falchi querzoli romano 2006 eif

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R E S E A R C H A R T I C L E

M. Falchi G. Querzoli G. P. RomanoRobust evaluation of the dissimilarity between interrogation windows

in image velocimetry

Received: 11 November 2005 / Revised: 14 February 2006/ Accepted: 31 March 2006 / Published online: 29 April 2006 Springer-Verlag 2006

Abstract Image velocimetry techniques, which extractmotion information by comparison of image regions,typically make use of cross-correlation to measure thedegree of matching. In this work, a novel measure of the

dissimilarity between interrogation windows is proposedwhich is based on a more robust estimator than cross-correlation. The method is validated on synthetic imagesand on two experimental data sets obtained from aperiodically pulsed jet and a backward-facing step. Theformer is a basically laminar flow, whereas the latter isfully turbulent. Both of them are characterized by re-gions of high velocity gradients. The efficiency of therobust image velocimetry (RIV) is compared with across-correlation algorithm (PIV). The analysis of re-sults shows that the RIV is less sensitive to the appear-ance and disappearance of particles, and to high velocitygradients and, in general, to noise, generating less spu-

rious velocity vectors. As a consequence RIV resolvesbetter the vorticity peaks at the center of the vortex ringsgenerated by the pulsed jet, obtaining, for a giveninterrogation window size, a higher spatial resolution.Moreover, in the analysis of the flow field generated bythe backward-facing step, the RIV performs better in theshear layer at the border of the recirculation region,leading to a more reliable estimation of Reynolds shearstress and horizontal velocity component.

1 Introduction

All the methods of fluid velocity measurement which arebased on image analysis follow the same basic idea:

dispersing tracer particles in the working fluid, takingseries of images of the moving particles and analyzingthe images in order to educe velocity.

There are two different classes of procedures that

extract velocity from images: the first class includes themethods that firstly identify particles on each imageand then associate the particles of one image to theparticles of the successive image (acquired after areasonably short time interval). These methods areusually referred to as particle tracking velocimetries(PTV) (Dalziel 1992; Querzoli 1996; Virant and Dra-cos 1997). The second class includes the methodswhich try to associate and compare portions of oneimage [the so called interrogation windows (IW)], toportions of the successive image under the so calledbrightness constancy constraint (BCC), i.e. theassumption that particles move between one frame and

the other conserving their luminosity (Horn andShunck 1981; Corpetti et al. 2002). Most of thesemethods are referred to as particle image velocimetries(PIV) (Adrian 1991; Willert and Gharib 1991). Algo-rithms of the first class have typically a high spatialresolution as far as each velocity sample is computedfrom the motion of a single particle rather then fromthe whole IW (which normally includes a number ofparticles, at least on the order of 10). On the otherhand, the algorithms of the second class are less sen-sitive to noise and image quality since they make useof the raw images and do not require the individualidentification of each particle; anyway it should be

noticed that, in order to compare IWs successfully,one, or both of them, must be transformed accord-ingly to a model of motion of the fluid chosen a pri-ori. For example, in classical PIV, one compares IWsthat are translated one each other; implying theassumption that the fluid velocity is uniform over thewhole examined region.

Given a series of Nf images, Ii(x,y), i = 1,..,Nf, takenat constant time intervals, dt, the general structure of thealgorithms of the latter class can be summarized asfollows:

M. Falchi G. P. RomanoDipartimento di Meccanica e Aeronautica,Universita` La Sapienza, Rome, Italy

G. Querzoli (&)Dipartimento di Ingegneria del Territorio,Universita` di Cagliari, Cagliari, ItalyE-mail: [email protected]

Experiments in Fluids (2006) 41: 279293DOI 10.1007/s00348-006-0148-3


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Identification of an appropriate interrogation win-dow, Wi, in the i-th image of a given sequence (when across-correlation camera is used the sequence consistsonly of two frames);

Assumption of a model of motion, F, which is able toaccount for the motion of the fluid within the IWduring the time interval between frames:Wi+1 = F(Wi, p1,..., pN). By means of this functionthe BCC equation can be written as:

Ii1Fx;y;p1; :::;pN Iix;y; with x;y 2 Wi; 1

where p1,..., pN are parameters describing the motion.In its simplest formulation, PIV considers simpletranslation: F(x, y, u, v) = (x + udt, y + vdt), whereu and v are the two component of the fluid velocity inthe IW. In this case, the above equation becomes:

Ii1x u dt;y v dt

Iix;y; withx;y 2 Wi: 2

In the real world, a number of violations of Eq. 1 occur,therefore we can only try to satisfy it in a statistical

sense, over the whole IW. Therefore, we have to: Define a measure of dissimilarity between image re-

gions, dW (where the subscript recall that the dissim-ilarity is computed statistically over an interrogationwindow W);

Solve an optimization problem that consists in findingthe values of the parameters p1,..., pN which minimizethe measure of dissimilarity:

dWIi1Fx;y;p1;:::;pN;Iix;y;

thus, the optimal parameters describe the fluid motionwithin the IW.

The choice of the IW and of the motion model are deeplyinterconnected. On one hand, a large IW is needed tosufficiently constrain the solution of the minimizationproblem and provide some insensitivity to noise. On theother hand, the larger the IW, the lower the probability forthe model of motion to be valid over the whole IW.Unfortunately, trying to solve this dilemma by adopting amore complicated model (which would be able to representadequately the fluid motion over a larger IW) would notnecessarily improve the solution since that model woulddepend on a large number of parameters. As a conse-quence, for a given IW size, the reliability of the dissimi-larity evaluation would decrease. That is why most of

researches prefer the simple translational model and try touse the smallest possible IWs (Tomasi and Kanade 1991).

In classical PIV algorithms, the opposite of the cross-correlation function is chosen as dissimilarity measure(indeed, in PIV one usually thinks in terms of cross-cor-relation maximization instead of minimizing the dissimi-larity). Let Wdenote the average of I over the IW:

Ih iW 1

NW

Xx;y2W

I x;y ;

where NW is the number of pixels over the IW, and letI = I W indicate the fluctuation around thataverage, the dissimilarity is therefore defined:

dW Ii1 F x;y;p1; :::;pN ;Ii x;y

1

NW

Xx;y2W

I0i1 F x;y;p1; :::;pN I0i x;y

; 3

since, as mentioned, the motion model assumed inclassical PIV is purely translational, the function to beminimized results in the usual cross-correlation formula:

dW Ii1 x du;y dv ;Ii x;y

X

x;y2W

I0i1 x du;y dv I0i x;y : 4

Initially, the cross-correlation was preferred as dis-similarity measure since it is calculable at a low com-putational cost by means of the Fast Fourier Transform(Adrian 1991). Nowadays, the computing power is sohugely increased that computational cost is not as cru-cial as at the beginning of the development of these

algorithms. Therefore, it is reasonable to explore otherkinds of measures, which are more effective in thecomparison of the IWs, in order to achieve a higheraccuracy in the measurement of the velocity field. Thispoint is the main objective of the present paper. Obvi-ously, the improvement in the effectiveness have to becompared with the increase in the computational cost toevaluate the global performance and the actual useful-ness of an algorithm.

Thus, a new measure is proposed and its effectivenessin velocity evaluation is compared versus a cross-corre-lation based algorithm (i.e. PIV). Tests are performed onan artificial image and on real image sets obtained from

two experiments: a periodically pulsed jet through asharp edged orifice, driven by a piston, and a backward-facing step. The former flow field has been selected dueto its interest for practical applications (jet propulsion ofsubmerged vehicles, hemodynamics of heart and largeblood vessels) and for the distinct presence of largevortex structures. Nevertheless, the flow is basicallylaminar and repeatable. On the contrary, the latter flowwas chosen since it is fully turbulent and exhibits a largerecirculation region, delimited by a sharp shear layer. Inorder to avoid that different strategies, carried out bydifferent algorithms, would affect the final result, and topoint out specifically the effect of the dissimilarity

measure, a structured program was designed ad hoc, andonly the module computing the dissimilarity was chan-ged during the tests.

2 A robust dissimilarity measure

If one omits to consider the computational cost, perhapsthe most obvious way to evaluate how different are twoimages, over an interrogation window, is the Euclidean

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distance or, which is the same, the sum of squared dif-ferences (SSD) of their gray levels. It corresponds tochoose a quadratic estimator, q, of the differences, D,between pixel intensities:q(D) = D2, and sum over the IW:

dW A;B A x;y B x;y 2

D EW

; 5

where, A and B are the arrays describing the gray levelsover the two images. The sum has been divided by the

total number of pixels in the IW in order to keep anuniform notation and to make the value insensitive tothe size of the region.

The SSD is not so different from the cross-correla-tion, and they are exactly equivalent if the IWs arenormalized so that they have unit standard deviationand null mean. To verify this statement, let A indicatethe normalized image:

A00 A Ah iWrW A

; 6

where rW (A) denotes the standard deviation of A over

the region W. Equation (5), applied to the normalizedimages A and B gives:

dW A00;B00 2 1 A00 B00h iW

: 7

It is straightforward to recognize in the last term inbrackets on the right-hand side, the cross-correlationbetween the normalized IWs. Considering two successiveframes, A = Ii, B = Ii+1, and choosing the transla-tional motion model, this term assumes the sameexpression as the classical PIV formulation reported inEq. 4.

It is well known that the squared difference is theoptimal estimator as far as the errors have a Gaussian

distribution, but it is also very sensitive to the presenceof outliers (Hubert 1981). As the magnitude of the errorincreases, the contribution to the dissimilarity increaseswithout bound, due to second power elevation. Thissensitivity is described by the influence function, W(D)(Hampel et al. 1986), defined as the derivative of theestimator:

W D @q D

@D:

It characterizes the effect that the gray level differ-ence, D, in a particular pixel has on the global dissimi-larity measure. For the SSD measure, the estimator isquadratic; therefore W is linearly increasing with D, asshown in Fig. 1 (green lines). As a result, the presence ofa few outliers can meaningfully increase the dissimilaritymeasure, possibly distorting the measure relative to themajority of the pixels.

Outliers are always present in image velocimetries.Indeed, there are many reasons for the presence ofoutstanding values in the differences between pixels ofcorresponding IWs. Some of them are related to theillumination-recording system; for example:

Appearance and disappearance of particles whichenter or exit the IW, or get in or out of the illuminatedplane;

Electronic and digital noise, due to video-camera andto the image compression during storage on a per-manent memory;

Variation of particle luminosity due to non-uniformillumination, or to reciprocal positions of emitting andreceiving optics.

Some others are related to the fluid flow itself, such as:

Violations of the motion model due to large velocitygradients within the IW;

Presence of scale of the motion smaller than thethickness of the light sheet, so that particles appear tomove differently in the IW since they are at differentdepth.

From the above lists, it is easy to realize that it is verydifficult, if not impossible, to remove all the causes of thepresence of outliers; thus, an outlier insensitive measure

of the dissimilarity is required. To this aim the use of theLorentzian estimator is here proposed:

q D ln 1 D

re

2 !; 8

where re is an heuristically tuned parameter whichshould correspond to the expected standard deviation ofthe pixel differences. It is a robust estimator introducedin the field of the computer vision, by Black and An-andan (1991); to our knowledge, it was never used influid velocity measurements. As clearly shown by theblue curves of Fig. 1, it does not increase so steeply as

the quadratic estimator. The influence, W(D), of a sam-ple, D, is about the same of quadratic estimator for smalldifferences, but as D increases it begins to grow moreslowly and it is a maximum for D = re. As the differ-ence increases furtherly, the influence decreases andtends to vanish for very large differences. This behaviorcorresponds to give larger importance to the pixelswhich are more or less similar, rejecting pixels which aretoo much different. When inserted in the minimizationprocedure described above, it results in finding a solu-tion that is in accordance with the behavior of themajority of the pixels and that is little affected by theminority of pixels even if they are very different. As a

consequence, the solution will fit the pixels that likelysatisfy the brightness constancy constrain and the modelof motion (which together give Eq. 1), even if a minorityviolates Eq. 1 because of appearing or disappearing ofparticles or one of the other reasons mentioned above(Black and Anandan 1996).

The computational cost of the evaluation of thecross-correlation on an IW via FFT is of order costPIV = Nr log(Nr) Nc log(Nc) (where Nr and Ncare the number of rows and columns of the interro-gation window, respectively). Lets assume that in the

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investigated flow, the velocity in the rows and columnsdirection is expected to belong to the interval [umin; umax]and [vmin; vmax], respectively. When using SSD or therobust estimation, one can limit the computation of thedissimilarity to the displacements that correspond to theabove intervals. Therefore, the computational cost inthis case is of order costRIV = Nr Nc Dnr Dnc (whereDnr = (umax - umin) dt and Dnc = (vmax vmin) dt are,respectively, the ranges of expected displacements in therows and columns directions and dt is the time interval

between the two shots). The ratio between them can beassumed as an indication of the relative effectiveness ofthe two algorithms in terms of time consumption. Itresults:

k costPIV

costRIV

log Nr log Nc

DnrDnc:

If the expected displacement ranges are equal to thewindow size (Dnr $ Nr, Dnc $ Nc), PIV is always faster(costPIV < costRIV); anyway, as the expected displace-ments decrease, the above ratio increases. Therefore, therobust dissimilarity (or the SSD) can be more convenientfor large IWs and small displacement ranges. During the

tests performed on the data-sets described in this paper,the ratio between the CPU time used by PIV and the oneused by RIV ranged from 0.5 (for the smallest IWs) upto 1.3 (for the largest IWs). Therefore, whatever thechoice of the IW size, the use of the RIV do not imply aprohibitive increase in the computation time.

In order to test the effect of the different estimators, asynthetic image from the Particle Image VelocimetryStandard Project (Okamoto et al. 2000, image n.1) wasused (Fig. 2, left); it represents the simulation of a 2Dwall shear-flow. There is no meaningful noise in the

image and the particles do not enter nor exit the framedarea in the third direction, as a consequence it is a quitesimple flow condition. An IW, 31 31 pixels in size, waschosen (white rectangle on Fig. 2), and the dissimilaritymeasure, dW, was computed for x and y displacementsranging from 15 to 15 pixels. The resulting maps,normalized so that the measured values range from 0and 1, are plotted in Fig. 3. The Lorentzian estimator,the SSD and the correlation obtained by FourierTransform (indicated with Fast Correlation) have beencompared.

All maps exhibit a sharp peak in (10,2), meaning thatfor this displacements the IWs are more similar, thismeans that this is likely the displacement of the fluid(within the IW) during the time interval between the twoframes. The maps are characterized also by another zoneof low dissimilarity in (7,12) which should be consid-ered as noise during the process of detection of theminimum dissimilarity.

The level of this false peak is above 0.65 with therobust estimator, it decreases to 0.6 with the SSD and to0.55 with the Fast Correlation. The general level of the

noise within the map is in agreement with the trenddescribed above, confirming that the Lorentzian esti-mator gives the best signal to noise ratio (meaning thatthe signal is the true peak and the noise the false peaks itmust be distinguished from). Presumably, the SSD per-forms better than Fast Correlation also because it con-siders shifted windows of constant size for eachdisplacement, therefore the first window coincide everytime with the white rectangle while the second is shiftedof the guessed displacement. Conversely, the Fast Cor-relation uses the same IW on both frames under the(unrealistic) assumption of spatial periodicity.

The image chosen for the previous tests was modified

in order to verify the sensitivity of the different methodsto a meaningful violation of the BCC. A 7 7 pixelsquare in the center of the IW has been set to themaximum of luminosity level (255) in the first frame,whereas the second frame has been left unmodified(Fig. 2, right). The size of the modified area has beenchosen as a typical value since it corresponds to thesudden disappearance of two or three particles, or aninstantaneous reflection which saturates a part of thefirst frame interrogation window. The resulting dissim-ilarity maps are shown in Fig. 4.

The map obtained with the Lorentzian estimator doesnot change so much, with the second peak only 0.1 lower

than the one found with the original images. On thecontrary, the value of the second peak on the mapsobtained with SSD and fast correlation, becomes hardlydistinguishable from the main one. As a matter of fact, itdecreases at less than 0.25 both with SSD and fast cor-relation. In this case, the advantage of (and the need for)using a robust estimator is clearly demonstrated; it isbased on the behavior of the majority of pixels and doesnot take into account the minority even if they give verylarge differences. Thus, the measure is not influencedby violation of the basic assumptions on intensity

0 1 2 3 4 5 60

1

2

3

4

5

Estimator

0 1 2 3 4 5 60

0.2

0.4

0.6

0.8

1

/e

Influence function

Fig. 1 Quadratic (green) estimator, q(D) = (D/re)2 and Lorentzian

(blue) estimator, q(D) = ln(1+(D/re)2), plotted as functions of the

difference of pixel intensity, D, and normalized by the expected

standard deviation, re

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conservation by spurious pixels (outliers) between (e.g.the white square artificially added in the first frame).

The SSD and cross-correlation are functions definedunivocally. Conversely, the Lorentzian depends on aparameter, re, which tunes how robust the estimator hasto be. As a matter of fact, it should equal the amplitudeof the expected differences between pixels fulfilling theBCC. The above maps have been computed with a valueequal to 26, that is about one half of the standarddeviation of the image gray levels (r = 43 0.1 forboth images).

In order to test the sensitivity of the solution onthe parameter re, the dissimilarity map given by the

Lorentzian estimator was computed for six differentvalues ofre, ranging from 2 to 128; results are plotted in

Fig. 5. If one assumes the level of the second peak as anindication of the signal to noise ratio, one should con-clude that the values of 26 is not optimal, since thevalues from 3 up to 13 behaves slightly better, but theresults are more or less similar to those obtained with 26.Further increases of the value deteriorate the S/N ratio,but also for re = 128, the Lorentzian estimator worksnoticeably better than SSD or cross-correlation. Theseresults indicate that the Lorentzian estimator performswell for a wide range of values of the parameter, eventhough the optimal seems to be at about 1/3 of the

Fig. 2 Test imagessuperimposed: in redthe first, in

green the second frame (left).The white square indicates theInterrogation Window. On theright the same image with arectangular artificially saturatedarea on the first frame

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SSD

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Fast Correlation

Fig. 3 Maps of dissimilarity with Lorentzian, sum of squareddifferences and fast correlation. Dissimilarity are subtracted of theminimum value and normalized by the maximum value. Contoursare drawn from 0 to 1 with a step of 0.05. For reference, the levels

0.7, 0.6 and 0.5 are drawn in red, green, and blue, respectively. Thelowest the dissimilarity the darker the background. Abscissa andordinate represent guessed displacement in the x and y direction,respectively

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Fast CorrelationFig. 4 Dissimilarity maps onthe modified image. Samecontour lines and axes as in

Fig. 3

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standard deviation of the gray level of the image. Ingeneral, the optimal value depends on the probabilitydistribution of the differences, which in turn is influ-enced also by the particular images under analysis.Anyway, Fig. 5 shows that the results are nearly optimalprovided a value below one half of the standard devia-tion is chosen. As a matter of fact, if the value of re ischosen too large, the estimator tends to behave as theSSD, and its reliability does not decreases so much. As aconsequence the user should not have any particularskill or experience to obtain good results since a finetuning for each kind of image is not required.

3 The velocimetry algorithm

It is well known that the performance of an image ve-locimetry algorithm is the result of the contribution ofmany different factors; the dissimilarity measure is onlyone of them. However, the aim of the present paper isnot the research of the best algorithm in absolute, but itsscope is limited to the effectiveness of such a measure.Therefore, in performing tests on real images, all resultswere obtained by means of the same identical procedure,except for the module computing dW.

A detailed description of the algorithm is out of thescope of the present paper, anyway, its main elementsare sketched in Fig. 6:

Pyramidal filtering of the images; Velocity extraction; Rejection of spurious vectors; Image warping.

Usually, to deal with large displacements, PIV algo-rithms increase the IW size. The idea of the pyramidal

filtering is to obtain the same result at a lower compu-tational cost by reducing the image size. It consists ofapplying a low-pass filter (Gaussian with r = 0.5 pixelin our case) and sub-sampling one pixel each four (whichmeans the side-length of the image is halved) (Burt andAdelson 1983). The low-pass filter avoids aliasing duringsub-sampling. The filtering is repeated a number oftimes, generating a pyramid of smaller and smallerimages, until, at the upper level, the IW can capture thelargest motion.

Velocity extraction is performed by finding the min-imum of the dissimilarity map. Sub-pixel approximationis obtained by the classical 1D, Gaussian interpolation(Westerweel 1997).

Spurius vectors are identified and rejected by meansof an iterative comparison with a filtered version of thefield, obtained by a Gaussian weighted average. Sampleswhich are too different from the filtered field are elimi-nated. The procedure differs from the classical iterativefilters (e.g. Nogueira et al. 1997) since it reconsiders theoriginal samples at each iteration, thus permitting torecover the samples that were initially rejected.

The validated field from one level is used to warp (bybilinear interpolation) the images at the lower level. Inthis way, an affine transformation of the IW is auto-matically accounted for during the next velocity

extraction and only the residual motions have to becomputed. Therefore, the IW size can be small even ifthe absolute displacements are large (the procedure isequivalent to use IW deformation and offset).

Velocity extraction, validation and image warping areperformed from the upper level of the pyramid, down tothe base level, which furnishes the final velocity estima-tion. Hereafter, when the dissimilarity map is computedusing the Lorentzian estimator, the above describedalgorithm will be called robust image velocimetry (RIV).

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Fig. 5 Dissimilarity maps onthe modified image (right ofFig. 2). Lorentzian estimatorwith different values ofre(indicated on the top of eachmap). Same normalizationcontour lines, and axes asprevious figures

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4 Case #1: Pulsed jet

4.1 Experimental set-up

To test the algorithm, a pulsed water jet generated by anorifice on a thin, planar wall was firstly investigated; asketch of the experimental set-up is presented in Fig. 7.The plate generating the jet is placed inside a parallel-epipedal vessel, 110 40 40 cm3. On the upstream

side of the vessel, a smaller, cubic chamber (side-length:20 cm) hosts the water inlet and the piston-cylinder de-vice which drives the flow. After that chamber, there is a40 cm long space with an honeycomb at the beginning inorder to regularize the flow. At the end of that chamberthere is a plate with a 3 cm wide, sharp-edged, orificegenerating the jet. The jet develops in a 60 cm longchamber. At the downstream end of that chamber,

10 cm before the downstream side of the vessel, a secondplate with a large (20 cm in diameter), central orifice, letthe water out without breaking the symmetry ofthe flow. This system has been set-up within a Euro-pean project (SMART-PIV, IST 2002 37548, http://www.smart-piv.com/) aiming to develop advanced Im-age Analysis procedures for biomedical applications.

The piston is driven by a linear motor controlled by apersonal computer. The motion of the piston generatesthe flow: when it moves backwards the volume of thefirst chamber is increased and the fluid flows in from theinlet; when it moves forwards, the fluid flows throughthe orifice. Both water inlet and outlet (indicated witharrows in Fig. 7) are connected to a constant-head tankthrough two one-way valves in order to prevent back-flows as much as possible.

The time program of the piston can be arbitrarilyshaped. An exponential-like law of motion have beenconsidered during the present tests (Fig. 8). The curve isrepeated continuously and the measurements are done,in phase, when a periodic regime is reached. In the testdescribed below, the period, T, has been kept constant at

1.0 s. The measure of the amplitude of the curves is gi-ven by the so called stroke volume, that is the volumemoved by the piston during one cycle. The stroke vol-ume, SV, is related to the total piston run, L, by meansof its cross-sectional area, Sp, SV = LSp. In the presentexperiment the stroke volume was 70 ml. Since the pis-ton is not directly connected to the orifice, and due tothe dynamics of the whole hydraulic circuit, the actualflow-rate through the orifice does not correspond exactlyto the assigned motion of the piston. This fact is clearlyvisible in Fig. 9, that shows the nominal flow rate (upperplot), that is the velocity of the piston times its surfacearea, Sp, in comparison with the actual flow rate mea-

sured at the orifice (lower plot).

Images

PyramidalFilter

Velocity estraction

Spuriousvectorsrejection

Image Warping

at the nextlevel of the

pyramid

Starting fromthe upper level

of the pyramid

Results

Is thelowerlevel?

Dissimilaritymaps

Search minima

Sub-pixelinterpolation

of minima

YES NO

Fig. 6 Flow chart of the velocimetry algorithm

40 cm60 cm

40cm

Pisto

inletoutlet

n

20cm

Honeycomb

illuminated plane

video-camera

Fig. 7 Sketch of the apparatus generating the pulsed jet

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/L

Fig. 8 Time law of the motion of the piston. Position, Xp, of thepiston is non-dimensionalized by the total travel length, L. Thetime is normalized by the period, T

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A vertical plane, aligned with the axis of the orifice isilluminated by a pulsed, Nd-Yag laser, capable of pro-ducing couples of pulses, 100 mJ each, with a small,adjustable, time interval between them (350 ls in thepresent experiment). The flow was seeded with non-buoyant particles, 10 lm in diameter. A cross-correla-tion camera, 1,376 1,040 12 bit in resolution, wasplaced orthogonal to the illuminated plane and acquiredcouples of images at the trigger signal provided by thepersonal computer controlling the linear motor. As aconsequence, series of images at the same phase of thepulsed-jet cycle could be taken. Given the light intensity,

the seeding resulted dense and uniform as shown inFig. 10, where one of the acquired images is shown. Thehardware (laser, optics, camera, interface board andacquisition software was provided by LaVision Gmbh).

Further details on the experimental set-up and on theoverall flow field behavior can be found in Falchi andRomano (2005) and in Romano (2005).

4.2 Flow field overview

The periodically pulsed jet generates series of vortexrings. The formation, evolution and decay of vortexrings has been studied extensively in the past (Shariffand Leonard 1992; Pullin 1979; Fabris Liepmann 1997;Gharib et al. 1998; Zhao et al. 2000; Krueger andGharib 2003). Anyway, depending on the time law of theflow-rate, the number of vortices per cycle, their inten-sity and their interaction can change. Therefore anoverall description of the time evolution of the flow fieldin the present case is given in Fig. 11, where velocity andvorticity are plotted. For each phase, 50 velocity fieldshave been acquired in different cycles and the averagewas performed. Velocities have been measured by RIV,with an IW size of 15 pixels, a distance between themeasuring points of 8 pixels, and an expected standard

deviation re = 26, (i.e. about one half of the standarddeviation of the image intensity).

At the beginning of the cycle, three weak vortex ringsare present on the field, and are moving leftwards. Theywere generated at the end of the previous cycle.The leading two vortices are observed while interactingreciprocally in the so-called leap-frogging (t/T =0.000.15). At t/T = 0.10 a vorticity sheet rolls up fromthe edges of the orifice, forming the main vortex ring(Lim and Nickels 1995). It is stronger, and moves fasterthan the previous ones. At t/T = 0.15 it has incorpo-rated the third of the vortices generated at the previouscycle and the vorticity sheet, that surround the trailing

jet, becomes unstable, generating a series of small,aligned, vortices (Zhao et al. 2000). At t/T = 0.23 thepinch-off process takes place, while the main vortexreaches the previous ones and turn them into a vorticitysheet rotating around the main vortex ring. At the end ofthe ejection (t/T = 0.33), the main vortex, that movesrapidly, is separated by the series of vortices generatedby the trailing jet. During the remaining part of thecycle, a residual flux generates the weak vortices that willbe found at the beginning of the following cycle. Thatflux is likely to be due to a residual high pressure in thechamber between the piston and the plate with the ori-fice.

4.3 Results

As described in the previous section, the pulsed flowthrough the orifice generates a series of vortex ringswhich travel downstream across the measuring section.As a test case for the detailed validation of the RobustImage Velocimetry, the flow at phase t/T = 0.25 waschosen. At this phase, 550 couples of images have beentaken.

0 0.2 0.4 0.6 0.8 1

-2

0

2

4

6x 10

-4

Nominalflow

rate(m3/s)

0 0.2 0.4 0.6 0.8 1

-2

0

2

4

6x 10

-4

Flow

rateattheorifice(m3/s)

t/T

Fig. 9 Nominal and actual flow rates as functions of the timenormalized by the period, T

Fig. 10 One of the images acquired during the pulsed jetexperiment

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To evaluate the quality of the velocity data, fourquantities have been considered: the average velocityand vorticity, the variance of the vorticity and the sumof the variances of the two components of the velocity.The overall behavior of these quantities, at the investi-gated phase, is drawn in Fig. 12. The values of Fig. 12

were computed by RIV, with an IW of 15 pixels, a8 pixel step between IWs, and an expected variancere = 26 (about one half of the standard deviation of theimages).

As in the plots presented in the previous section, thevorticity map shows clearly the well developed vortex

Fig. 11 Vorticity (color-map) and velocity (vectors) field at six instants. In the lower left corner the flow-rate during the cycle is plotted; thered circle indicates the point within the cycle. Vorticity is non-dimensionalized by the period T

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ring, with a trailing jet, surrounded by a series of nearly

equispaced, small, vortex rings. This velocity field, withlarge coherent structures and intense vortices, is ideal toevaluate the effectiveness of the RIV. The flow, at thisstage of evolution, is basically laminar, therefore thevariance both of the velocity and vorticity are due,partly to the small variability of the phenomenon fromone cycle to the other, but mostly to the spurious vectorsin the measured velocity field. Both contributions tendto increase the variances in the regions of high spatialgradients: therefore mainly at the center of the vortices.

The values of the above mentioned quantities havebeen obtained both by fast correlation (PIV) and robustestimation (RIV), for IW sizes of 11, 16, 32 pixels (PIV)

and 11, 15, 32 pixels (RIV), with a grid-step of 8 pixelsexcept for the case of IW size of 32 pixels, which wascomputed with a 16 pixel, grid-step. Anyway, theamount of overlapping should not affect the results sinceit influences mainly the identification of the spuriousvectors, but in these tests the tolerance of validation hasbeen intentionally kept large enough to keep the per-centage of intervention very low, with the aim to avoidthat the post-processing could mask the effects of a lessefficient dissimilarity measure. When the window sizewas not a power of two the correlation was computed by

the discrete Fourier transform instead of FFT. The ex-

pected standard deviation of the Lorentzian estimatorwas always set to 10. For a quantitative comparison,values along the segment drawn in top-left plot ofFig. 12, have been plotted in Figs. 13, 14, and 15.

In Fig. 13, the distribution of vorticity of the lower,main vortex is shown as obtained from the differentcomputations. Two main effects are apparent:

Fig. 12 Maps of vorticity (top-left), variance of the vorticity (top-right) and mean square velocity fluctuation (bottom). Lengths aremeasured in pixels and times in intervals between frames

0 50 100 150 200 250 300 350 400 450 500

0

0.1

0.2

0.3

0.4

pixels

RIV 11

RIV 15

RIV 32

PIV 11

PIV 16

PIV 32

Fig. 13 Vorticity profile along the black line drawn in the top-leftplot of Fig. 12. The legend indicates the method used and the IWsize. Same units as in Fig. 12

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interval between the couples was 4 ms. In this way, themeasured velocity field were not too much correlatedone each other.

5.2 Flow field overview

During the experiment the flow was almost steady,therefore, time averages will be presented. For the dis-cussion, the quantities more sensitive to the presence ofuncorrelated noise have been chosen. In particular themean horizontal velocity and the Reynolds shear stress.Their maps, in the region downstream the step, areplotted in Fig. 17. These data have been obtained usingRIV with an IW size of 15 pixels and an expectedstandard deviation re = 8. Variances have been notconsidered since they are affected by two concurrenteffects: on one hand, increasing the sensitivity and

resolution of the measurements will increase the vari-ances, since the small scale fluctuations and suddenvelocity changes will be followed better; on the otherhand, a higher level of spurious vectors due to a worsequality of the measurement gives the same result, and itis difficult to discern between the two effects.

The general structure of the flow is clearly seen in theplot showing the mean vertical velocity. In the lowerpart of the flow there is a zone of negative velocitycorresponding to the recirculation region. Above thestep (x/Hs = 0), the horizontal velocity is a maximuminitially in the centerline. Downstreams, the maximumtends to move downwards as the flow expands. Someimage border effects are observed at the right-end of theplot, where the velocity decreases artificially as a con-sequence of the loose of the particles going out of theframed area between the two snapshots of a couple. Thedistribution of the Reynolds shear stress exhibits highpositive levels in the region where the interface betweenthe free flow and the recirculation becomes unstable andthe KelvinHelmholtz structures grow while being ad-vected. The high Reynolds stress region starts, near the

step, as a thin layer and grows downwards until itreaches the lower wall. The darker blue zone in the up-per part of the flow corresponds to slightly negativeReynolds shear stresses.

5.3 Results

A detailed comparison between cross-correlation androbust estimation has been done on a vertical cross-section at x/Hs = 1.0. That section was chosen since it is

Fig. 16 One of the images acquired during the backward-facingstep experiment

Fig. 17 Fluid flow downstreamthe step. a Mean horizontalvelocity; b mean Reynolds shearstresses. Values are made non-dimensional by the centerlinevelocity Uc, and step height Hs

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relatively close to the step, and the shear layer, betweenthe upper, free flow and the recirculation region, is stillthin and sharp. As a consequence, the resolution and theaccuracy of the measurement technique results crucialfor the successful evaluation of the flow statistics.

Figure 18 shows the vertical profile of the meanhorizontal velocity obtained using both PIV and RIV,for different IW sizes. Data are compared with the re-sults on a similar experiment performed by Kasagi et al.(1993). The flow was at Re = 5540, and the velocity wasmeasured by PTV.

Robust image velocimetry results are in very goodagreement with PTV for all the window sizes. Twomeaningful differences are noticed. Firstly, near theupper wall, for y/Hs > 2.5, RIV data coincide betweenthem, but are slightly low in comparison with PTV. As amatter of fact, the profile obtained by Kasagi et al. has asudden, little increase in that zone, whereas RIV datacontinue smoothly their decreasing trend toward thewall. Therefore, that discrepancy could be ascribed tosome difference in the incoming flow. Secondly, theprofile computed with a 32 pixel IW seems to underes-

timate the velocity in the recirculation region(y/Hs < 1.0). That difference is observed also in the PIVdata obtained with the same window size, as a conse-quence, it is likely to be an effect of the lack of resolutiondue to the window size.

Particle image velocimetry profiles behave more orless as RIV ones in the recirculation region, but aresensitive to the IW size in the upper region of free flow(y/Hs > 1.0). The mean horizontal velocity at y/Hs = 2.0 is underestimated up to 25% with a IW sizeof 7 pixels. The difference decreases for increasingwindow size, but is perceptible also with an IW of32 pixels. Responsible for the lower values is the

fraction of spurious vectors (higher with smaller IWs):as far as they are uncorrelated and, presumably, withzero mean, they tend to decrease the value of theglobal average.

The above interpretation is confirmed by the analysisof the profiles of Reynolds shear stress plotted inFig. 19. PIV results are very noisy, especially for smallIW sizes, suggesting an elevated level of spurious vec-tors. Only for an IW of 32 pixels the profile exhibits aclear trend without too much fluctuations. Unfortu-nately, that window size determines an excessivesmoothing. As a consequence, the sharp peak aty/Hs = 1.0 is almost completely missed: the measuredpeak values are, one third of the PTV data and aboutone half of the values obtained by robust estimation.This is a typical limitation of PIV-like measurements:one would need small windows to achieve the requiredresolution, but as far IW size decreases the quality of themeasurements decreases dramatically, therefore theoverall results do not improve correspondingly. Itshould be noticed that a good validation algorithmcould solve that problem, but in this case, the validationwas intentionally let loose, in order to point out thedifferences due to the dissimilarity measure.

The level of noise in the RIV data is much lower,some fluctuations are seen only with the IW of 7 pixels

and the peak is always well distinguishable. In addition,though at 32 pixels the peak Reynolds shear-stresses arelargely underestimated, the other three profiles seem totend to a well defined limit value. IW sizes of 11 and7 pixels give nearly the same value, whereas the 15 pixelswindow gives a value that differs only of 10%. The limitvalue is about 70% of the peak value furnished by thePTV, but, anyway twice than the best obtained by PIV,thus confirming what suggested by the analysis of thepulsed jet results: RIV is more robust to noise andachieves a better resolution in the measurements.

6 Conclusions

The observation that a number of violations of thehypotheses of the image velocimetry based on window

-0.5 0 0.5 10

0.5

1

1.5

2

2.5

3

/Uc

y/Hs

Kasagi et al.PIV07PIV11PIV15

PIV32

-0.5 0 0.5 10

0.5

1

1.5

2

2.5

3

/Uc

y/Hs

Kasagi et al.RIV07RIV11RIV15

RIV32

Fig. 18 Vertical profiles of non-dimensional horizontal velocityat x/Hs = 1.00. On the leftresults from PIV, on the rightresults from RIV. The numbersin the legend indicate the IWsize. Black dots represent PTVdata after Kasagi et al. (1993)

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comparison occurs in the real acquisitions, suggested theidea that the robust statistics could be successfully ap-plied to the evaluation of the velocity fields. As a con-

sequence, the RIV has been proposed. It is based on aLorentzian estimator, instead of the classical cross-cor-relation used in PIV.

The term robust indicates that, in comparison tothose of cross-correlation based algorithms, the resultsare much less dependent on differences between imagepairs, due to the so-called outliers (i.e. on almost indi-vidual incoherence between corresponding IWs causedby electronic, digital or optical noise, incoming andoutcoming tracer particles, large velocity gradients, orhigh depth of focus).

The differences among classical PIV based on cross-correlation evaluation and the proposed RIV were tested

on synthetic images and on two experiments: a pulsed jetand a backward-facing step.

The choice of the pulsed jet flow allowed to comparethe detection of expected high velocity and vorticity peaksin presence of low variances (the flow field is laminar forthe large part of the cycle) made by the different algo-rithms, whereas the backward-facing step experimentpermitted the comparison in a case of turbulent flow, witha higher background noise level in the images.

To perform this comparison in a right way, the pre-processing, pyramidal search, image warping and vali-dation sections of the processing software are exactly thesame; only the section containing the algorithm for

displacement evaluation was changed.Results on the pulsed jet indicate that a clear

advantage (up to 30%) is obtained in vorticity peakextraction when RIV is used instead of PIV; this increaseis a maximum when small interrogation windows areused (lower than 16 pixels). A large improvement (up to100%) is obtained also in the reduction of variances ofvorticity and velocity.

Analysis on the backward-facing step data indicatesthat, also with higher noise and turbulence levels, RIVperforms better than PIV both in mean velocity and

Reynolds shear-stress evaluation. In particular, theimprovement stems from the smaller window size that itis possible to use with the robust velocimetry without

increasing the fraction of failed measures.Both experiments confirm that the proposed algo-

rithm is less sensitive to noise in the acquired images andto violation of the BCC and/or to the model of motion.As a consequence, it generates less spurious vectors. It isimportant to point out that this is not obtained by aquestionable velocity vector post-processing, rather bymeans of a robust approach in analyzing and comparingsequence of images that was introduced in the field ofthe computer vision.

Acknowledgements The authors wish to thank the support bySmart-PIV IST 2002 37548 European Project and in particular

LaVision Gmbh for the hardware provided.

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