Image analysis through the Wigner distribution function
Gabriel Crist6bal, Julian Besc6s, and Javier Santamaria
The Wigner distribution (WD) for discrete images is computed and applied for texture analysis. First, theWD for different test images with spatial frequency and gray level information content is computed, and themost important features of the WD are outlined. Likewise, the original image with 1-D variation is recoveredfrom the discrete WD; local filtering operations have been performed on the distribution, and a filteredversion of the original image is then obtained. Since the WD entails simultaneous representation of theintensity distribution and of the spatial interdependence, it seems specially adequate for processing oftextured information. In this way, texture recognition is performed through the extraction of features fromthe WD, and the results are compared with other methods. Finally, to avoid the great computational effortthat requires this kind of representation, a hybrid optical-digital processing system is proposed for thetexture recognition process.
1. Introduction
It is well known that the Fourier power spectrumshows us the global distribution of the energy of asignal as a function of spatial frequency and producesan image representation that is invariant to shifts.Unfortunately the shift invariance of the power spec-trum is obtained at the expense of the loss of phaseinformation contained in the Fourier transformationof an image. However, one is often more interested inthe local distribution of the energy, computing thepower spectrum over subregions of an image.1 TheWigner distribution (WD) gives a local or regionalimage representation and, unlike the Fourier transfor-mation, has no phase associated with it; yet the WD,like the Fourier transformation, is a reversible trans-formation, and thus the phase information is implicitlyencoded.
Although the Wigner distribution was first definedby Wigner2 for a thermodynamic description in quan-tum mechanics, only in relatively recent times has itbeen found to be useful in many areas of science andengineering.3-6 The WD simultaneously describes afunction in space and spatial frequency domain, andthe interest in image representation is motivated to a
G. Crist6bal is with Facultad de Informatica, Campus Montegan-cedo, 28660 Madrid, Spain; the other authors are with Institute ofOptics, Serrano 121, 28006 Madrid, Spain.
large extent by the fact that the WD possesses a num-ber of attractive properties for image analysis. In thisway, we have mainly considered the application of theWD in that sense.
In Sec. II, the Wigner distribution function is de-fined for continuous and discrete signals, and the mostrelevant properties are included. Also, the distribu-tion for different 1-D variation test images with spatialfrequency and gray level information content is com-puted. Afterward, the WD is calculated for 2-D im-ages that include textural information.
In Sec. III, the WD is computed for space-variantdegraded objects. Also, a test image with 1-D varia-tion has been first recovered from the discrete WD,and different local filtering operations are performedover the distribution, giving a filtered image output byinverse transformation.
In Sec. IV, we have considered the application of theWD for texture discrimination and classification in thestatistical approach. Several textural features havebeen extracted from the WD and evaluated for classifi-cation purposes in the case of Brodatz textures.
Finally, a hybrid optical-digital processing system isproposed for image analysis based on optical imple-mentation of the WD in Sec. V to reduce the highcomputational cost that requires the WD computa-tion.
11. Wigner Function Distribution
A. Definition and Properties
The Wigner distribution (WD) introduced by Wig-ner 2 in 1932 as a phase representation in quantummechanics gives a simultaneous representation of asignal in space and spatial-frequency variables. It
belongs to a large class of bilinear distributions knownas the Cohen class7 8that includes Woodward's9 ambi-guity function and Rihaczek's10 complex energy densi-ty function. The Wigner distribution might be inter-preted as a local or regional spatial-frequencyrepresentation of an image. However, it presents twomain advantages with respect to other local represen-tations. First, the WD is a real valued function andencodes directly the Fourier phase information. Sec-ond, the election of the appropriate window size, whichdepends on the kind of analyzed information, is notrequired for the computation of the WD.
On the other hand, the WD of a 2-D image is a 4-Dfunction that involves Fourier transformation for ev-ery pixel of the original image. As a consequence,calculation of the WD of discrete images requires animportant number of computations that can limit, inprinciple, the range of applications. The problem canbe mitigated by using optical processors for the genera-tion and display of the WD even for 1-D signals. Inthis way, several optical arrangements have been pro-posed recently in the literature to produce the WD of1-D and 2-D signals 1l-17 as well as different opticalvisualizations have been shown to display this 4-Dfunction.
In this section, we consider the auto-Wigner distri-bution function of 1-D signals for notational simplic-ity, the extension to 2-D signals being straightforward.The WD Wf(x,w) of the signal f(x) is given by
W1(XW) = f (X+ f (x- )exp(-jwxo)dxo. (1)
The auto-Wigner distribution gives a generalized au-toconvolution of nonzero frequencies. 18
A complete set of properties of the WD was formu-lated by Claasen and Mecklenbrauker.8 The follow-ing are the most relevant for image processing applica-tions.
1. Realness
The WD of any real or complex function is real as aFourier transform of the Hermitian function: f(x +xo/2)f*(x - xo/2). However, it is not possible to inter-pret this distribution as a density energy distribution,because in general the WD is not always positive. Thisis in accordance with the Heisenberg uncertainty prin-ciple, which prohibits an arbitrarily sharp space dis-crimination with an arbitrarily sharp frequency dis-crimination. 8
2. Space Integration and Frequency IntegrationThe integration of Wf over the spatial variable at a
fixed frequency gives the energy spectral density atthat frequency, and the integration of Wf over thefrequency variable at a fixed spatial point gives thelocal power at that point:
J Wf(x,w)dx = IF(w)12; (2)
2 J Wf(x,w)dw = lf(X)I. (3)
3. Autoconvolution
Wf(xO) = f (x + )f* x-X') dx, (4)
(5)WF(wO) = F + F - dw2 .
B. Discrete Wigner Distribution
Although the WD was proposed in the past for con-tinuous variable functions, the concept of the discreteWD was adapted to discrete variable functions byClaasen and Mecklenbrauker.8 An alternate discretedefinition that maintains the generalized convolutionproperty was given by Brenner. 4 The well-known dis-crete Fourier transform (DFT) is given by
N-1F(O) = f(n) exp(-jnO), k = 0,1,. . .,N-1,
n=O
(6)
where n and 0 = 27rk/N are the spatial and frequencyvariables, respectively. The DFT is periodic with pe-riod 27r, i.e., F(0)-F(0+ 27r). The discrete Wigner dis-tribution according to Ref. 8 is defined by
N-iWf(n,O) = 2 E f(n + k)f*(n - k) exp(-j2kO).
k=O(7)
If we have N samples of the function f(x), 0 < k < N -1, the right-hand side of the Eq. (7) can be interpretedas the N-point FFT of the sequence:
g(n,k) = f(n + k)f* (n - k) (8)
for fixed n.The discrete WD is periodic in the frequency vari-
able with period 7r, i.e., Wf(n,0) = Wf(n,0 + 7r). Thediscrepancy between the periodicities could be avoid-ed by discharging the factor of 2 in the exponent in Eq.(7), but this has the drawback that the components in fat 0 occur at 20 in the WD. The presence in the Eq. (7)of the product f(n + k)f*(n - k) produces that thesimultaneous apparition of the even and odd samplesoccurs separately. This can result in aliasing, unlessthe signal is oversampled by a factor of 2.
Two problems frequently arise in practice when wecompute the discrete WD version of Eq. (7): aliasingand leakage. They can be reduced by the applicationof low-pass filtering and spatial apodization, respec-tively.1 8 1 9 Figure 1 shows the procedure for the calcu-lation of the WD in the case of a 1-D image. It presentsa composite sinusoidal test image and the sequence forthe WD computation corresponding to the center ofthe high frequency region.
C. Wigner Distribution Representation
The WD computation of a signal involves the highprice of doubling the input function dimensionality.That is why several optical methods have been pro-posed recently to produce the WD of 1-D and 2-Dsignals, and different optical visualizations have beenshown to display the 4-D function generated from a 2-D image. Bamler and Glunder13 display the 4-D func-tion as 2-D slices; Easton et al.15 use the Radon trans-
Fig. 2. (a) Sinusoidal grating object of 64 pixels/period; (b) WD of(a) for the x direction and the autoconvolution profile ( = 0) with
zoom 2X.
Fig. 1. WD generation procedure for the center of the high-fre-quency sinusoidal region.
formation to produce the Fourier transform anddisplay the WD as 1-D slices of the 4-D WD. On theother hand, Jacobson and Wechsler20 have proposedconformal mapping techniques to obtain invariant im-age representations.
Here, the WD for different 1-D test images withspatial frequency and gray level information content iscomputed, and the most important characteristics ofthe WD are outlined.2'
1. Test ImagesFigure 2(a) shows a 256 X 256 truncated sinusoidal
grating (1/2 + 1/2 cos27rx/64) with 64 pixels/period,where x represents the spatial coordinate. Figure 2(b)shows with zoom 2X the corresponding modulus of theWD for the x direction with the local spectra displayedalong the y axis. The autoconvolution of the originalgrating is displayed along the x axis ( = 0) with animportant variation of the intensity as represented bythe profile curve. Along the y axis appears not onlythe fundamental spatial frequency of the grating butalso harmonics corresponding to a (sin)2 signal. TheWD is also periodic with a period one-half of the origi-nal grating object.
Figures 3(a) and (b) show a composite sinusoidalgrating with 32 and 16 pixels/period, a 0-1 grey levelrange, and its corresponding WD with zoom 2X. Inthis case, each main spatial frequency associated withthe different regions of the object appears in the WD.They are dominant mainly in the central zones of thetwo regions, while the two main frequencies arepresent in the WD of the center of the object, as expect-ed.
Figure 4 shows the WD corresponding to a rectangu-lar window test. In this case, the WD can be seen as asine function composition where the zeros are shifted
a b
Fig. 3. (a) Composite sinusoidal grating of 32 and 16 pixels/period;(b) WD of (a) for the x direction, and autoconvolution profile ( = 0),
with zoom 2X.
Fig. 4. WD for a rectangular window test in the x direction and theautoconvolution profile (a = 0) with zoom 2X.
as we move from the center to the border (image prod-uct FFT for different widths of the rectangle test).
Finally, Fig. 5(a) shows an aerial image (1.2X zoom)with two main tree species, Eucalyptus globulus andPinus Pinea. Figure 5(b) shows on top the imageproduct
g(mnk,l) = f(m + k,n + 1)f*(m - k,n - 1) (9)
previously expressed by Eq. (8) at two representativepoints (-72,0) of the pinus region and (+72,0) of theeucalyptus region. On the bottom is displayed the
Fig. 5. (a) Aerial test image with two main tree species: pinus (left)and eucalyptus (right) with zoom 1.2X; (b) image products at thepoint (-72,0) of the pinus region (above left) and at the point (+72,0)of the eucalyptus region (above right) and WDs in the two consid-
ered points (bottom). The origin is at the center of the picture.
WD of the two points, i.e., W(-72,0,u,v) andWf(+72,0,u,v), respectively. The textured structureof both species is kept in the product images, and,therefore, it can also be observed in the WD of the twopoints. The WD shows a more important contributionof high frequencies in the pinus region than in theeucalyptus region.
Ill. Local Image Filtering Through the WD
Let us consider a linear image formation system inwhich an input signal f(x,y) gives an output signalg(x,y). This operation can be described by the Fred-holm superposition integral:
g(x,y) = | J f(Q,n)h(x,yZ)dtdni. (10)The function h is termed the point spread function(PSF) and gives the radiant energy distribution in theimage plane of a point in the object plane. There aremany image formation systems which present PSFswhich vary with the position in the object plane(space-variant PSFs). In these cases, the modelingprocess of image degradations presents a major diffi-culty. Aberrations, atmospheric turbulence, and de-focusing are a few examples of the most frequent deg-radations that can have different amounts of blur indifferent regions of the picture. In these cases, theWD local adaptation seems to make this distributionappropriate for the restoration of images with space-variant degradations.22
Figure 6(a) shows a grating in focus on the right-hand side and progressively defocused on the left.Figure 6(b) shows the image products and the WD atpoints (-32,0) and (+32,0). The WD reflects a differ-ent behavior range for the object regions due to thespatially variant defocusing. A greater extension ofthe WD and a more complete structure are obtained at(+32,0) as consequence of the greater definition of thefocused part in contrast with the defocused part.
For the local filtering application we consider the 1-D rectangular grating of Fig. 7(a) as a test image.First, the modulus squared of the original image isrecovered from the WD through the use of property[Eq. (3)] regarding summation over the frequencies.
Fig. 6. (a) Grating with progressive defocusing toward the left asinput for WD; (b) image products at points (-32,0),(32,0) from the
center and the WD at these points.
a b
nH Ht
C
Fig. 7. (a) Rectangular grating test; (b) recovered image from thediscrete WD; (c) results of local filtering operations: low-pass filter-
ing (half left) and high-pass filtering (half right).
Figure 7(b) shows the recovered image modulus fromthe discrete WD by integration of the distribution overthe spatial frequency. Next, two local filtering opera-tions are performed on the Wigner distribution: low-pass filtering on the left half of the image and high-pass filtering in the right half. Figure 7(c) shows theresults of these operations after coming back to theoriginal domain: a smoothing is obtained on the left-hand side and an edge sharpening on the right handside. Although the filtered WD is not a true WD, theintroduction of different masks on the WD can beapplied for local filtering purposes.
IV. Texture Analysis Through the WD
The goal of image analysis is the region extraction orthe object description from images. Textural proper-ties of image regions are often used for classification
and segmentation purposes. Texture is the term em-ployed to characterize the surface of an object, and it isone of the important features used in object identifica-tion and recognition of an image. The term texturecould be defined as the arrangement or spatial distri-bution of intensity variations in an image.23 The twomajor characteristics of a texture are its coarseness anddirectionality. Since the spatial frequency domainrepresentation contains explicit information about thespatial distribution of an image, one expects to obtainuseful textural features in the frequency domain.However, the texture methods that embody spatialfrequency information have met mediocre results,24 25
mainly because each frequency component containsglobal information. Even reducing the window size tocontain homogeneous texture subimages was not anadvantage in the discrimination performance com-pared with the co-occurrence methods.
Recently it has been suggested that the visual dis-crimination ability of texture is achieved locally.26
The WD encodes the main characteristics of the im-ages including the spectral local variation and can givea better texture discrimination. In this way, we haveconsidered the application of the WD for texture dis-crimination and classification in the statistical ap-proach.
The WD based method for texture classification hasbeen applied to four Brodatz texture fields27: sand(top left), straw (top right), cotton canvas (bottomleft), and raffia (bottom right) (Fig. 8). The texturesamples were digitized and converted into 256 X 256picture arrays quantized into 128 gray levels. Differ-ent test window sizes have been proposed in the litera-ture to obtain good statistical resolution.28 29 The WDincludes from the definition spatial and frequency in-formation, and therefore, the selection of windows isnot required. However, we have computed the WD ata specified window size to reduce the computation timeand make available comparison of the texture analysisresults to other methods. In our study, we have usedN = 16 regions of 64 X 64 pixels for each texture.
A. Feature Extraction
The first-order statistics of all textures have beennormalized to uniform distributions so that differ-ences in luminance and contrast are eliminated in thediscrimination process. After performing histogramequalization on the textures, a complete homogeneityof the data is reached, and any result from the compu-tation is solely due to the structural interrelationshipof the elements rather than unwanted biases. Figure8(a) shows the four preprocessed Brodatz textures andFig. 8(b) its corresponding Fourier transform.
Several textural features have been extracted fromthe WD of Brodatz textures and evaluated accordingto textural distance measures as well as by discrimi-nant analysis. The method used for texture charac-terization requires the computation of the auto-WD atN = 16 different points of the texture separated by 2pixels and the corresponding selection of features fromsuch distributions to obtain the texture feature vector.
Fig. 8. (a) Preprocessed Brodatz textures (clockwise from top left):sand, straw, raffia, and cotton canvas; (b) Fourier spectra of tests (a).
The auto-WD.obtained for each point has been sam-pled in I = 16 annular regions and H = 8 sectors in asimilar way to the power spectrum methods.30 Thefeatures extracted from the WD are the following:
mean frequency w = i=i __I m _ V(lm)inringi
E E Wf(l,m)I m
E" 7,1p - wiI2Wf2(Im)1
contrast - =1 I n V (I'm) in ring
E W(1,m)I m
E[ Wf2(,'M)]
directionality W3 = 1Im (')isetrh;
E E Wf2(1,M)I
directionalityvariation w4
H
h=1
(d - w3)2Wf2(1,m)
(11)
(12)
(13)
V (Im) in sectorh
W(l m)I m
(14)
homogeneity W5 = IW I ,"w(OO) W1(0,0)1
(15)
in which W2(0,O) is the mean value of the WD at the Nselected points, and p,0 are the polar coordinates in thespatial frequency domain. The feature w is a mea-sure of the mean spatial frequency content of the im-age. It will have low values for images with importantbackgrounds and will increase as sharp details arepresent in the image. The contrast feature w2 is ameasure of the contrast or the amount of local intensi-ty variations present in the image. The features W3
and W4 give the directionality and the local variation ofthe directionality, respectively. Finally, W5 is a mea-sure of the homogeneity of the image. These featureshave been extracted from sixteen WD samples andrepresented by a 5-D vector:
There are two quantitative methods for the evalua-tion of statistical texture measures. The first is aclassification method which involves the measurementof classification errors, and the second is the figure ofmerit method, which usually involves a distance func-tion to provide a measure of separation between twotexture classes.31
The texture features proposed in Sec. IV.A havebeen evaluated for comparison purposes with othermethods according to their Bhattacharyya distance (Bdistance). The B distance is a scalar function of theprobability densities of the features of two classes thatare monotonically related to the Chernoff bound of theprobability of classification error when a Bayes classi-fier is used.31 Faugeras and Pratt have suggested theevaluation of the texture features according to theirBhattacharyya distance for comparison purposes.29
For Gaussian densities, the B distance between apair of texture classes Si and S2 becomes
B(S 1 ,S2 ) = (m - 2) 2 2]'(ml- m2)T
+1 [I 1/2 (E + 2)I (17)2 LIF'1 1i/2 1E,11/2
where mi and Fi represent the feature mean vector andfeature covariance matrix of the i class. For equallylikely texture field pairs, a B distance of 4 or greatercorresponds to a classification error bound of _1%.31Figure 9(a) shows sixteen product images corre-sponding to the cotton canvas texture at points(128,48),(128,50),.. .,(128,80) and Fig. 10(a) the prod-uct images for points of the straw texture at thesepoints. The corresponding Fourier transform or WDat each point is displayed in Figs. 9(b) and 10(b),respectively, that show the effects of the object period-icity.
Let us consider the WD feature extraction method(WDFEM) based on the features 4.1-4.5 proposed.The performance of the WDFEM was compared withthat of the spectral energy feature extraction method(SEFEM) and the co-occurrence matrix feature ex-traction method (CMFEM). In SEFEM, the energieswere summed within five concentric rings centered ineach 64 X 64 spectra, producing a 5-D vector:
f = x/2444] (18)
The second texture representation (CMFEM) wasbased on five features extracted from the gray level co-occurrence matrix: energy, entropy, correlation, localhomogeneity, and inertia for a pixel interdistance (d =2), giving a 5-D vector:
C = [C1,C2,C3,C4,C517 (19)
The results of the pairwise B-distance computation forthe three methods considered have been tabulated in
Fig. 9. (a) Product images corresponding to the individualizedcotton canvas texture at points (-16,0),(-14,0),...,(14,0); (b) WD at
these points.
aFig. 10. (a) Product images corresponding to the individualized
straw texture at points (-16,0),(-14,0); (b) WD at these points.
Table I. In both SEFEM and CMFEM, the sand-straw pair is the most similar and the cotton-raffia pairthe least similar. In contrast, in the WDFEM thesand-raffia pair is the most similar and the sand-strawpair the least similar. The B distance has been com-puted using the statistical package IMSL32 running ina VAX 11/750 computer.
In addition to the pairwise linear discriminants, wehave considered multiple discriminant analyses to ob-tain the 2-D subspace which maximizes the Fisherratio between class to within class.33 Projections ofthe samples on this plane give a scatter diagram of thefour texture classes, which is a convenient visualiza-
Table 1. Comparative Results for Various Textural Feature ExtractionMethods
Fig. 11. (a) Two-dimensional scatter diagram for the summedspectral energy features; (b) Id. for the co-occurrence method and (c)
Id. for the WD method.
tion of the class discriminability. Figure 11(a) showsthe 2-D scatter diagram for the summed spectral ener-gy features, and Figs. 11(b) and (c) show the scatterdiagrams for the co-occurrence matrix and WD fea-tures, respectively. Although the clusters are morecompact in the summed energy plot and in the co-occurrence matrix plot, there is also more overlap be-tween the classes than in the WD feature extractionmethod. The summed spectral energy and the co-occurrence diagrams show a considerable amount ofoverlap between the sand and straw classes, as expect-
a b
Fig. 12. (a) Cottom canvas-raffia texture pair; (b) (from left toright and top to bottom) WD at points (0,-16),(0,-14),. . .,(0, 14).
ed from Table I, showing also the co-occurrence dia-gram a scattered raffia plot.
C. Texture Discrimination
The first step of image analysis is the segmentationof an image into regions which are homogeneouslytextured. Two types of methodology are widely usedto attempt to solve the segmentation problem. In theedge detection method, parts of the picture where atransition occurs from one uniform region to anotherare searched. In the second method, region growingstarts from small uniform regions and expands them asfar as possible without altering their uniformity.
A texture pair is constructed putting two texturepatterns together on one image. The significance ofthe texture pairs comes from the fact that any textureanalysis problem can be broken down into an equiva-lence texture pair discrimination problem.34 A simple1-D texture discrimination procedure based on theWD computation between texture pairs is suggested.The texture discrimination process involves threeclasses of texture pixel, edge pixel, near-edge pixel, andinterior pixel.35 The proposed method is based on thecomputation of differences between the WD in adja-cent points along a selected direction and the WD*mean in the texture edge. Figure 12(a) shows a 256 X256 texture pair (cotton-canvas-raffia), and Fig. 12(b)shows from left to right and top to bottom the WD at 16points, from the center of the cotton canvas region tothe center of the raffia region. The window size select-ed was a 64 X 64 pixel to obtain a good statisticalresolution. The WDs obtained for the extreme pointsare similar to those of the two isolated textures [Figs.9(b) and 10(b)], and they become mixed for the inter-mediate points.
To detect the edge between texture pairs, we selectN = 16 points in the texture edge. Then the WD,W.(ij) = W(xy.,ij) is computed. The parameter ewfor a given point (Xd,yd) is then defined by36
k k
ewd = E 7 1W0 (ii) - WnM(iIj)1I'i=1 j=l
(20)
where WBM is the WD mean associate to edge texture:N
Fig.13. Edge operator ewvs the interepixel distanced correspond-ing to sand-straw, straw-raffia, cotton canvas-raffia, and sand-cotton canvas texture pairs, respectively. The range of ewvariationhas been normalized to obtain a representation between the 0-1
margin.
and N = 16 is the number of points selected in thetexture edge. Figures 13(a)-(d) show for four texturepairs a plot of ew vs the interpixel distance d; it appearsas a pronounced minimum correspondent to the tex-tural borders. This fact permits one to indicate thatthe WD method might give good discrimination abilitybetween edge pixels and near-edge pixels.
V. Hybrid Optical-Digital System
Combining electronic technology with optical sys-tems as a means of applying the fast processing abilityand parallelism of optics to a wider range of problemswas proposed by Huang and Kasnitz and subsequentlydeveloped by Casasent and others.35 The main advan-tages of optical systems in contrast to digital sytsemsare: (a) parallel computation of 2-D information, (b)ability to operate at very high data rates, and (c) theFourier transform is simple to perform. The deficien-
cies of optical systems happen to be the strong pointsof some electronic systems. The main advantages ofdigital systems are: (a) program and control flexibili-ty and (b) the ability to make decisions.
A hybrid optical-digital processing system is pro-posed for image analysis based on the WD in which thegreat computational effort that requires the WD calcu-lation is avoided by using an optical processor, andfurther processing is performed digitally. Figure 14presents the schematic diagram of the hybrid proces-sor where the WD is generated through the opticalprocessor and stored in the computer for subsequentanalysis. The generation of the WD in the setupsdescribed in Ref. 13 is obtained in the following way.A laser beam is spatially filtered to reduce noise andcollimated. The beam goes through a beam splitterand illuminates the object transparency. The intro-duction of a lens and a plane mirror allows one to
Fig. 14. Schematic diagram of the hybrid processor.
Fig. 16. Spectra obtained with the hybrid processor in the texturetransitions raffia-cotton canvas.
I Experimental arrangement of teyise
Fig. 15. Experimental arrangement of the hybrid system.
obtain after reflection in the mirror the same objecttransmittance but rotated 180° with respect to theoriginal one. A new transmission by the transparencyyields the product image, which after reflection in thebeam splitter is Fourier transformed by a lens. In thisway, the Wigner distribution for the center of theobject transparency is obtained at the phototube cam-era plane, if the diaphragm placed on the object iscentered with the optical axis. A shift of the objectand diaphragm produces an opposite shift of the re-flected image, and so the system yields the WD at thenew object point. Hence the WD for different objectpoints is obtained by shifting the object transparencyto those points and sequential recording of the lightdistribution at the camera plane.
Figure 15 shows the experimental arrangement ofthe optical system which is connected through the TVcamera with a VICOM image processor. Figure 16presents the results obtained with the hybrid proces-sor in the texture transitions raffia-cotton. The spec-tra are very similar to those generated via digital com-putation, except for the noise introduced by thecamera acquisition process.
VI. Conclusions
The Wigner distribution function gives a simulta-neous image representation in the space and spatialfrequency domains that encodes the main image char-acteristics. Consequently, this paper considers theapplication of the WD to several areas of digital imageanalysis.
Image representation through the WD has been per-formed for images with 1-D and 2-D variation, display-ing the spectral local variations in a continuous way.The experimental results are especially interesting fortextured images and images with spatial-variant deg-radations. Likewise, recovering the original imagefrom the WD has also allowed us to perform differentlocal filter operations over the distribution and obtaina local filtered version of the original image.
The application of the WD for texture discrimina-tion and classification in the statistical approach hasbeen considered. Several textural features have beenextracted from the WD and computed for the Brodatztextures case. The performance of these WD featureshas been evaluated according to their Bhattacharyyadistance and discriminant analysis. The proposedWD method results in a percentage of classificationerror smaller than 1% and is superior to band-wedgeFourier methods. The use of the WD for texture dis-crimination has also been considered and applied byusing an edge detection approximation.
Finally, the introduction of a hybrid optical-digitalprocessing system has been tested to avoid the greatcomputational effort that requires the computation of
the WD for discrete images. The WD is obtainedthrough the optical processor, and further processingcan be performed through the digital system.
This work was partially supported by the SpanishAdvisory Commission for Scientific and Technical Re-search under grant 358. We also thank Ana Plaza forher collaboration in the general development of thework.
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