 043026 450 436 425 429 0252 052 550 527 875 870 33754 043 600 596 650 640 0255 043 625 611 650 650 0256 043 500 516 725 719 21257 051 575 552 900 886 337589 043 500 513 700 675 17510 043 575 545 700 685 137511 052 625 605 875 853 25012 043 550 532 675 663 12513 043 575 559 650 654 087514 043 575 553 650 649 10015 043 575 554 675 659 0875





1a 675 656 775 767 1001b 600 591 725 728 1254 700 698 700 690 0005 750 747 700 702 minus02589 675 668 750 739 07510 650 653 750 743 07513 650 650 725 709 037514 650 645 725 711 05015 675 663 7375 731 050











1a 550 5411b 525 5184 475 4895 500 5179 525 49910 5375 53413 5125 51614 500 52015 500 524






















EXPERIMENT 2


Methods



Results





079 925 048 750043 575 043 725039 500 029 475







079 925 048 700043 600 043 675031 475 029 475



Discussion






079 925 048 725043 650 043 6875023 425 029 500








073 925 048 7375043 700 043 700016 3875 029 475








079 925 048 775048 700 043 725043 6375 029 575



079 925 048 775043 675 043 7375016 375 029 475



l = I r + a






079 925 048 7875048 725 043 725043 650 029 600



079 925 048 7875043 700 043 725016 375 029 475







079 925 048 7875048 725 043 725043 675 029 600



079 925 048 800043 700 043 725016 375 029 475








22 575 543 725 717 162523 600 596 675 674 062524 650 650 6875 681 0312525 700 689 700 697 0125











Tabl

e7

Med

ian

and

mea

nM

unse

llm

atch

esa

ndth

eH

odge

sndashL

ehm

ann

esti

mat

oro

btai

ned

for

the

diam

onds

inF

igs

26ndash2

8

Fig

ure

Alt

ered

Hod

gesndash

Una

lter

edH

odge

sndashnu

mbe

rL

ehm

ann

Leh

man

nes

tim

ator

esti

mat

orL

ight

stri

pD

ark

stri

pL

ight

stri

pD

ark

stri

p

Med

ian

Mea

nM

edia

nM

ean

Med

ian

Mea

nM

edia

nM

ean

266

756

697

375

735

062

56

375

625

725

736

112

527

700

689

725

727

037

56

506

497

257

260

7528

700

690

725

722

037

56

756

677

257

310

625








163 211 253 295 332



000 009 015 020 023








GENERAL DISCUSSION


























CONCLUSION


Acknowledgement


NOTES





















REFERENCES
























































































1a 043 475 487 725 735 23751b 043026 450 436 425 429 0252 052 550 527 875 870 33754 043 600 596 650 640 0255 043 625 611 650 650 0256 043 500 516 725 719 21257 051 575 552 900 886 337589 043 500 513 700 675 17510 043 575 545 700 685 137511 052 625 605 875 853 25012 043 550 532 675 663 12513 043 575 559 650 654 087514 043 575 553 650 649 10015 043 575 554 675 659 0875





1a 675 656 775 767 1001b 600 591 725 728 1254 700 698 700 690 0005 750 747 700 702 minus02589 675 668 750 739 07510 650 653 750 743 07513 650 650 725 709 037514 650 645 725 711 05015 675 663 7375 731 050











1a 550 5411b 525 5184 475 4895 500 5179 525 49910 5375 53413 5125 51614 500 52015 500 524






















EXPERIMENT 2


Methods



Results





079 925 048 750043 575 043 725039 500 029 475







079 925 048 700043 600 043 675031 475 029 475



Discussion






079 925 048 725043 650 043 6875023 425 029 500








073 925 048 7375043 700 043 700016 3875 029 475








079 925 048 775048 700 043 725043 6375 029 575



079 925 048 775043 675 043 7375016 375 029 475



l = I r + a






079 925 048 7875048 725 043 725043 650 029 600



079 925 048 7875043 700 043 725016 375 029 475







079 925 048 7875048 725 043 725043 675 029 600



079 925 048 800043 700 043 725016 375 029 475








22 575 543 725 717 162523 600 596 675 674 062524 650 650 6875 681 0312525 700 689 700 697 0125











Tabl

e7

Med

ian

and

mea

nM

unse

llm

atch

esa

ndth

eH

odge

sndashL

ehm

ann

esti

mat

oro

btai

ned

for

the

diam

onds

inF

igs

26ndash2

8

Fig

ure

Alt

ered

Hod

gesndash

Una

lter

edH

odge

sndashnu

mbe

rL

ehm

ann

Leh

man

nes

tim

ator

esti

mat

orL

ight

stri

pD

ark

stri

pL

ight

stri

pD

ark

stri

p

Med

ian

Mea

nM

edia

nM

ean

Med

ian

Mea

nM

edia

nM

ean

266

756

697

375

735

062

56

375

625

725

736

112

527

700

689

725

727

037

56

506

497

257

260

7528

700

690

725

722

037

56

756

677

257

310

625








163 211 253 295 332



000 009 015 020 023








GENERAL DISCUSSION


























CONCLUSION


Acknowledgement


NOTES





















REFERENCES
























































































1a 675 656 775 767 1001b 600 591 725 728 1254 700 698 700 690 0005 750 747 700 702 minus02589 675 668 750 739 07510 650 653 750 743 07513 650 650 725 709 037514 650 645 725 711 05015 675 663 7375 731 050











1a 550 5411b 525 5184 475 4895 500 5179 525 49910 5375 53413 5125 51614 500 52015 500 524






















EXPERIMENT 2


Methods



Results





079 925 048 750043 575 043 725039 500 029 475







079 925 048 700043 600 043 675031 475 029 475



Discussion






079 925 048 725043 650 043 6875023 425 029 500








073 925 048 7375043 700 043 700016 3875 029 475








079 925 048 775048 700 043 725043 6375 029 575



079 925 048 775043 675 043 7375016 375 029 475



l = I r + a






079 925 048 7875048 725 043 725043 650 029 600



079 925 048 7875043 700 043 725016 375 029 475







079 925 048 7875048 725 043 725043 675 029 600



079 925 048 800043 700 043 725016 375 029 475








22 575 543 725 717 162523 600 596 675 674 062524 650 650 6875 681 0312525 700 689 700 697 0125











Tabl

e7

Med

ian

and

mea

nM

unse

llm

atch

esa

ndth

eH

odge

sndashL

ehm

ann

esti

mat

oro

btai

ned

for

the

diam

onds

inF

igs

26ndash2

8

Fig

ure

Alt

ered

Hod

gesndash

Una

lter

edH

odge

sndashnu

mbe

rL

ehm

ann

Leh

man

nes

tim

ator

esti

mat

orL

ight

stri

pD

ark

stri

pL

ight

stri

pD

ark

stri

p

Med

ian

Mea

nM

edia

nM

ean

Med

ian

Mea

nM

edia

nM

ean

266

756

697

375

735

062

56

375

625

725

736

112

527

700

689

725

727

037

56

506

497

257

260

7528

700

690

725

722

037

56

756

677

257

310

625








163 211 253 295 332



000 009 015 020 023








GENERAL DISCUSSION


























CONCLUSION


Acknowledgement


NOTES





















REFERENCES





























































































1a 550 5411b 525 5184 475 4895 500 5179 525 49910 5375 53413 5125 51614 500 52015 500 524






















EXPERIMENT 2


Methods



Results





079 925 048 750043 575 043 725039 500 029 475







079 925 048 700043 600 043 675031 475 029 475



Discussion






079 925 048 725043 650 043 6875023 425 029 500








073 925 048 7375043 700 043 700016 3875 029 475








079 925 048 775048 700 043 725043 6375 029 575



079 925 048 775043 675 043 7375016 375 029 475



l = I r + a






079 925 048 7875048 725 043 725043 650 029 600



079 925 048 7875043 700 043 725016 375 029 475







079 925 048 7875048 725 043 725043 675 029 600



079 925 048 800043 700 043 725016 375 029 475








22 575 543 725 717 162523 600 596 675 674 062524 650 650 6875 681 0312525 700 689 700 697 0125











Tabl

e7

Med

ian

and

mea

nM

unse

llm

atch

esa

ndth

eH

odge

sndashL

ehm

ann

esti

mat

oro

btai

ned

for

the

diam

onds

inF

igs

26ndash2

8

Fig

ure

Alt

ered

Hod

gesndash

Una

lter

edH

odge

sndashnu

mbe

rL

ehm

ann

Leh

man

nes

tim

ator

esti

mat

orL

ight

stri

pD

ark

stri

pL

ight

stri

pD

ark

stri

p

Med

ian

Mea

nM

edia

nM

ean

Med

ian

Mea

nM

edia

nM

ean

266

756

697

375

735

062

56

375

625

725

736

112

527

700

689

725

727

037

56

506

497

257

260

7528

700

690

725

722

037

56

756

677

257

310

625








163 211 253 295 332



000 009 015 020 023








GENERAL DISCUSSION


























CONCLUSION


Acknowledgement


NOTES





















REFERENCES








































































































EXPERIMENT 2


Methods



Results





079 925 048 750043 575 043 725039 500 029 475







079 925 048 700043 600 043 675031 475 029 475



Discussion






079 925 048 725043 650 043 6875023 425 029 500








073 925 048 7375043 700 043 700016 3875 029 475








079 925 048 775048 700 043 725043 6375 029 575



079 925 048 775043 675 043 7375016 375 029 475



l = I r + a






079 925 048 7875048 725 043 725043 650 029 600



079 925 048 7875043 700 043 725016 375 029 475







079 925 048 7875048 725 043 725043 675 029 600



079 925 048 800043 700 043 725016 375 029 475








22 575 543 725 717 162523 600 596 675 674 062524 650 650 6875 681 0312525 700 689 700 697 0125











Tabl

e7

Med

ian

and

mea

nM

unse

llm

atch

esa

ndth

eH

odge

sndashL

ehm

ann

esti

mat

oro

btai

ned

for

the

diam

onds

inF

igs

26ndash2

8

Fig

ure

Alt

ered

Hod

gesndash

Una

lter

edH

odge

sndashnu

mbe

rL

ehm

ann

Leh

man

nes

tim

ator

esti

mat

orL

ight

stri

pD

ark

stri

pL

ight

stri

pD

ark

stri

p

Med

ian

Mea

nM

edia

nM

ean

Med

ian

Mea

nM

edia

nM

ean

266

756

697

375

735

062

56

375

625

725

736

112

527

700

689

725

727

037

56

506

497

257

260

7528

700

690

725

722

037

56

756

677

257

310

625








163 211 253 295 332



000 009 015 020 023








GENERAL DISCUSSION


























CONCLUSION


Acknowledgement


NOTES





















REFERENCES























































































079 925 048 750043 575 043 725039 500 029 475







079 925 048 700043 600 043 675031 475 029 475



Discussion






079 925 048 725043 650 043 6875023 425 029 500








073 925 048 7375043 700 043 700016 3875 029 475








079 925 048 775048 700 043 725043 6375 029 575



079 925 048 775043 675 043 7375016 375 029 475



l = I r + a






079 925 048 7875048 725 043 725043 650 029 600



079 925 048 7875043 700 043 725016 375 029 475







079 925 048 7875048 725 043 725043 675 029 600



079 925 048 800043 700 043 725016 375 029 475








22 575 543 725 717 162523 600 596 675 674 062524 650 650 6875 681 0312525 700 689 700 697 0125











Tabl

e7

Med

ian

and

mea

nM

unse

llm

atch

esa

ndth

eH

odge

sndashL

ehm

ann

esti

mat

oro

btai

ned

for

the

diam

onds

inF

igs

26ndash2

8

Fig

ure

Alt

ered

Hod

gesndash

Una

lter

edH

odge

sndashnu

mbe

rL

ehm

ann

Leh

man

nes

tim

ator

esti

mat

orL

ight

stri

pD

ark

stri

pL

ight

stri

pD

ark

stri

p

Med

ian

Mea

nM

edia

nM

ean

Med

ian

Mea

nM

edia

nM

ean

266

756

697

375

735

062

56

375

625

725

736

112

527

700

689

725

727

037

56

506

497

257

260

7528

700

690

725

722

037

56

756

677

257

310

625








163 211 253 295 332



000 009 015 020 023








GENERAL DISCUSSION


























CONCLUSION


Acknowledgement


NOTES





















REFERENCES























































































079 925 048 700043 600 043 675031 475 029 475



Discussion






079 925 048 725043 650 043 6875023 425 029 500








073 925 048 7375043 700 043 700016 3875 029 475








079 925 048 775048 700 043 725043 6375 029 575



079 925 048 775043 675 043 7375016 375 029 475



l = I r + a






079 925 048 7875048 725 043 725043 650 029 600



079 925 048 7875043 700 043 725016 375 029 475







079 925 048 7875048 725 043 725043 675 029 600



079 925 048 800043 700 043 725016 375 029 475








22 575 543 725 717 162523 600 596 675 674 062524 650 650 6875 681 0312525 700 689 700 697 0125











Tabl

e7

Med

ian

and

mea

nM

unse

llm

atch

esa

ndth

eH

odge

sndashL

ehm

ann

esti

mat

oro

btai

ned

for

the

diam

onds

inF

igs

26ndash2

8

Fig

ure

Alt

ered

Hod

gesndash

Una

lter

edH

odge

sndashnu

mbe

rL

ehm

ann

Leh

man

nes

tim

ator

esti

mat

orL

ight

stri

pD

ark

stri

pL

ight

stri

pD

ark

stri

p

Med

ian

Mea

nM

edia

nM

ean

Med

ian

Mea

nM

edia

nM

ean

266

756

697

375

735

062

56

375

625

725

736

112

527

700

689

725

727

037

56

506

497

257

260

7528

700

690

725

722

037

56

756

677

257

310

625








163 211 253 295 332



000 009 015 020 023








GENERAL DISCUSSION


























CONCLUSION


Acknowledgement


NOTES





















REFERENCES























































































079 925 048 725043 650 043 6875023 425 029 500








073 925 048 7375043 700 043 700016 3875 029 475








079 925 048 775048 700 043 725043 6375 029 575



079 925 048 775043 675 043 7375016 375 029 475



l = I r + a






079 925 048 7875048 725 043 725043 650 029 600



079 925 048 7875043 700 043 725016 375 029 475







079 925 048 7875048 725 043 725043 675 029 600



079 925 048 800043 700 043 725016 375 029 475








22 575 543 725 717 162523 600 596 675 674 062524 650 650 6875 681 0312525 700 689 700 697 0125











Tabl

e7

Med

ian

and

mea

nM

unse

llm

atch

esa

ndth

eH

odge

sndashL

ehm

ann

esti

mat

oro

btai

ned

for

the

diam

onds

inF

igs

26ndash2

8

Fig

ure

Alt

ered

Hod

gesndash

Una

lter

edH

odge

sndashnu

mbe

rL

ehm

ann

Leh

man

nes

tim

ator

esti

mat

orL

ight

stri

pD

ark

stri

pL

ight

stri

pD

ark

stri

p

Med

ian

Mea

nM

edia

nM

ean

Med

ian

Mea

nM

edia

nM

ean

266

756

697

375

735

062

56

375

625

725

736

112

527

700

689

725

727

037

56

506

497

257

260

7528

700

690

725

722

037

56

756

677

257

310

625








163 211 253 295 332



000 009 015 020 023








GENERAL DISCUSSION


























CONCLUSION


Acknowledgement


NOTES





















REFERENCES























































































073 925 048 7375043 700 043 700016 3875 029 475








079 925 048 775048 700 043 725043 6375 029 575



079 925 048 775043 675 043 7375016 375 029 475



l = I r + a






079 925 048 7875048 725 043 725043 650 029 600



079 925 048 7875043 700 043 725016 375 029 475







079 925 048 7875048 725 043 725043 675 029 600



079 925 048 800043 700 043 725016 375 029 475








22 575 543 725 717 162523 600 596 675 674 062524 650 650 6875 681 0312525 700 689 700 697 0125











Tabl

e7

Med

ian

and

mea

nM

unse

llm

atch

esa

ndth

eH

odge

sndashL

ehm

ann

esti

mat

oro

btai

ned

for

the

diam

onds

inF

igs

26ndash2

8

Fig

ure

Alt

ered

Hod

gesndash

Una

lter

edH

odge

sndashnu

mbe

rL

ehm

ann

Leh

man

nes

tim

ator

esti

mat

orL

ight

stri

pD

ark

stri

pL

ight

stri

pD

ark

stri

p

Med

ian

Mea

nM

edia

nM

ean

Med

ian

Mea

nM

edia

nM

ean

266

756

697

375

735

062

56

375

625

725

736

112

527

700

689

725

727

037

56

506

497

257

260

7528

700

690

725

722

037

56

756

677

257

310

625








163 211 253 295 332



000 009 015 020 023








GENERAL DISCUSSION


























CONCLUSION


Acknowledgement


NOTES





















REFERENCES























































































079 925 048 775048 700 043 725043 6375 029 575



079 925 048 775043 675 043 7375016 375 029 475



l = I r + a






079 925 048 7875048 725 043 725043 650 029 600



079 925 048 7875043 700 043 725016 375 029 475







079 925 048 7875048 725 043 725043 675 029 600



079 925 048 800043 700 043 725016 375 029 475








22 575 543 725 717 162523 600 596 675 674 062524 650 650 6875 681 0312525 700 689 700 697 0125











Tabl

e7

Med

ian

and

mea

nM

unse

llm

atch

esa

ndth

eH

odge

sndashL

ehm

ann

esti

mat

oro

btai

ned

for

the

diam

onds

inF

igs

26ndash2

8

Fig

ure

Alt

ered

Hod

gesndash

Una

lter

edH

odge

sndashnu

mbe

rL

ehm

ann

Leh

man

nes

tim

ator

esti

mat

orL

ight

stri

pD

ark

stri

pL

ight

stri

pD

ark

stri

p

Med

ian

Mea

nM

edia

nM

ean

Med

ian

Mea

nM

edia

nM

ean

266

756

697

375

735

062

56

375

625

725

736

112

527

700

689

725

727

037

56

506

497

257

260

7528

700

690

725

722

037

56

756

677

257

310

625








163 211 253 295 332



000 009 015 020 023








GENERAL DISCUSSION


























CONCLUSION


Acknowledgement


NOTES





















REFERENCES























































































079 925 048 7875048 725 043 725043 650 029 600



079 925 048 7875043 700 043 725016 375 029 475







079 925 048 7875048 725 043 725043 675 029 600



079 925 048 800043 700 043 725016 375 029 475








22 575 543 725 717 162523 600 596 675 674 062524 650 650 6875 681 0312525 700 689 700 697 0125











Tabl

e7

Med

ian

and

mea

nM

unse

llm

atch

esa

ndth

eH

odge

sndashL

ehm

ann

esti

mat

oro

btai

ned

for

the

diam

onds

inF

igs

26ndash2

8

Fig

ure

Alt

ered

Hod

gesndash

Una

lter

edH

odge

sndashnu

mbe

rL

ehm

ann

Leh

man

nes

tim

ator

esti

mat

orL

ight

stri

pD

ark

stri

pL

ight

stri

pD

ark

stri

p

Med

ian

Mea

nM

edia

nM

ean

Med

ian

Mea

nM

edia

nM

ean

266

756

697

375

735

062

56

375

625

725

736

112

527

700

689

725

727

037

56

506

497

257

260

7528

700

690

725

722

037

56

756

677

257

310

625








163 211 253 295 332



000 009 015 020 023








GENERAL DISCUSSION


























CONCLUSION


Acknowledgement


NOTES





















REFERENCES























































































079 925 048 7875048 725 043 725043 675 029 600



079 925 048 800043 700 043 725016 375 029 475








22 575 543 725 717 162523 600 596 675 674 062524 650 650 6875 681 0312525 700 689 700 697 0125











Tabl

e7

Med

ian

and

mea

nM

unse

llm

atch

esa

ndth

eH

odge

sndashL

ehm

ann

esti

mat

oro

btai

ned

for

the

diam

onds

inF

igs

26ndash2

8

Fig

ure

Alt

ered

Hod

gesndash

Una

lter

edH

odge

sndashnu

mbe

rL

ehm

ann

Leh

man

nes

tim

ator

esti

mat

orL

ight

stri

pD

ark

stri

pL

ight

stri

pD

ark

stri

p

Med

ian

Mea

nM

edia

nM

ean

Med

ian

Mea

nM

edia

nM

ean

266

756

697

375

735

062

56

375

625

725

736

112

527

700

689

725

727

037

56

506

497

257

260

7528

700

690

725

722

037

56

756

677

257

310

625








163 211 253 295 332



000 009 015 020 023








GENERAL DISCUSSION


























CONCLUSION


Acknowledgement


NOTES





















REFERENCES
























































































22 575 543 725 717 162523 600 596 675 674 062524 650 650 6875 681 0312525 700 689 700 697 0125











Tabl

e7

Med

ian

and

mea

nM

unse

llm

atch

esa

ndth

eH

odge

sndashL

ehm

ann

esti

mat

oro

btai

ned

for

the

diam

onds

inF

igs

26ndash2

8

Fig

ure

Alt

ered

Hod

gesndash

Una

lter

edH

odge

sndashnu

mbe

rL

ehm

ann

Leh

man

nes

tim

ator

esti

mat

orL

ight

stri

pD

ark

stri

pL

ight

stri

pD

ark

stri

p

Med

ian

Mea

nM

edia

nM

ean

Med

ian

Mea

nM

edia

nM

ean

266

756

697

375

735

062

56

375

625

725

736

112

527

700

689

725

727

037

56

506

497

257

260

7528

700

690

725

722

037

56

756

677

257

310

625








163 211 253 295 332



000 009 015 020 023








GENERAL DISCUSSION


























CONCLUSION


Acknowledgement


NOTES





















REFERENCES

























































































Tabl

e7

Med

ian

and

mea

nM

unse

llm

atch

esa

ndth

eH

odge

sndashL

ehm

ann

esti

mat

oro

btai

ned

for

the

diam

onds

inF

igs

26ndash2

8

Fig

ure

Alt

ered

Hod

gesndash

Una

lter

edH

odge

sndashnu

mbe

rL

ehm

ann

Leh

man

nes

tim

ator

esti

mat

orL

ight

stri

pD

ark

stri

pL

ight

stri

pD

ark

stri

p

Med

ian

Mea

nM

edia

nM

ean

Med

ian

Mea

nM

edia

nM

ean

266

756

697

375

735

062

56

375

625

725

736

112

527

700

689

725

727

037

56

506

497

257

260

7528

700

690

725

722

037

56

756

677

257

310

625








163 211 253 295 332



000 009 015 020 023








GENERAL DISCUSSION


























CONCLUSION


Acknowledgement


NOTES





















REFERENCES



























































































163 211 253 295 332



000 009 015 020 023








GENERAL DISCUSSION


























CONCLUSION


Acknowledgement


NOTES





















REFERENCES






















































































GENERAL DISCUSSION


























CONCLUSION


Acknowledgement


NOTES





















REFERENCES










































































































CONCLUSION


Acknowledgement


NOTES





















REFERENCES


































































































REFERENCES




















































































adelson s tile and snake illusions: a helmholtzian type of ... · adelson s illusions. indeed, the...

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