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S. Gabarda and G. Cristóbal Vol. 25, No. 9 /September 2008 /J. Opt. Soc. Am. A 2309

Discrimination of isotrigon textures using theRényi entropy of Allan variances

Salvador Gabarda* and Gabriel Cristóbal

Instituto de Óptica “Daza de Valdés” (CSIC), Serrano 121, Madrid-28006, Spain*Corresponding author: [email protected]

Received January 22, 2008; revised April 25, 2008; accepted May 27, 2008;posted July 28, 2008 (Doc. ID 91961); published August 20, 2008

We present a computational algorithm for isotrigon texture discrimination. The aim of this method consists indiscriminating isotrigon textures against a binary random background. The extension of the method to theproblem of multitexture discrimination is considered as well. The method relies on the fact that the informa-tion content of time or space–frequency representations of signals, including images, can be readily analyzedby means of generalized entropy measures. In such a scenario, the Rényi entropy appears as an effective tool,given that Rényi measures can be used to provide information about a local neighborhood within an image.Localization is essential for comparing images on a pixel-by-pixel basis. Discrimination is performed through alocal Rényi entropy measurement applied on a spatially oriented 1-D pseudo-Wigner distribution (PWD) of thetest image. The PWD is normalized so that it may be interpreted as a probability distribution. Prior to thecalculation of the texture’s PWD, a preprocessing filtering step replaces the original texture with its localizedspatially oriented Allan variances. The anisotropic structure of the textures, as revealed by the Allan vari-ances, turns out to be crucial later to attain a high discrimination by the extraction of Rényi entropy measures.The method has been empirically evaluated with a family of isotrigon textures embedded in a binary randombackground. The extension to the case of multiple isotrigon mosaics has also been considered. Discriminationresults are compared with other existing methods. © 2008 Optical Society of America

OCIS codes: 150.1135, 100.2000, 100.4992.

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. INTRODUCTIONextures whose third-order correlation functions arequal to that of uniformly distributed noise patterns areenoted as isotrigon. Isotrigon patterns are a wide varietyf binary (black-and-white) and ternary (black, white, andean luminance gray) texture patterns with spatial cor-

elation properties and reproducible characteristics [1]. Inhis paper, we focus on the case of binary textures only.pecifically, isotrigon textures are created by a recursionrocedure in which the product of three pixels determineshe value of a fourth. Textures with correlation functionsased on triplets of pixels (hence trigons) have been de-cribed by Julesz et al. [1]. Systematic methods for pro-ucing isotrigon textures have been developed by Mad-ess et al. [2]. Purpura et al. [3] have suggested thatncoding features such as edges or contours require cor-elations among three or more points. Isotrigon texturesre useful for studying human sensitivity to these corre-ations. Maddess and Nagai [4] determined human dis-rimination performance for several classes of isotrigonextures and compared these experimental results withhe outputs of two statistical discrimination models.hese models employed versions of the Allan variance [5]

n order to incorporate early stages of orientation process-ng into a scheme that examines fourth- or higher-orderorrelations in order to discriminate isotrigon textures.ne of the models was based upon a global variance mea-

ure, and the other was based upon a localized varianceith an orientation bias. To do this, they employed lineariscriminant analysis (LDA) and quadratic discriminantnalysis (QDA) [6]. LDA and QDA are used in statistics

1084-7529/08/092309-11/$15.00 © 2

or determining the combination of features that bettereparate two or more classes of objects or events. For a de-ailed description of LDA and QDA, see [6]. They hypoth-sized that an obvious way to improve texture detectionerformance by these statistical methods is to increasehe number of the orientations used to analyze texturemages, but this would greatly increase the computationalost. However, directionality is a fundamental feature inhe case of textures, and it will play a key role in ournalysis. In recent years, various methods have been pro-osed to tackle the problem of multiorientation analysisf images [7–10].

This paper presents a new method for discriminatingsotrigon textures from a binary background. The exten-ion to the case of multitexture discrimination has alsoeen considered. Here the discrimination is based not onhe statistics of the standard autocorrelation function butn directional measurements of the entropy shown by thellan variances of the textures. Entropy is measured

hrough the pseudo-Wigner distribution (PWD) of localpatially oriented variances of the textures. Allan vari-nces have shown to be an efficient preprocessing step toapture directional anisotropy when dealing with isot-igon textures [4] and have been effectively used in thexperimental examples described in this paper. The PWDncludes important features, being especially suitable forignal analysis, providing a straightforward pixelwise,oint space–frequency representation of a given texture,dding the possibility of being both spatially and direc-ionally localized when used in a 1-D version. Moreover,eneralized Rényi entropy measurements can provide

008 Optical Society of America

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2310 J. Opt. Soc. Am. A/Vol. 25, No. 9 /September 2008 S. Gabarda and G. Cristóbal

uantitative criteria to evaluate the importance of theoint space–frequency information around a given pixelocation.

To summarize, the aim of our method is to achieve aomputational algorithm for isotrigon texture discrimina-ion, sharing some special characteristics with the visualystem, such as spatial localization, restricted spatial fre-uency, and orientation tuning. We do not intend for ourethod to reveal the way the visual system behaves; we

imply take recognized mathematical tools such as theényi generalized entropy and the PWD to implement aorkable algorithm with the already mentioned visual

haracteristics.This paper is structured as follows: The basic math-

matical description and theoretical background of theethod are described in Section 2. Experimental results

re given in Section 3. Discussions of several related as-ects are presented in Section 4, and, finally, conclusionsre drawn in Section 5.

. DESCRIPTION OF THE METHODhe isotrigon texture analysis method is based on threeonsecutive stages. The first is a processing stage thatrovides the Allan variance. Second, the PWD of the pre-ious Allan variance is computed. Third, the Rényi en-ropy is extracted from the PWD.

. Allan Variancehe simplest isotrigon textures have third-order statistics

dentical to those of coin-flip textures, i.e., textures whoseixel values are jointly independent random variablesith equal probability of taking the value 0 or 1. Globally

onsidered (the meaning of global refers to the associationf a single value of entropy for the whole image), there iso difference between the entropy of a binary noise tex-ure and the entropy of an isotrigon texture. The classicefinition of Shannon entropy, H�S�=−�i=1

q P�si�log2�P�si��,epends only on the probability of the different symbolsnd not in the way they are arranged. However, consider-ng 2-D image data sets instead of single pixels andquating the domain types with Shannon’s symbols, it cane shown [4] that the entropy is much less for isotrigonatterns. Many image transforms such as curvelets [7],ountourlets [8], wedgelets [9], or directionlets [10] haveeen proposed to better represent the directionality thats inherent to images. By measuring the entropy in differ-nt directions, that is, by measuring the anisotropy, sig-ificant image differences can be determined. Hence, a di-ectional entropy measure turns out to be an effective toolor identifying such differences. Isotrigon textures usuallyhow some spatial anisotropic activity that can be re-ealed by extracting local and oriented variances. The Al-an variance has gained some interest as a measure ofharacterizing time series and 1/ f� noise [5] and becomesspecial case of local oriented variance when applied over-D data sets. The conventional Allan variance is just thealf-sum of the squared differences between adjacentoints of a 1-D series of samples x�k�:

AV�n� =1

2�N − 1� �k=−N/2N/2−1

�x�n + k + 1� − x�n + k��2. �1�

he factor 1/2 ensures that if the random variables x�k�,=n−N /2 , . . . ,n+N /2−1, are pairwise uncorrelated, eachith variance �2, then E�AV�n��=�2. The estimator for

he Allan variance [Eq. (1)] is unbiased for division byN−1� instead of N.

Introducing �p ,q� as a new notation for pixel n, the Al-an variance can be extended to 2-D matrices as follows:

AV�,��p,q� =1

2�N − 1�2 �k=−N/2N/2−1

�l=−N/2

N/2−1

�x�p + k + �,q + l + ��

− x�p + k,q + l��2. �2�

The interpretation of Eq. (2) is actually the differenceetween the square contrast and the correlation [11].Here a particular pixel �p ,q� represents the central one

f a set of pixels �x�k , l�� in the neighborhood. The valuesf � ,� are limited to those indicated in Table 1 for the onlyour orientations that result from considering adjacentixels exclusively when applied to 2-D discrete images.The nature of isotrigon textures, whose pixel values are

alculated following a law that utilizes only adjacent pix-ls, justifies the use of a short length sequence in the cal-ulation of the Allan variance. Figure 1 compares the Al-an variance of a particular example of an isotrigonexture calculated using a horizontally oriented sampleith that of a binary noise texture. Note that the Allanariance estimate (Fig. 1(d)) of the binary noise looks likerandom product as well, whereas the Allan variance es-

imate of the isotrigon texture (Fig. 1(b)) seems to capturehe anisotropy of the original (Fig. 1(a)).

. One-Dimensional Pseudo-Wigner Distributionpace–frequency information of a given image can be ob-ained by associating the gray-level spatial data to one ofhe well-known spatial/spatial-frequency distributions12]. For this application, the Wigner distribution [13] haseen selected. The spatial/spatial-frequency representa-ions configure a family of functions introduced by Cohenn the 1-D signal analysis [14]. Any member of the familyan be regarded as a linearly filtered version of theigner distribution [12]. We have based our preferences

n considering the Wigner distribution as the paramountistribution among the Cohen class. Some of the proper-ies of the Wigner distribution—such as its real character,hich implicitly encodes the phase information—make itasier to handle for practical applications than other

Table 1. Possible Parameter Values in Eq. (2)

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patial/spatial-frequency representations that are com-lex.Once the PWD has been selected apparently as theost suitable for this application, any specific pixel n of

he image can be associated with a vector containing its-D PWD, calculated in a window of length N. The use ofwindowed 1-D transform for a 2-D signal can be justi-

ed by considering the three main aspects of the problem.irst, by using a 1-D PWD, data can be arranged in anyesired direction over a 2-D image; second, calculationime is greatly decreased compared to a 2-D version of theWD; and third, the 1-D PWD is an invertible function,nd thus the information is preserved.A discrete approximation of the Wigner distribution

roposed by Claasen and Mecklembräuker [15–17], simi-ar to Brenner’s expression [18], has been chosen for thisroblem:

ig. 1. A. Particular sample of an isotrigon texture (even zigzaorizontal window analysis (note that the result reveals an anisollan variance of the binary noise calculated in the same horizoinary noise is isotropic and all directionalities are equivalent)8 pixels.

W�n,k� = 2 �m=−N/2

N/2−1

z�n + m�z*�n − m�e−2i�2�m/N�k. �3�

In Eq. (3), n and k represent the time- and frequency-iscrete variables, respectively, and m is a shifting pa-ameter, which is also discrete. Here �z�n�� is a 1-D se-uence of data from the image, containing the gray valuesf N pixels, aligned in the desired direction. Equation (3)an be interpreted as the discrete Fourier transformDFT) of the product z�n+m�z*�n−m�. Here z* indicateshe complex conjugate of z. This equation is limited to apatial interval �−N /2 ,N /2−1� (the PWD’s window), giv-ng a local character to the information. By scanning themage with a 1-D window of N pixels, i.e., by shifting theindow to all possible positions over the image, the full

12�512 pixels. B. Allan variance of the given texture, using anature with a privileged direction tilted 45°). C. Binary noise. D.heme (anisotropy is not detectable, as expected, indicating thatnce calculations have been obtained by applying Eq. (2) for N

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ixelwise PWD of the image is produced. The window cane tilted in any direction to obtain a directional distribu-ion.

. Rényi Entropy Measuresntropy is a measure of information in a given set of data.ince textures can be considered as representations of theurfaces of objects and it is well established that surfacesre often composed of features with multiple orientations,he question arises as to whether image-data-gathering-riented channels transmit different amounts of informa-ion about textures and real-world images. Entropy cane applied as a global measure or as a local one, suggest-ng that examining the entropy at different orientationsor directions) in an image may be a useful approach tohe problem of texture discrimination. The direct applica-ion of directional entropy measures can reveal orienta-ion biases in a given texture class, but in some cases, areprocessing step that involves the calculation of the Al-an variances of the textures is required. Whether a pre-rocessing step is necessary or not, directional entropyeasurements may be useful for detecting the anisotropy

f 2-D distributions. For this reason, the Rényi entropytands out as the most relevant entropic measure in thisontext.

A thorough review of the existing Rényi measures,hich we summarize here, can be found in [19]. The en-

ropy measure was initially proposed independently byhannon [20] as a measure of the information content perymbol, coming from a stochastic information source, andy Wiener [21] as a measure of randomness. Later, Rényi22] extended this notion to yield the generalized entropy.he use of the Rényi entropy measures in the context ofime–frequency distributions was introduced by Williamst al. [23,24], with a significant contribution by Flandrint al. [25] in establishing the properties of these mea-ures. The Rényi entropy measure applied to a discretepace–frequency distribution P�n ,k� has the form

R� =1

1 − �log2��

n�

kP��n,k� , �4�

here n and k represent the spatial and frequency vari-bles, respectively; ��2 are values recommended forpace–frequency distribution measures. Although Rényieasures of joint space–frequency distributions formally

esemble the original entropies, they do not have theame properties, conclusions, and results as the classicalhannon entropy. For instance, the positivity condition�n ,k��0 will not be always preserved, along with thenity energy condition, �n�kP�n ,k�=1, which are classi-al requirements for considering P a regular probabilityistribution. In order to reduce a distribution to the unityignal energy case, some kind of normalization must beone [24]. The normalization can be done in various ways,eading to a variety of possible measure definitions19,25].

Quantum mechanics [26] inspires a normalizing stepy associating the space–frequency distribution P of aiven position n with a wave function. We have selectedhe already mentioned quantum normalization for this

articular problem of isotrigon texture discrimination,ue to its excellent response, as will be illustrated later inection 3.A wave function is a mathematical tool used in quan-

um mechanics to describe any physical system. The val-es of the wave function are complex numbers that, whenultiplied by their complex conjugates, give the probabil-

ty distribution of the possible states of the system. In aimilar way, we may derive a probability distribution bynterpreting W�n ,k� in Eq. (3) as the wave function of theixel states.In such a case, W̆�n ,k�=W�n ,k�W*�n ,k� is the probabil-

ty distribution of W. Here W* represents the complex con-ugate of W. A normalizing step must follow to satisfy theormalizing condition �n�kW̆�n ,k�=1.The general case in Eq. (4) with �=3, identifying P

ith the normalized PWD [W in Eq. (3)], and interpretinghe measure on a pixelwise basis, leads us to

R3�n� = −1

2log2��

kW̆3�n,k� . �5�

Here W has been normalized by

W̆�n,k� =W�n,k�W*�n,k�

�k�W�n,k�W*�n,k��.

An accurate uncertainty measure, especially in situa-ions when position and momentum are highly correlated,as been proposed by Süßmann [27], known as the Rényi–üßmann entropy. Süßmann proposed an entropy mea-ure that employs the Wigner distribution after squaringhe distribution values [28]. Dealing with images, theypical variables are position and frequency. Images, con-idered as random processes, show a high correlation be-ween position and frequency in neighborhood pixels, thexception being made at the borders of the objects in themage. These properties have to be considered when anppropriate model has to be built. The Wigner distribu-ion was proposed by Wigner as a phase–space represen-ation in quantum mechanics [13]. Subsequently, Ville de-ived in the area of signal processing the sameistribution that Wigner derived some years before [29].fter that, phase–space methods became widespread inn increasing number of applications due mainly to theirntuitive character and to their being comprehensivelyroad [30].

. EXPERIMENTAL RESULTShe tests that are going to be described below have beenerformed by generating texture samples at random eachime with a MATLAB function due to Maddess [4]. We as-ume that there is enough image diversity, based in theumber of trials and in the random differences among theamples used in every particular experiment. Also, theethod previously described involves several parameters

hat must be set up to have accurate results. Anisotropyay present many different schemes. In particular, isot-

igon textures involve a strong correlation among neigh-oring pixels. Presumably, the window size of the PWDan be small and has been fixed after some experimental

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rade-off with the results. Orientations are restrained tohe directionalities available with the Allan variance.

The discriminating process requires a test image andwo reference images. The test image contains the targetsotrigon texture embedded in a binary random back-round. One of the reference images is an arbitraryample of the isotrigon texture under test; the second ref-rence image is an arbitrary sample of the binary randomoise in the background of the test image. In the ex-mples that follow, several digital images of 512512 pixels with 8 bits/pixel have been considered. The

hree previously mentioned images are subjected to a pre-rocessing step consisting of substituting their originalixel values for their local Allan variances calculated ac-ording to Eq. (2) for N=8 and using the right orientation,hich is determined empirically (as described below). The

our possible orientations, 0°, 45°, 90°, and 135°, will beeferred as AV0, AV45, AV90, and AV135, respectively. Also,ombinations of the variances are examined by squaring

ig. 2. Example of detection of a test texture (even zigzag) of 5round. B. Result of the application of the Allan variance to A witpplying the Rényi entropy over a PWD of B tilted 45° and a win

ums and differences. The results are biased to have ainimum value over zero (+1 was determined to be a

uitable bias by trial and error). Biasing the values is nottrivial operation, because the PWD and the Rényi en-

ropy are nonlinear operations and the result dependspon the range in which the values are set. Hence, bias-

ng the values influences the sensitivity of the process. Its not critical, but operations with all values close to zerore not recommendable.Allan variances are then submitted to a PWD calcula-

ion process to obtain their joint spatial-frequency pixel-ise distributions. The PWDs have been taken using a-D scheme and calculated by a sliding window of N8 pixels, as indicated by Eq. (3), and oriented according

o the spatial anisotropy shown by the Allan variances ofhe textures. The PWD is oriented to have a minimum ofhe Rényi entropy. Sometimes the orientation is visiblee.g., 45° for even zigzag texture in Fig. 1); in other cases,easures must be compared among the possible orienta-

12 pixels. A. Texture embedded in a binary random noise back-orizontal orientation �N=8 pixels�. C. Entropy map. D. DM afterf N=8 pixels.

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ions. Once the Rényi entropies of the pixelwise distribu-ions have been measured, a decision map (DM) that de-ermines the texture discrimination is derived.

The DM is derived as follows. Every pixel in the imageas a different N-component PWD vector associated with

t; the amount of information related to a given pixel ofhe image is measured by the Rényi entropy of this PWD,ccording to Eq. (5). Distances can be measured pixel toixel by using the Rényi entropies associated with theest and the reference images, i.e.,

Di�n� = �Ri� − �RT�n��N, i � �1,2�. �6�

Here Ri is a matrix indicating the Rényi entropy of aiven reference texture i (i=1 noise sample, i=2 textureample) in a pixelwise scheme. Brackets indicate meanalues corresponding to the samples of the textures. Fi-ally, �RT�n��N indicates the mean value of the Rényi en-ropy measured in the test image, calculated in the neigh-orhood of N�N pixels around the position n. Theaximum of �D1�n� ,D2�n�� is used to determine the DM:

DM�n� = argi

max�Di�n��, i � �1,2�. �7�

As only two results are possible, a binary DM is de-ived. The white pixels can be set to indicate the locationf the isotrigon texture and the black ones to indicate theinary noise. Figure 2 shows an example of a particularexture. Here a 45° tilted PWD of AV0 has been used toeasure the entropy.The algorithm can be summarized as indicated below:Three images are involved: test image (TI), texture ref-

rence (TR), and noise reference (NR). For each image,erform the following steps:(1) Take a pixel n.(2) Consider the neighborhood of pixel n, consisting inpixels centered in n and aligned in any of the four pos-

ible directions, 0°, 45°, 90°, and 135°.(3) Use Eq. (2) to calculate the four Allan variances and

heir derived squared sums and differences (see Table 2),ssociated with pixel n.(4) Repeat the calculations with all the pixels in the

hree input images.(5) Use Eq. (3) to calculate the PWDs of all the AVs ob-

ained above, using the same set of orientations.

Table 2. Discrimi

Texture

Even

AV PWD S Sp

1 Box AV0 90° 0.9548 0.9986Triangle �AV45−AV90�2 90° 0.9490 0.99673 Cross AV45 135° 0.9755 0.98944 Zigzag AV0 45° 0.9698 0.99555 Oblong AV90 0° 0.9657 0.9931

6 Tee �AV0−AV45�2 0° 0.9654 0.98237 Wye �AV45−V135�2 0° 0.9451 0.99668 Foot �AV45−AV90�2 90° 0.9561 0.99599 El �EV0−AV90�2 90° 0.9433 0.9624

aTextures used to determine the quality of the method, indicating the best directioalues that are determined after the ROC process. S, sensitivity; Sp, specificity.

(6) Draw a set of DMs by considering all the relatedroducts (TI, TR, and NR after the same combination ofVs and PWDs) by means of Eqs. (5)–(7).(7) Select the most discriminating DM to perform the

egmentation.The method has been applied to a set of isotrigon tex-

ures, of 512�512 pixels in size, to determine its dis-rimination capabilities. The textures used in this experi-ental section belong to the nine families of isotrigon

extures tested by Maddess and Nagai [4]. Each memberf the nine families contains two version’s termed evennd odd. Therefore, the total number of cases considereds 18. The directionality of the different textures was dis-overed after applying systematically all the possible Al-an variances (or combinations) of them. Squaring of theums and differences of the AVs has been used whenimple Allan variances did not show a directional prefer-nce. An example of this situation is illustrated in Fig. 3.

This experiment has been statistically evaluated byonsidering 10 different samples of each of the 18 differ-nt isotrigon textures selected. The local analysis haseen performed using a window of 8�8 pixels in theeighborhood of the target pixel to evaluate the Allanariance �N=8�. Also a 1-D windowing scheme of N8 pixels has been used to evaluate the Rényi entropy

hrough the oriented 1-D PWDs. The segmentation per-ormance has been determined by comparing the DM withhe mask used to build the test samples. The quality pa-ameter used has been the percentage of correct decisionPCD), defined as

PCD = 100 �number of accurate pixels

total number of pixels. �8�

Texture and noise areas have been computed sepa-ately. Two indices have been derived from the PCD mea-ured in the DM, considering the two different areas (tex-ure and noise) independently. The first one is calledensitivity (S) and is defined as the probability of cor-ectly recognizing an isotrigon field as a texture. Thether is called specificity (Sp) and is defined as the prob-bility of reporting random when the texture was ran-om. The results have been interpreted through a re-eiver operating characteristic (ROC) by plotting theensitivity versus (1−specificity). The last one is usually

n Performancea

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AV PWD S Sp S&Sp

7 AV0 0° 0.9747 0.9960 0.98539 �AV45−AV90�2 90° 0.9637 0.9893 0.97655 �AV45−AV135�2 45° 0.9602 0.9840 0.97237 AV45 0° 0.9663 0.9950 0.98074 AV90 90° 0.9742 0.9964 0.98538 �AV45−AV135�2 90° 0.9565 0.9943 0.97549 �AV45+AV135�2 0° 0.9700 0.9874 0.97870 �AV45+AV90�2 90° 0.9684 0.9917 0.98019 �AV0+AV90�2 90° 0.9624 0.9742 0.9683

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eferred to as the false positive rate. This plotting is nor-ally used for binary classifier systems when the dis-

rimination threshold is varied. Figure 4 shows the ROCor the odd cross texture versus random noise. The high-st simultaneous quality performance is selected (S&Sp)o determine the threshold values [i.e., �R � values in Eq.

quaring of the differences of Allan variances. A. Target isotrigonf Allan variances �AV45−AV0�2 obtained from A �N=8 pixels�. C.lted 90° and a window of N=8 pixels.

ig. 5. Comparison of the results of this method for nine fami-ies of even isotrigon textures versus the most relevant QDA re-ults given by Maddess and Nagai [4].

ig. 3. Example of detection of a isotrigon texture (even triangle) by sexture embedded in a binary random noise background. B. Squaring o

ig. 4. Example of ROC for the odd cross texture versus randomoise. The highest simultaneous quality is selected to determinehe threshold values, i.e., �Ri� values in Eq. (9). The experimentndicated by a black circle provides the best threshold values toiscriminate this texture from binary noise, using the direction-lities indicated in Table 2 for this texture.

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6)] as, for example, the experimental result indicatedith a black circle in Fig. 4. Note that to discriminate a

exture under test, we can choose among an infinitely di-erse number of texture and noise representatives to beompared with the test texture. Which texture and noiseamples must be used is decided by the ROC process, se-ecting among all the experiments the one with the maxi-

um S&Sp value. Table 2 shows the expected quality per-

ig. 6. A. Image consisting of five isotrigon textures. B. Segmenriangle). D. Segmentation of region “2” (even zigzag). E. Segmentegions “2, 3, and 4” can be segmented straightforwardly by usntropy for the parameters given in Table 2 as region “4” but hasy eliminating the other four, as no suitable directionality was obets method [33,34]. Figure 6(h) courtesy of M. G. Forero.

ig. 7. A. Image consisting of five isotrigon textures. B. Segmenong). D. Segmentation of region “2” (even wye). E. Segmentationegions can be segmented straightforwardly by using the directiectly identified. Regions “0, 2, 3, and 4” have been recognized wM by a complex wavelets method [33,34]. Figure 7(h) courtesy

ormances when using the optimal threshold valuesreviously determined, after applying the method de-cribed here to the same set of test textures used by Mad-ess and Nagai [4]. Results for even isotrigon texturesersus binary random noise are compared in Fig. 5 withhe most relevant QDA referenced by [4].

The method can be extended to the case of discriminat-ng multiple isotrigon textures. For example, Fig. 6 pre-

of region “0” (even oblong). C. Segmentation of region “1” (evenf region “3” (even cross). F. Segmentation of region “4” (even box).e directionalities indicated in Table 2. Region “0” has the samesolated by difference with this one. Region “1” has been obtained. G. DM after previous single results. H. DM by a complex wave-

of region “0” (even El). C. Segmentation of region “1” (even ob-on “3” (even tee). F. Segmentation of region “4” (even zigzag). Alles indicated in Table 2. Only regions “1” and “4” have been per-e degree of uncertainty. G. DM after previous single results. H.. Forero.

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ents a mosaic of five different textures. Using the dis-rimination directions indicated in Table 2 for theseextures and the algorithm previously described, texturesave been segmented. Figures 6(b)–6(f) present the seg-entation results for such isotrigon mosaics. The input to

he algorithm consists of a set of the 18 isotrigon texturesonsidered (9 even and 9 odd) and the test image “A” thatppears at the top left corner of Fig. 6. The result is a la-eled DM indicating the position where every textureatches the entropy of the test image. Differences in en-

ropy between textures are not always sufficiently dis-riminative, hence textures as, for example, even triangleFig. 6(a), region 1) cannot be segmented, because the en-ropy measured is similar to the entropy measured inther regions. Hence, it requires segmentation by elimi-ation when the other regions have been already recog-ized. Also, coincidence of entropy in a given Allan com-ination is possible, as, for example, even oblong (region) and even box (region 4). In this case, region “0” haseen isolated by subtracting region 4. Figure 7 presentsnother example of multitexture isotrigon discrimination.ere regions 1 and 4 are clearly defined, but the remain-

ng regions present a certain amount of uncertainty.ingle discrimination results have been combined to drawhe complete DMs for these examples (i.e., Figs. 6(g) and(g)). A quantitative comparative study of the classifica-ion accuracy has been performed by considering textureairs as shown in Table 3. One alternative method chosenas a Gabor filtering technique optimally designed

hrough a genetic algorithm for best performance in tex-ure segmentation [31,32]. Each Gabor filter has beenuned to each texture to satisfy maximum discrimination,here a preprocessing stage based on a second-order Sh-nnon entropy has been considered for better perfor-ance. This study does not pretend to be exhaustive and

nly illustrates the limitations of other traditional meth-ds in the discrimination of those high-order textures.able 3 shows that the above-mentioned Gabor-based fil-ering method has been outperformed by the method in-roduced in this paper for the textures indicated in theable. A similar performance can be expected for otherexture pairs, including even/odd combinations. On thether hand, it is worth mentioning that the currentethod can produce a few false positives located in the

ackground (see, e.g., Fig. 3(c)), which presumably coulde avoided by means of some modification of the methodhat has not been investigated yet. For example, oneould use a measure other than max�D1�n� ,D2�n��, suchs a threshold difference, and then derive the ROCs by

Table 3. Percentage of Correct Decisions [PCD, EqThree Different S

Texture Pair Markov Method [35] Gabo

Cross–Zigzag 100Zigzag–Oblong 88

Zigzag–El 87Cross–El 71

Mean 86

arying the threshold. However, the max criterion com-ines the advantages of its simplicity and accuracy. In ad-ition to that, some postprocessing could be applied forliminating isolated points, such as through median fil-ering. On the other hand, methods based on multiscaleecompositions such as the complex wavelets failed toive an acceptable DM, as revealed by Figs. 6(h) and 7(h),hich show the results obtained with such a technique

33,34].The current method has been compared with the Meas-

ex test suite, which is a framework for quantitative com-arisons of texture classification algorithms [35]. Threeifferent texture classification algorithms have been con-idered in this study: Gauss random Markov field, Gabornergies, and fractal dimension (see Table 3). Gray-levelo-occurrence matrices were not included in this studyue to the bilevel characteristics of the isotrigon textures.f all competing approaches, the Markov-based methoderforms the best. However, the current method outper-orms other existing methods for the mean texture dataet (see Table 3).

. DISCUSSIONe have implemented a biological-inspired workable al-

orithm capable of discriminating isotrigon textures. Ourethod mimics some simple properties of the visual sys-

em such as spatial/spatial-frequency localization and ori-ntation tuning, and it displays some parallelism with theonlinearities that have been observed in the visual path-ays. Tyler [36] and later Schumer and Ganz [37] cor-

oborated that the human brain is performing spatial-requency analysis as an initial stage before patternecognition takes place. In this paper, discrimination ofnsembles of isotrigon textures is attained using fourth-rder correlation approaches as described in [3]. Experi-ental results due to Purpura et al. [3] show that there

xist mechanisms in the monkey visual cortex that caniscriminate between textures differing in certain fourth-rder spatial correlations. Purpura et al. [3] concludedhat the visual cortex must have mechanisms capable ofetecting spatial correlations involving three or moreoints in the image. Linear spatiotemporal filtering can-ot detect spatial correlations higher than second order.owever, oriented filters, combined with a special set ofonlinearities, may be a key computational component inhe cortex area V1 for detecting these higher-order corre-ations. Thus, the proposed method can be regarded as a

of Isotrigon Even Texture Pairs Corresponding tontation Methods

ring Method [35]

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Allan/RényiMethod

(This Paper)

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omputational model that implements some essential sa-ient features of the visual system, such as nonlinearity,rientation selectivity, multiple spatial correlation mecha-isms, and spatial/spatial-frequency localization.We now consider the rationale behind our approach.

he rationale of the method described here is to imple-ent such fourth-order mechanisms through the com-

ined use of the Allan variance and the PWD computa-ion. The Allan variance constitutes a neurally plausibleay to implement quadratic interactions [3]. The PWDnalysis provides a quadratic nonlinearity order as well.herefore our method provides a fourth-order spatial cor-elation approach, which appears to be among the earliestnformation extracted by our visual brain and that consti-utes the basic core mechanism for isotrigon texture dis-rimination [38]. Some criticism can be raised about theptimal window size to be used in the experiments. Theindow size is a compromise among spatial-frequency

ontents, spatial localization, and computational complex-ty. The local entropy associated to each pixel is computedsing a 1-D-oriented window of 8 pixels, extracted fromn 8�8 neighborhood of each pixel, and following a pre-iously selected orientation. It could be possible that theptimization of the parameters involved in the algorithmhould originate an even better discrimination, but thistudy is beyond the scope of the present paper. Thus, theumber of pixels used in our experiments has been chosenollowing an empirical procedure. Unless the local domains 8�8 pixels, calculations at the pixel level are per-ormed using a 1-D-oriented window of 8 pixels. Hence,nly 8 pixels are used to determine the entropy of a givenixel and consequently to establish the discriminationeasure. The number of pixels used is an even number

ccording to the definition assumed for the PWD. Theomplexity of the calculations can be reduced by usingmaller windows, but the accuracy of the method is notxpected to be better. Herein, the pixel entropy is associ-ted with spatial frequencies. Localized spatial frequen-ies are measured by considering a neighborhood of pix-ls. In such a sense, there is always an implicitonfiguration of pixels, i.e., the oriented window used toerform the calculations.In summary, our method is a texture discriminationethod tuned exclusively for isotrigon ensambles. It has

een designed as a “blind method,” because it ignores howhe textures have been generated. Hence, the problem ofdentifying the textures when generating rules are knownn advance is not an issue here. A comparison with otherexture discrimination methods has been presented. Weave selected the set of textures already studied by Mad-ess and Nagai [4] in order to have a qualitative anduantitative reference with our results. The examplesresented indicate that sharp boundaries are obtained foroise–texture discrimination and comparable to the sizef the window in the texture mosaics (see Figs. 6 and 7).

. CONCLUSIONSnew method for discriminating isotrigon textures ver-

us binary noise and isotrigon textures of different typesas been developed and evaluated. This new method isased in a particular Rényi entropy measure, through the

llan variances of the textures. The method has beenvaluated by means of a systematic study applied to 18ifferent textures, showing that this method makes pos-ible a high discrimination performance similar to and inome cases better than other known methods such asDA. The method has been extended to the multitextureiscrimination task. Results have shown that the methodrovides a good discrimination performance even in suchases. Although the multitexture discrimination problemas not yet been completely solved by this technique, theesults are significantly better than other discriminationethods, such as wavelets, indicating that further re-

earch based in this approximation may lead to a full dis-rimination of isotrigon multitextures.

Also, this method provides new avenues to deal withhe problem of discriminating high-order textures in gen-ral and, more important, shares relevant characteristicsith the visual system. Some of the most commonly rec-gnized properties of cortical cells are spatial localization,estricted spatial frequency, and orientation tuning. Ourethod relies on these properties, and it provides a

ourth-order correlation procedure. Although its psycho-hysical validation is beyond the scope of the present pa-er, it opens up new research directions to analyze thesychophysical response of the Rényi directional entro-ies for the discrimination of isotrigon stimuli. Compre-ensive comparisons with other segmentation techniquesre left for further work. In addition to that, further re-earch is required to analyze the new trilevel isotrigonextures developed by Maddess et al. [2]. Also, compara-ive measures other than the sensitivity and the specific-ty that have been used here can be considered as valu-ble.

CKNOWLEDGMENTShis research has been supported by the followingrojects: TEC2004-00834, TEC2005-24739-E, TEC2005-4046-E, and 20045OE184 sponsored by the Spanishinistry of Science; and PI040765 sponsored by the Span-

sh Ministry of Health. We thank Ted Maddess for fruitfuliscussions and for providing us with the MATLAB code toenerate the isotrigon textures presented in this paper.e thank Manuel Forero and M. E. Barilla for providing

s with Figs. 6H and 7H.

EFERENCES1. B. Julesz, E. N. Gilbert, and J. D. Victor, “Visual

discrimination of textures with identical third-orderstatistics,” Biol. Cybern. 31, 137–140 (1978).

2. T. Maddess, Y. Nagai, A. C. James, and A. Ankiewcz,“Binary and ternary textures containing higher-orderspatial correlations,” Vision Res. 44, 1093–1113 (2004).

3. K. P. Purpura, J. D. Victor, and E. Katz, “Striate cortexextracts higher-order spatial correlations from visualtextures,” Proc. Natl. Acad. Sci. U.S.A. 91, 8482–8486(1994).

4. T. Maddess and Y. Nagai, “Discriminating isotrigontextures,” Vision Res. 41, 3837–3860 (2001).

5. D. W. Allan, “Statistics of atomic frequency standard,” Proc.IEEE 54, 223–231 (1966).

6. R. A. Johnson and D. W. Wichern, Applied MultivariateStatistical Analysis (Prentice Hall, 1992).

7. J. L. Starck, E. Candes, and D. Donoho, “The curvelet

1

1

1

1

1

1

1

1

1

1

2

22

2

2

2

22

2

2

3

3

3

3

3

3

3

3

3


transform for image denoising,” IEEE Trans. ImageProcess. 11, 670–684 (2002).

8. M. N. Do and M. Vetterli, “The contourlet transform: anefficient directional multiresolution image representation,”IEEE Trans. Image Process. 14, 2091–2106 (2005).

9. D. Donoho, “Wedgelets: nearly minimax estimation ofedges,” Ann. Stat. 27, 859–867 (1999).

0. V. Velisavljevic, B. Beferull-Lozano, M. Vetterli, and P. L.Dragotti, “Directionlets: anisotropic multidirectionalrepresentation with separable filtering,” IEEE Trans.Image Process. 15, 1916–1933 (2006).

1. A. Cumani, “Edge detection in multispectral images,”Comput. Vis. Graph. Image Process. 53, 40–51 (1991).

2. L. D. Jacobson and H. Wechsler, “Joint spatial/spatial-frequency representation,” Signal Process. 14, 37–68(1988).

3. E. Wigner, “On the quantum correction for thermodynamicequilibrium,” Phys. Rev. 40, 749–759 (1932).

4. L. Cohen, “Generalized phase–space distributionfunctions,” J. Math. Phys. 7, 781–786 (1966).

5. T. A. C. M. Claasen and W. F. G. Mecklenbräuker, “TheWigner distribution—a tool for time–frequency analysis.Part I. Continuous-time signals,” Philips J. Res. 35,237–250 (1980).

6. T. A. C. M. Claasen and W. F. G. Mecklenbräuker, “TheWigner distribution—a tool for time–frequency analysis.Part II. Discrete-time signals,” Philips J. Res. 35, 276–300(1980).

7. T. A. C. M. Claasen and W. F. G. Mecklenbräuker, “TheWigner distribution—a tool for time–frequency analysis.Part III. Relations with other time–frequencytransformations,” Philips J. Res. 35, 372–389 (1980).

8. K. H. Brenner, “A discrete version of the Wignerdistribution function,” EURASIP J. Appl. Signal Process.2005, 307–309.

9. L. Stankovic, “A measure of some time–frequencydistributions concentration,” Signal Process. 81, 623–631(2001).

0. C. E. Shannon and W. Weaver, The Mathematical Theory ofCommunication (University of Illinois Press, 1949).

1. N. Wiener, Cybernetics (Wiley, 1948).2. A. Rényi, “Some fundamental questions of information

theory,” in Selected Papers of Alfréd Rényi, P. Turán, ed.(Akadémiai Kiadó, 1976), pp. 526–552 (1976). [Originallypublished in Magy. Tud. Akad. Mat. Fiz Tud. Oszt. Kozl.,10, 251–282 (1960)].

3. W. J. Williams, M. L. Brown, and A. O. Hero, “Uncertainty,

information and time–frequency distributions,” Proc. SPIE1566, 144–156 (1991).

4. T. H. Sang and W. J. Williams, “Rényi information andsignal dependent optimal kernel design.” in Proceedings ofthe IEEE International Conference on Acoustics, Speech,and Signal Processing (IEEE, 1995), Vol. 2, pp. 997–1000.

5. P. Flandrin, R. G. Baraniuk, and O. Michel,“Time–frequency complexity and information,” inProceedings of the IEEE International Conference onAcoustics, Speech, and Signal Processing (IEEE, 1994), Vol.3, pp. 329–332.

6. R. Eisberg and R. Resnick, Quantum Physics (Wiley, 1974).7. G. Süßmann, “Uncertainty relation: from inequality to

equality,” Z. Naturforsch. 52, 49–52 (1997).8. J. J. Wlodarz, “Entropy and Wigner distribution functions

revisited,” Int. J. Theor. Phys. 42, 1075–1084 (2003).9. J. Ville, “Théorie and applications de la Notion de Signal

Analytique,” Cables Transm. 2A, 61–74 (1948).0. D. Dragoman, “Applications of the Wigner distribution

function in signal processing,” EURASIP J. Appl. SignalProcess. 10, 1520–1534 (2005).

1. B. Yegnanarayana, P. Pavan Kumar, and S. Das, “One-dimensional Gabor filtering for texture edge detection,”Proceedings of Indian Conference on Computer Vision,Graphics and Image Processing (IEEE, 1998), pp. 231–237.

2. MATLAB code for segmentation and classification ofmultitexture images is available from http://www.cse.iitk.ac.in/~amit/courses/768/00/rajrup/.

3. P. de Rivaz and N. Kingsbury, “Complex wavelet featuresfor fast texture image retrieval,” Proceedings of IEEEConference on Image Processing (IEEE, 1999), pp. 25–28.

4. M. E. Barilla, M. G. Forero, and M. Spann, “Color-basedtexture segmentation,” in Information Optics, 5thInternational Workshop, G. Cristóbal, B. Javidi, and S.Vallmitjana, eds. (AIP, 2006), pp. 401–409.

5. G. Smith and I. Burns, “MeasTex image texture databaseand test suite” (1997). Available online at http://www.texturesynthesis.com/meastex/meastex.html.

6. C. W. Tyler, “Stereoscopic tilt and size effects,” Perception4, 287–192 (1975).

7. R. Schumer and L. Ganz, “Independent stereoscopicchannels for different extents of spatial pooling,” VisionRes. 19, 1303–1314 (1979).

8. T. Maddess and Y. Nagai, “Lessons from biologicalprocessing of image texture,” in International CongressSeries (Elsevier, 2004), Vol. 1269, pp. 26–29.

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