a wavelet characterization of high-resolution ndvi patterns for precision agriculture

8/12/2019 A Wavelet Characterization of High-resolution NDVI Patterns for Precision Agriculture

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JAG Volume 3 - Issue 2 - 2001

A wavelet characterization of high-resolution NDVIpatterns for precision agriculture

Virginie Epinatl, Alfred Stein 2, Steven M de Jong3 and Johan Bournal

1 Laboratory of Soil Science and Geology, Wageningen University, Wageningen, The Netherlands

2 Wageningen University, Mathematical and Statistical Methods group, Wageningen, The Netherlands and International Institute forAerospace Surveys and Earth Sciences, PO Box 6, 7500 AA Enschede, The Netherlands (e-mail: [email protected])

3 Wageningen University, Laboratory for geoinformation and remote sensing, Wageningen, The Netherlands

KEYWORDS: Wavelets, NDVI, Precision Agriculture,

Spatial Statistics, Remote Sensing

ABSTRACT

This paper presents a quantitative analysis of patterns visible inhigh-resolution NDVI images obtained f rom airborne remote sens-ing. Attention focuses on the use of wavelets to distinguish pat-

terns of interest for precision agriculture at several scales. A generalprocedure for analyzing these images is presented and applied to asingle field in the Netherlands, monitored at four different days

during one growing season. Wavelet decomposition of the imagesis capable to reveal and quantify patterns present at different reso-

lution levels and directions and to filter information that is less rele-vant for precision agriculture applications. Wavelet approximationwith different wavelet functions is useful within the backward-look-ing and the forward-looking approaches of decision-making byallowing adaptation of the analysis to the characteristics of the

available images or maps and to the possibilities of the existing site-specific instruments, respectively.

INTRODUCTIONIn current practices of precision agriculture, managementactivities are guided by changes in crop and soil condi-tions that likely vary within the field. For instance, mod-ern technological equipment allows changing applicationrates of agro-chemicals from place to place. In decision-making for precision agriculture, we commonly distin-

guish a backward-looking approach that aims at explain-ing differences which occurred during a growing season,from a forward-looking approach that aims at using thedifferences within the current growing season to supportthe farmer in decision making [Bouma et al, 19991.Within both approaches, an increasing need exists toquantify patterns of crop and soil conditions during thegrowing season.

Remotely sensed images may be useful for this purpose,as geometric and spectral resolutions are being refinedand the number of spectral bands is expected to further

increase in the years to come [Goetz et al, 19851. In par-

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titular, multispectral high-resolution remote sensing maygreatly contribute to precision farming by its possibilityto assess leaf area development and crop cover at fieldscale during the growing season [Clevers, 19971. Usingmodern agricultural simulation models, one would then

be able to translate these into management recommen-dations for the farmer, who may in turn apply an amountof nutrients and pesticides better suited to the conditionsat the specific locations. Analysis of crop and soil condi-tion patterns derived from remote sensing hence aims toimprove decision making for precision agriculture.

As current practices and modern technological equip-ment hardly allow for continuous changes in manage-ment, identification and quantification of soil and cropcondition patterns is becoming of increasing importance.The use of remote sensing for precision agriculture

requires a quantitative assessment of crop variables suchas leaf area index (LAI) or cover [Moran et a/, 19971,characterization of the patterns, and in a later stage theirintegration over a period of time and their comparison.Patterns are here defined as the non-random distributionand arrangement of low and high values. In previousstudies, fractals have been used to quantify vegetationpatterns on remote sensing images [de Jong & Burrough,19951. Also, geostatistical methods have successfullybeen used to quantify and compare spatial variability ofproperties of importance for precision agriculture[Bouma & Verhagen, 19981. In particular, Stein et a/

[I9971 used the cross-correlation function to comparespatial variability patterns of millet yield and soil data.Wavelets are a promising tool to quantify patterns byallowing pattern identification at various resolution levelsand directions.

Wavelets have been applied in the past to analyze images[Mallat, 19891. They have found many applications inremote sensing, such as removing speckle noise fromradar images [Horgan, 19981, merging high spectral res-olution images with high spatial resolution images[Yocky, 1996; Zhou et a/, 19981, and texture analysis and

classification [Zhu & Yang, 19981. Here we use wavelets


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Wavelet characterization of high-resolution NDVI patterns JAG Volume 3 - Issue 2 - 2001

to analyze remote sensing images useful for precision

agriculture as they allow to decompose the image intodifferent levels of resolution. The information at one par-ticular level may be important to understand the nature

of the pattern and to adjust agricultural equipment to

observed features, whereas wavelets may filter out dis-

turbing artifacts at other levels.

The objective of this study is to quantify spatial patterns

of a set of four NDVI images derived from remote sens-ing at the field scale during a single growing season. In

particular, we explore use of wavelets to detect main fea-tures in the patterns of a winter wheat field in the

Netherlands during the 1997 growing season. Moreover,

sensitivity for the type of a particular wavelet function isanalyzed. Finally, we discuss how to include quantifiedpatterns in decision making for precision agriculture.

MATERIALS AND METHODS

STUDY AREAResearch was conducted on a 100 ha-commercial farm

situated in Voorne Putten, a former island in the south-

west of the Netherlands. A single winter wheat field of

14.7 ha (Figure 1) was selected for the study based on itssize, the variability of soil characteristics and availabilityof detailed yield data. This field has been described pre-

viously by van Alphen & Stoorvogel [1998]. Soils, origi-

nating from marine deposits, are classified as Typic

Fluvaquents [Soil Survey Staff, 19941. Their texture variesfrom fine loam to heavy-clay loam. Organic matter con-tent in the upper layer (25-30 cm) varies between 2 per-

cent and 10 percent. Winter wheat was sown onNovember 16, 1996, and harvested between August 9and August 14, 1997. The growing season was charac-

terized by wet climatic conditions.

Within the field, nine points were located covering vari-

ous soil conditions for detailed measurements (Figure 1).At these locations crop characteristics were measured on

2m cl 200 4mm

FIGURE1: The study area and the sampling points.

May 30 and July 11. Crop density was estimated bycounting plants in a 1 m fraction of several rows nearby

the points. Dry matter weight and leaf area were mea-

sured for a sample of 10 plants randomly selected with-in a vicinity of 1 m from these locations. From this, crop

biomass per surface unit and leaf area index (LAI) wereestimated by correcting by the plant density.

Final local yields and corresponding point locations were

determined with a grain mass flow sensor and a GPSmounted on a combine harvester. Measurements were

made on 3 different dates: August 9, 13 and 14

because of the unequal development of the crop within

the field.

REMOTE SENSING

Remote sensing observations were made with the Dutch

multispectral airborne imaging scanner CAESAR (Charge-

Coupled Device Airborne Experimental Scanner for

Applications in Remote sensing) [Pouwels, 19871 at analtitude of 3 km. Reflectance was measured in the green

(550*15 nm), the red (670*15 nm) and the near-infrared

(870*25 nm) spectral bands on four clear days in 1997:April 1, May 30, July 11 and August 7. Radiometric cor-rection was done using the known spectral characteris-

tics of panels laid on the ground. Pre-processing resulted

in four reflectance images with a spatial resolution of0.75 m x 0.75 m and a geometric accuracy between0.49 m and 1 . I 1 m. Reflectance in the spectral bands

670*10 nm and 870*10 nm was also measured in the

field using a hand-held radiometer on May 30 and July

11. This instrument measures simultaneously incidentand reflected radiation 2 m above ground using sensorsdirected vertically upwards and downwards, respectively,

for each spectral band. Its field of view has a diameter of

approximately 1 m. Replicate measurements were made

at 5 locations in a radius of 2 m around the nine points.Calibration was done by measuring incident radiation ineach spectral band with both sensors.

Plant cover patterns were approached by computing theNormalized Difference Vegetation Index (NDVI) defined

as:

NDVI = (NIR - R)/(NZR + R) (1)

where NIR is the reflectance in the spectral band cen-tered at 870 nm, and R the reflectance in the spectralband centered at 670 nm.

WAVELETS

We consider a digital image of a size M x N pixels in thehorizontal and vertical direction respectively. This imageis denoted by F&y), with the same spatial resolution rin both directions. In a wavelet analysis, a two-dimen-sional image is approximated by a linear combination of

two-dimensional wavelet functions [Bruce & Gao, 19961.

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Waveiet characterization of high-resolution NDVI patterns JAG Volume 3 - Issue 2 - 2001

These wavelet functions are the simple product of one-dimensional wavelet functions, one in the horizontal X-direction and one in the vertical Y-direction. For eachdirection two one-dimensional wavelet functions exist: awavelet function $(.) is used to represent smooth (lowfrequency) parts of signals and a wavelet function cp(.)their detailed (high frequency) parts. Consequently, four

possible two-dimensional wavelet functions emerge,denoted by Q x,y), cph x,y>, rpv x,y) and @(x,Y), respec-tively:

@(XPY)= 4W44Y)

@(su) = 4WrpM

P(X>Y) = cP(4.44Y)(2)

rpd(%Y) = rp(~).cpW

where the superscript h denotes that rph(~,y) identifiesfeatures in the horizontal direction by capturing variationin the vertical direction by using r&). Similarly, rpv(~,y)

identifies features in the vertical direction by using rp(x)and cpd(x,Y) in the diagonal direction. The function$(x,y) represents the smooth and low frequency part ofthe image. In this study, we used both the Haar wavelet,and the symmlet s8 constructed by Daubechies, present-

b(x) 4(Y) &I P(Y)

a b

e

ed in Figure 2 [Daubechies, 19921. The Haar waveletfunction 9(x) equals 1 for x E [O,l], and is 0 elsewhere,the function q(x) equals 1 for x E [0,1/z], -1 for x E[t/2,1], and 0 elsewhere. They have a compact support(zero outside a finite interval), are orthogonal and sym-metric, but not continuous. Figures 2a-d show how thetwo-dimensional wavelet functions emerge from the one

dimensional ones. Symmlet wavelets are defined in asimilar way [Daubechies, 19921, are also orthogonal buthave a compact support that is larger than that of theHaar wavelet, are continuous and nearly symmetric.Construction of the two-dimensional wavelet functionsfrom the one-dimensional ones is illustrated in FiguresZe-h.

Next, a first choice for the level of detail, indicated by thenumber of resolution levels J is to be made in advance.By definition, the resolution level of the original imageequals 0, in this case corresponding to a spatial resolu-

tion r. Therefore, a resolution level j corresponds to aresolution of 2i.r. Then wavelet basis functions aredefined with respect to a scaling factor 2-j forj = I,..., Jand a translation vector (m,n) into the (x,y) direction.They are generated from the wavelet functions:

d

FIGURE : Haar fky) (a), hfx,yj (b), $x,fi (c), %,yJ (d) and symmlet fk y) e), Q,y) (f), jv(x,y) (g), d(x,y) (h) basis wavelet functions.

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(3)

where di; is the direction (h, v or d). The wavelet approx-

imation F(x,y) of the image F(x,y) then equals the sumof two-dimensional wavelet functions at different scalesand locations:

where Sj,m,n and d, & are the wavelet transform coeffi-cients. Their absolute value can be considered as a mea-sure of the contribution of the corresponding waveletbasis function to the image approximation. CoefficientsSj,m,n approximate the smooth part of the image at levelj and difm,n, d&,and djfm,n represent at that level devi-

ations from it into the different directions. Wavelet func-tions are applied at a broad range of scales (from 1 to J)at a large number of pixels (for each combination ofand n) and into 3 directions. Notice that coefficients sonly occur at level J and that smooth variation at lowerlevels j is captured by coefficients d at level j+Z. Awavelet approximation includes the smooth function onlyat level Jand the detailed wavelet functions at all levels.The coefficients Sj,m,n are given by:

(5)

A similar expression applies to the coefficients d &. Inthe practice of image analysis, integrals are replaced byfinite summations. The coefficients are grouped into so-called crystals, also referred to as subbands [Daubechies,19921, according to resolution level j and, for coeffi-cients d, direction. Contribution of the separate crystalsto the reconstructed image is measured by the percent-age of energy by crystal, defined as the sum of squaresof the coefficients of that crystal, divided by the sum ofsquares of the pixel values of the whole image.Anisotropic autocorrelation of the coefficients into dif-ferent directions reveals the presence of spatial structure

at various scales.

The two-dimensional wavelet approximation (Equation 4)can also be expressed as the sum of 3*J+l image com-ponents corresponding to different resolution levels anddifferent directions:

(6)

where Sj(X,y) is the smooth image and ~,?(x,y) aredetail images at resolution level j, showing object edgesin the horizontal, the vertical and the diagonal direction.

Sj(x,y) is a smooth approximation of the image at the

coarsest resolution level J. The smooth approximation ofan image at a resolution level j-1 can be derived fromthe sum of the smooth and detail images at resolutionlevel j:

s,_,x, Y) = S/(X?Y) + 107 (x7 >dir

The smooth images S(x,y), j =l , .,J, are multi-resolutionapproximations of the image.

COMPARING WAVELET COEFFICIENTSThe correlation coefficient was used as a simple

approach to compare smooth wavelet transform coeffi-cients at a given resolution level. The coefficient was esti-

mated using a 5x 5 pixel window shifted over the twocrystals, where one pixel represents a wavelet coefficientat position (m,n), and assigned to the central pixel.

RESULTSNDVI IMAGESFigure 3 presents four rotated NDVI images derived fromthe CAESAR observations and Table 1 gives theirdescriptive statistics. Figure 4 represents the develop-ment stage of the crop at the image acquisition dates.Values of 0, 1 and 2 indicate emergence, flowering andmaturity of the crop, respectively. The first two imageswere taken during the vegetative phase while the lasttwo images correspond to the reproductive phase.Between the second and the third observations, a dis-ease caused by a fungus (fusarium) broke out and lodg-

ing occurred. As can be seen in Figure 3, all imagesshow management features, like tracks in the 21 direc-tion with the EW direction, old field boundaries and anet of former canals perpendicular to these. On theimage of April 1 (NDVI=0.33*0.074), three main regionscan be distinguished. Region A located at the upper partof the field presents higher NDVI values, region B at thelower part shows lower values, and region C at thelower left part of the field has again higher values(Figure 1). The boundary between regions A and B cor-responds to the border of former parcels with differentland uses: grassland on A and cropland on B. On May 30

and July 11 NDVI values are very high (0.983kO.026) and(0.920*0.020), respectively, indicating that the soil is inmost places fully covered by the crop. Regions A and Bcan still be distinguished, but region C is now absent.The image on July 11 shows a pattern opposite to thepattern on April 1, with slightly different regions. RegionA presents lower values especially at the left of the field,region B higher values and region C shows again lowervalues. The lower values in region A can be explained bythe breakout of fusarium and lodging in that part of thefield. Two parallel black lines correspond to missingspectral values. On August 7, just before harvesting,

NDVI values have reduced to 0.459kO.074, indicating

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I ~~~

0 2;0 400 600 800a

090 092 094 096 098 100

,I /

,

0 200 400 600 Emc

I, I-

O 200 400 600 800

b

TABLE 1: Summary statistics of Caesar NDVI on April 1, May 30,July 11, August 7 (n = 261238)

April 7 May 3 July 11 Augus t 7

Minimum 0.029 0.480 0.553 0.2521st quartile 0.269 0.979 0.912 0.404Median 0.323 0.989 0.925 0.460Mean 0.327 0.983 0.920 0.4593rd quartile 0.373 0.996 0.933 0.510Maximum 0.763 1 .ooo 0.999 1 .oooStandard deviation 0.074 0.026 0.020 0.074

that discoloring of the crop has started. The three samezones can be distinguished with higher values at theupper part, lower at the lower part, and lowest at thelower left part of the field.

COMPARISON BETWEEN HAND-HELD RADIOMETER MEA-

SUREMENTS AND CROP CHARACTERISTICS

Summary statistics of the crop characteristics presentedin Table 2 show that the average plant density almost didnot vary between May 30 and July 11. Above-groundbiomass increased during this period, mainly due to thedevelopment of the ears. However, the average greenLAI decreased, due to the discoloring of the crop that is

FIGURE 3: NDVI images on April 1 (a), May 30 (b), July 11 (c), August 7(d).

0

d

200 400 600 800

I------l I)-

FIGURE 4: Development stage of the crop at the image acquisi-tion dates (represented by arrows)

on July 11 in its ripening phase (Figure 4). Table 3 showsthat NDVI derived from hand-held radiometer measure-ments is significantly correlated with plant density at the0.05 level on May 30, and at the 0.1 level on July 11. Noclear relation between this vegetation index and the LAIor above-ground biomass can be found, probablybecause a complete soil cover by the crop is reached inmost places, and because of the high LAI values (Table

2). As a result, the NDVI curve tends then to saturate and

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TABLE 2: Summary statistics of crop characteristics and hand-held radiometer on May 30 and July 11 n = 9)

Minimum Mean Maximum Standarddeviation

NDVI 0.723 0.743 0.757 0.011Plant densityplantslmz) 152 186 213 20

May 30Green LAI 3.72 6.25 9.54 1.92

Above-groundbiomass g/mz) 559 785 1079 155

NDVI 0.666 0.679 0.697 0.009

Plant densityplants/mz) 144 183 212 25

July 11Green LAI 2.75 4.14 6.34 1.07Above-groundbiomass (g/m2) 1298 1771 2261 259

/ / I I I0 200 400 600 800

FIGURE5: Yield map (t/ha)

TABLE 3: Correlation between NDVI derived from hand-held radiometer and crop characteristics on May 30 andJuly 11

Plant density

Green LAI

Above-ground biomass

May 300.690**

0.106

-0.204

July 110.646*

0.533

0.241

signific ant at the 0.1 level ; ** signific ant at the 0.05 level.

the area where the disease and lodging occurred, andhigher yields in the southern part of the field. Local lower

values at the beginning and end of the transects are most

probably errors due to flow processes within the com-bine. Other outliers might be caused by errors related tophysical properties of the combine and the yield mea-

surement system, to in-field differences such as crop

moisture content, and to the operator such as speed

changes and varying cutting width [Thylen et al, 19971.

NDVI variations within the field tend to be small.Discrepancy in correlations between NDVI and LAI on one

side and between NDVI and above-ground biomass on

the other side can be explained by the fact that theabove-ground biomass includes non-green material, ie,

dead leaves, stems and ears, scattered over several layersthroughout the vertical plant profile.

WAVELET ANALYSIS

Summary statistics of the NDVI derived from the hand-held radiometer observations at the 9 points on May 30

and July 11 (Table 2) show smaller means and standard

deviations than those derived from the CAESAR images.Possible explanations are the limited number of points atwhich reflectance using the hand-held radiometer was

measured, the differences in bandwidth of the observa-

tions in the red and near-infrared wavelengths, in radio-metric corrections and in spatial resolution according to

the remote sensing device. We assumed that relationsbetween hand-held radiometer NDVI and crop character-

istics are also valid for CAESAR NDVI images.

For the wavelet analysis we rotated the images over anangle of 21, thus setting horizontally the main manage-

ment direction, that is the direction of dominant butirrelevant tracks. Pixels outside the field were assigned

the average NDVI value to avoid bias. Six resolution lev-els were chosen, which give a coarsest resolution of 48

m on the ground. In this study, wavelet transform coeffi-cients and multi-resolution approximations were comput-

ed with the symmlet wavelet s8 for all NDVI images, aswell as with the Haar wavelet for the image of July 11, In

the first case, errors in image reconstruction and artifacts

at its boundaries were reduced by reflecting the originalseries at the boundaries and periodically extending them.In the second case, extension of the image was done by

assigning zeros to the wavelet transform coefficients at

the boundaries of the image. We now first focus on the

image of July 11, being the image with the clearestobservable pattern.

July 11

YIELD MAP

Figure 5 presents a rotated yield map derived from theoriginal 12,449 point data with the nearest neighbor

method applied to a 0.75 m resolution grid. Yield valuesin the original data set vary between 0.1 t/ha and 22 t/ha,with an average of 8.48 t/ha and a standard deviation of1.93 t/ha. The distance of the measurement points totheir nearest neighbor is on average 2.11 m and varies

between 0 m and 3.92 m. The map shows lower yields at

We notice for the wavelet transform coefficients result-

ing of a decomposition with the s8 wavelet that:

a. Horizontal crystals mainly depict the tracks, especiallyat resolution levels between 2 and 4, as well as thearea with the disease and lodging in region A. Whenmoving from level 4 to level 5 the regions becomemore pronounced, while tracks disappear, as the scale

of resolution becomes coarser (Figure 6).

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Wavelet characterization of high-resolution NDVI patterns

b.Vertical crystals mainly reproduce the disease zone inregion A at all resolution levels, as well as the two linesat levels 1 to 3 and the boundary of older parcels atthe finest levels.

c. Diagonal crystals capture in general less variation thanhorizontal and vertical crystals. Higher coefficients arelocated at region A. Old boundaries can be distin-guished at resolution level 2.

Pattern features that are of minor importance for preci-sion farming are therefore most prominent at the finestlevels, and mainly into horizontal and vertical directions.The interesting change in resolution occurs between lev-els 3 and 4 where the pattern of the main regions is nowmore prominently being quantified. We will now focuson levels 3 and 4.

Wavelet transform coefficients and autocorrelations areplotted in Figure 7 for level 3 and in Figure 8 for level 4,corresponding to 6 m and 12 m on the ground, respec-

tively. At level 3 we notice that horizontal crystals pre-sent clear spatial correlation in the horizontal direction,and have a periodicity of approximately 75 m in the ver-tical direction, mainly due to presence of tracks. The ver-tical crystal shows a pattern with strong correlations inthe vertical direction, and alternating correlations in thehorizontal direction. This is probably caused by the twolines and the field boundary. The remainder of the pat-tern is composed of higher coefficients in region A withvariation that is mainly noise, as represented by analmost flat correlation function. The diagonal crystalshows mainly noise, with some pattern visible in region


A, leading to an almost flat autocorrelation function.

At level 4, the vertical crystal now shows some morestructure, with clear dark areas at the top left, bottom leftand right parts of the crystal (Figure 8). This more promi-nent pattern type is present as well in an autocorrelationfunction that shows more structure than at level 3. Onthe other hand, the diagonal crystal shows some struc-tured variation, that, however, is not represented by theautocorrelation function, that mainly remains flat, butalternates in the horizontal direction for small distances.

We next turn towards the multi-resolution approxima-tions with the s8 wavelet (Figure 9). At the coarsest res-olution level (level 6), the 3 main zones corresponding toregions A, B and C occur. From resolution level 5 (- 24m) onwards, a grid-like structure appears. This structurerefines and is most obvious at resolution level 4 (- 12 m).Differences in image texture between regions A and C

occur from level 3 onwards.

We have seen that the wavelet analysis using the s8wavelet functions reveals patterns such as the regions A,B and C that are of primary interest for precision agricul-ture. These regions might display differences in crop con-ditions that can be treated in a location specific way.Moreover, patterns that are visible but are caused byobjects that are less relevant to precision farming are fil-tered out from a specific level of detail onwards. Use ofthe s8 wavelet, therefore, contributes to analyzingprocesses that are relevant for precision agriculture.

d

FIGURE 6: Matrix of absolute s8 wavelet transform coefficients of the NDVI image of July 11 in horizontal direction at resolution lev-

els 1 (a), 2 (b), 3 (c), 4 (d), 5 (e) and 6 (6.

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Sensi t iv i ty to the type of wavelet

Wavelet analysis was also carried out for the image ofJuly 11 using the Haar wavelet (Figure 10). We foundthat similar patterns for all crystals emerge as for the s8

wavelet, although distribution of energy into crystals dif-

fers according to wavelet type, as coefficients d t M nda and dim contribute more to the total energy

(Table 4). Multi-resolution approximations obtained with

Haar wavelets present a blocky pattern that does not as

easily allow for recognition of the 3 regions A, B and C(Figure IO), and fewer artifacts than approximations with

~8. The relative error L2 between the original image andthe reconstructed image at resolution level 0 though isnow lower (2 x 1 O- 15 as compared to 3 x 1O- 3 for ~8.

b

(_ ; Vi : _,, .i : .

, , , -; ,I Iii,:;,

.

. \.. ,. : ,I ,,_:.

,_ ,,

l:__. j

n

f

FIGURE 7: Matrix of absolute s8 wavelet transform coefficients and corresponding autocorrelation functions of the NDVI image ofJuly 11 at resolution level 3 (12 m) in horizontal (a, b), vertical (c, d) and diagonal (e, f) directions.

a

b d

FIGURE 8: Matrix of absolute s8 wavelet transform coefficients and corresponding autocorrelation functions of the NDVI image of

July 11 at resolution level 4 (12 m) in horizontal (a, b), vertical (c, d) and diagonal (e, f) directions.

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Wavelet characterization of high-resolution NDVI patterns JAG Volume 3 - issue 2 - 2001

Comparison between yield and My 11 NDVi smooth the original NDVl image at level 3. Clear positive correla-wave/et coefficients tions exist in the upper part of the image, whereasYield and NDVI smooth wavelet transform coefficients strong negative correlations exist in the southern part ofwere compared at identical resolution levels by estimat- the image, indicating the similarity of the overall yielding the correlation coefficient within a 5 x 5 pixels-win- and NDVI patterns in the first location, and the inversiondow shifted on the crystals. Figure presents the cor- in the second location.relation coefficient for comparison of the yield map and

a

d

b

FIGURE : Approximations of the NDVI image of July 11 with s8 at resolution levels 6 (a), 5 (b), 4 (c), 3 (d), 2 (e) and 1 (f).

a

d

b

FIGURE o: Approximations of the NDVI image of July 11 with Haar at resolution levels 6 a), 5 b), 4 c), 3 d), 2 e) and 1 f).

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TABLE 4: Percentage of energy by crystal for NDVI images onApril 1, May 30, July 11 and August 7

Apr i l 1 May 30 J u l y 11 Au gu s t 7 J u l y 11s8 58 s8 s8 Haar

d: 0 . 0 2 98 0 . 0 0 23 0 . 0 0 05 0 . 0 3 77 0 . 0 0 14

dl " 0 . 0 1 6 3 0 . 0 0 15 0 . 0 0 06 0 . 0 1 83 0 . 0 0 10

df 0 . 0 1 0 3 0 . 0 0 11 0 . 0 0 02 0 . 0 1 49 0 . 0 0 03

d 0 . 0 8 44 0 . 0 0 42 0. 0 0 19 0.1029 0 . 0 0 30

d s " 0 . 0 2 42 0 . 0 0 19 0 . 0 0 13 0 . 0 1 67 0 . 0 0 16

d 2" 0 . 0 0 55 0 . 0 0 03 0 . 0 0 01 0 . 0 0 4 1 0 . 0 0 03

d 3" 0 . 1 8 24 0 . 0 0 49 0 . 0 0 41 0 . 1 2 64 0 . 0 0 46

di ' 0 . 0 4 25 0. 0 0 35 0.0019 0 . 0 3 11 0 . 0 0 14

d: 0 . 0 1 2 9 0 . 0 0 04 0 . 0 0 04 0 . 0 0 77 0. 0 0 04

d 4" 0 . 1 5 56 0 . 0 0 54 0 . 0 0 40 0 . 1 0 87 0 . 0 0 36

di ' 0 . 0 7 17 0 . 0 0 44 0 . 0 0 15 0 . 0 4 58 0 . 0 0 17

d 4" 0 . 0 1 61 0 . 0 0 04 0 . 0 0 05 0 . 0 1 06 0. 0 0 0 5

dsh 0 . 1 7 14 0 . 0 0 30 0 . 0 0 26 0 . 0 9 74 1 . 0 3 95

d s " 0 . 0 6 73 0 . 0 0 11 0 . 0 0 09 0 . 0 4 87 0 . 7 6 85

d: 0 . 0 1 74 0 . 0 0 04 0 . 0 0 03 0 . 0 1 06 0 . 0 0 74

dsh 0 . 1 6 62 0 . 0 0 14 0 . 0 0 09 0 . 0 9 98 0 . 4 8 76

d 8' 0 . 0 47 5 0 . 0 0 12 0 . 0 0 03 0 . 0 3 43 0 . 0 0 08

d f f 0 . 0 2 43 0 . 0 0 01 0 . 0 0 02 0 . 0 1 26 0 . 0 0 03

S6 9 8 . 8 5 4 3 9 9 . 9 6 2 4 9 9 . 9 7 77 9 9 . 1 7 18 9 7 . 6 7 59

Ot h e r i ma g es

On April 1, May 30 and August 7, s8 wavelet transform

coefficients (not shown) depict some tracks by the hori-

zontal crystals from resolution level 2 onwards.Boundaries of older parcels are especially visible on verti-

cal crystals of the first and the last images, at resolutionlevels 3 to 5. Diagonal crystals contain, as on July 1 1 ess

high coefficients than the other crystals. On the oppositeof the image of July 11, the lower part of the field

(regions B and C) present more high coefficients than the

upper part of the field (region A) from resolution level 2on May 30. Coefficients in horizontal crystals at resolu-

tion levels 3 and 4 present a periodic horizontal auto-cor-relation, while coefficients in vertical crystals present a

vertical auto-correlation. Auto-correlation is negligible inthe diagonal crystals.

Approximations at resolution levels 6 and especially 5 (48m and 24 m on the ground) show a clear separation

between region A and region B on the images of April 1and August 7. The pattern due to tracks, older parcelsand cdals is the most obvious on all images at a resolu-tion& 12 m.

DISCUSSION AND CONCLUSIONSA wavelet transform of the four NDVI images proved to

be a good method to quantify patterns at different reso-lution levels and directions. Historical management fea-

tures were separated at fine resolution levels. Patterns


reflecting natural soil variability or historical develop-ments that are important for precision agriculture clearly

show up at coarser resolution levels. Multi-resolutionapproximations allowed for detecting these features in

the images at different resolution levels.

For the backward-looking approach for decision making,

the symmlet wavelet appears to be most useful, as itssmoothness makes it well suited for the analysis of con-tinuous patterns such as NDVI. Multi-resolution approxi-mations allow to make quantitative comparisons at dif-

ferent resolution levels. Differences in plant developmentthat could be caused by differences in soil conditions or

elevation are observed and quantified. Wavelets there-

fore provide a tool to quantify these patterns, and henceto relate scale specific patterns between different vari-

ables or different days. As such their use may lead to a

better understanding of processes that have causedthese patterns. Also, characterization of areas in the field

with similar plant development during the growing sea-son or between growing seasons might benefit from acombination of remote sensing images and yield maps

derived from a combine harvester.

In the forward-looking approach, the property ofwavelets that they approximate original remote sensing

images at a range of different resolutions can be used tothe advantage. Precision agriculture technology does not

allow for continuous applications yet. Hence, multi-reso-

lution approximations using the Haar wavelet translateNDVI images to the precision of the available technology.

On the other hand, the coefficients allow comparing

emerging plant growth patterns as observed on remotesensing images. Observed patterns are a coarse approxi-mation to the cropping pattern on the field and as such

may help to adapt management practices to the actualconditions.

a

Y

Y

Y

Y

.

1

E

FIGURE II: Smooth wavelet t ransform coefficients of yield (a)and July 11 NDVI (b) at resolution level 3 and corresponding

correlation coefficient (c).

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In comparison with other pattern analysis techniques, themain benefits of a wavelet approximation are that sever-al resolutions and directions are split out, and help tocharacterize an individual image. Fractals quantify varia-tion over a range of scales but do usually not allow fordiscrimination between variations occurring at differentscales. Geostatistics are useful for interpolation of indi-

vidual variables and making univariate and bivariate spa-tial dependence functions. Moreover, geostatistical simu-lations are useful in analysis of error propagation.Though, a decomposition of an image as in this study isoutside the range of current geostatistical methods.Wavelets being a modern tool for spatial analysis providein a way a useful addition to geostatistical methods.

Using the wavelet characterization of high-resolutionNDVI patterns allows to make a quantitative comparisonat different scales between images and hence betweenvariables that reveal variation at different scales. If, for

example, an NDVI image mainly shows variation at oneparticular scale, corresponding to a specific resolution j ,then quantifying the relation with other variables andimages showing variation at the same scale may lead touseful insight into causes of spatial variability. Such awavelet characterization therefore quantitatively relatesvariability between different images, instead of entirelyfocusing on average values that have been used so far.This is a step forward in precision agriculture where themain focus is within-field variability.

ACKNOWLEDGEMENTSThis study is embedded within a large-scale research pro-ject funded by the Dutch Board for Remote Sensing(BCRS). The research of V. Epinat is funded by the Earthand Life Sciences Research Council of the NetherlandsOrganization for Scientific Research (NWO-ALW). Thevan Bergeijk family members are kindly acknowledgedfor supporting this research on their farm.

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RESUMECet article presente une analyse quantitative de formes visibles

dans des images NDVI obtenues a partir de la teledetectionaerienne. Lattention est concentree sur Iutilisation donde-lettes pour distinguer des formes presentant un inter& pourune agriculture de precision a plusieurs echelles. Une procedureg&Wale pour analyser ces images est presentee et appliquee aun seul champ aux Pays-Bas, controlee durant quatre jourspendant une saison de croissance. La decomposition par onde-lettes de ces images est capable de reveler et quantifier desformes presentes a differents niveaux de resolution et directionset de filtrer Iinformation qui est moins importante pour desapplications dune agriculture de precision Une approximationpar ondelettes avec differentes fonctions dondelettes est utiledans les approches de decision avec un regard vers ie passe etun regard vers Iavenir en permettant une adaptation de Iana-

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lyse aux caracteristiques des images ou cartes disponibles et auxpossibilites des instruments existants specifiques du site, respec-tivement.

R SUM N

Este articulo presenta un analisis cuantitativo de 10s patronesvisibles en imdgenes NDVI de alta resolution obtenidas por tele-detection aeroportada. La atencion se concentra en el use depequenas ondas (wavelets), para distinguir patrones de inter&en agricultura de precision a diferentes escalas. Se presenta unprocedimiento general para analizar estas imagenes; el mismoha sido aplicado a una parcela agricola en 10s Paises Bajos, l acual se monitoreo en cuatro diferentes dias durante una esta-

cion de crecimiento de 10s cultivos. La descomposicion de laspequenas ondas en las imdgenes es capaz de revelar y cuantifi-car patrones presentes a diferentes niveles de resolution y endiferentes direcciones, y de filtrar la information que es menosrelevante, para aplicaciones en agricultura de precision. Estaaproximacion basada en diferentes funciones de ondas peque-Aas es titil en 10s enfoques regresivos y progresivos de la tomade decisiones, porque permite adaptar el andlisis a las caracteris-ticas de l as imdgenes o de 10s mapas disponibles y a las posibili-dades de 10s instrumentos existentes en 10s sitios especificos,respectivamente.

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a wavelet characterization of high-resolution ndvi patterns for precision agriculture

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