Unsupervised Texture Segmentation Using Gabor Filters

7/29/2019 Unsupervised Texture Segmentation Using Gabor Filters

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Unsupervised Texture Segmentation Using GaborFilters1

Ani1 K. JainDepartment of Computer Science

Michigan State UniversityEast Lansing, MI 48824-1027, U.S.A.

j ai n@cpswh. cps. msu. edu

Abstract

We presenf a texture segmentation algorithm inspired bythe multi-chann el filtering theory fo r visual information pro-cessing in the early stages of human visual system. Thechannels are characterized by a bank of Gabor filters thatnearly uniformly coversthe spatial-frequency domain. Wepropose a systematic filter selection scheme which is basedon reconstruction ofthe nput im age from the filtered images.Texture features are obtained by sub jecting each (selected)filtered image to a nonlinear transformation and computinga measure of energy in a window around each pixel.Anunsupervised square-emr clustering algorithmis then usedto integrate the feature images and produce a segm entation.A simple procedure to incorporate spatial adjacencyinfor-mation in the clustering process is also proposed. We reportexperiments on imageswth natural texturesas well as arti-ficial textures with identical 2nd- and 3 rd-order statistics.

Introduction

Image segmentation is a difficult yet very important task in manyimage analysis or compu ter vision applications. Differences inthe mean gray level or in color in small neighborhoods aloneaxenot always sufficient for image segmentation. Rather, one hasto rely on differences in the spatial arrangement of gray valuesof neighboring pixels- hat is, on differences in texture. Theproblem of segmenting an image based on textural cues is referredto as texture segmentation p roblem .

The diversity of natural and artificial textures makes it im-possible to give a universal definition of texture.A large numberof techniques for analyzing image texture has been proposed inthe past two decades [ l l , 221. In this paper, we focus on a par-ticular approach to texture analysis which is referred to as themulti-channel filtering approach. This approach is inspired by amulti-channel filtering theory for processing visual information inthe early stages of the human visual system. First proposed byCampbell & Robson [4], the theory holds that the visual systemdecomposes the retinal image into a number of filtered images,each of which contains intensity variations over a narrow range

of frequency (size) and orientation. Th e psychop hysical experi-ments that suggested such a decomposition used various gratingpattems as stimuli and were based on adaptation techniques [4].Subsequent psychophysiological experiments provided additionalevidence supporting the theory[lo].

This work was supported in part by the National Science Foundationinfrastructure grant CDA-8806599, nd by a grantfromE. . Du Pont De Nemours& Company Inc.

Farshid FarrokhniaDepartment of Electrical Engineering

Michigan State UniversityEast Lansing, MI 48824-1226, [email protected]

The multi-channel filtering approach to texture analysis is in-tuitively appealing because the dominant spatial-frequency com-ponents of different texturesare different. An important advantageof the multi-channel filtering approach to texture analysis is thatone can use simple statistics of gray values in the filtered images astexture features. Th is simplicity is the direct result of decomp osingthe original image into several filtered images with limited spec-tral information. The main issues involvedin the multi-channel

filtering approach to texture analysis are:1) functional character-ization of the channels an d the number of channels,2) extractionof appropriate texture features from the filtered images.3) th e re-lationship between channels (depen dent vs. independen t), and 4)integration of texture features from different channels to producea segmentation. Different multi-chann el filtering techniques thatare proposed in the literature differ in their approach to oneormore of the above issues.

We use a bank of Gabor filters to characterize the chan-nels. We show that the filter set forms an approx imate basis fora wavelet transform, with the Gabo r function as the wavelet. Wepropose a systematic filter selection scheme which is based onreconstruction of the input imag e from the filtered images. Each(selected) filtered imageis subjected to a bou nded non linear trans-formation that behaves as a blob detector . Th e combination ofmulti-channel filtering and the nonlinear stages canbe viewed as

performing a multi-scale blob detection. Texture discriminationis associated with differences in the attributes of these blobs indifferent regions. A statistical approach is then used where theattributes of the blobs are captured by texture features defined bya measure of energy in a small window around each pixel ineach response image. This process generates on e feature imagecorresponding to each filtered image (see Figure 1). The size ofthe window for each response image is determined using a simpleformula involving the radial frequency to which the correspond-ing filter is tuned. A squ are-error clustering algorithm is then usedto identify the texture categories. A simple procedure for inclu-sion of contextual (spatial adjacency) information in the clusteringprocess is also proposed.

2 Channel Characterization

We represent the channels with a bank of two-dimensional Gaborfilters. A two-dimensional Gabor function consists of a sinusoidalplane wave of some frequency and orientation, modulated by atwo-dimensional Gaussian env elope.A canonical Gabo r filter inthe spatial domain is given by

9OCH2930-6~/000(M014$01.W 1990 IEEE
mailto:[email protected]:[email protected]:[email protected]:[email protected]


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Input Image

Bank of Gabor Filters

(l .r ]ilteredmagesNonlinearitv

Local 'Energy' Computat ion

Square-Error ClusteringI

Segmented Image

Figure 1: An overview of the tex ture segmentation algorithm.

where uo and q5 are the frequency and phase of the sinusoidal planewave along the z-axis (i.e. the0" orientation), and u z an d u y arethe space constants of the Gaussian envelope along the z- andy-axis, respectively. A Gabor filter with arbitrary orientation,00 ,can be obtained via a rigid rotation of theI-y coordinate system.These two-dimensional functions have been shown to be good fitsto the receptive field profiles of simple cells in the striate cortex

The frequency- and orientation-selective properties of a Gab orfilter are more explicit in its frequency domain representation.With 4 = 0, the Fourier transform of the Gabor function in(1) isreal-valued and given by

[18,71.

where u u = l /2iruz, U, = l /2 i ruy, and A = 2xu,u,. The Fourierdomain representation in (2) specifies the amount by which the fil-ter modifies or modulates each frequency component of the inputimage. Such representationsare, herefore, referred to as modula-tion transfer functions(MTF). Figure 2 shows an even-symmetricGabor filter and its MTF in a 64 x 64 array.

Texture segmentation requires simultaneous measurementsinboth the spatial and the spatial-frequency domains. Filters withsmaller bandwidths in the spatial-frequency domain are more de-sirable because they allows us to make finer distinctions amongdifferent textures. On the other hand, accurate localization oftexture boundaries requires filters thatare localized in the spatialdomain. However, the effective width of a filter in the spatialdomain and its bandwidth in the spatial-frequency domainare in -versely related. An important property o f Gab or filters is that theyhave optimal joint localization,or resolution, in both the spatialand the spatial-frequency domains[8].

The use of Gabo r filters in texture analysis is not new. Forexample, Tumer [21 ] used a fixed set of Gabor filters and demo n-

c

Figure 2: (a) An even-symmetricGabor filter in the spatial do-main. (b) CorrespondingMTF. T h e origin is at ( r , ) = (32,32) .

scrated their potential fo r texture discrimination. Similarly, Perry& Lowe 1191 us e a fixed set of Gabor filters in their texture seg-mentation algorithm. Boviket at. 11 1 have used complex Gaborfilters, where the real part of each filter is an even-symmetric Ga-bor filter (i.e., 4 = 0) and the imaginary part is an odd-symmetricGabor filter (i.e., 4 = n/2). Instead of using a fixed set of filters,Bovik er al . apply a simple peak finding algorithm to the powerspectrum of the image in order to determine the radial frequen-cies of the appropriate Gabor filters.In our texture segmentationalgorithm, we model the channels with a fu ed set of Gabor filtersthat preserve almost all the information in the input image.

2.1 Choice of Filter Parameters

We implement each Gabor filter as a discrete realization of theMTF in (2). We use four values of orientation00 : O", 45", go",and 135". For an image array with a width of N, pixels, whereN , is a power of 2, the following valuesof radial frequency u oare used:

Note that the radial frequenciesare 1 octave apart. (The frequencybandwidth, in octaves, from frequencyf l to frequency fi is givenby log,(f2/fl).) We let the orientation and frequency bandwidthsof each filter be 45" an d 1 octave, respectively. Several experi-ments have shown that the frequency bandwidth of simple cellsin the visual cortex is about1 Octave [20]. P sychophysical exper-iments show that the resolution o f the orientation tuning ability ofthe human visual system is as high as5 degrees. Therefore, ingeneral, finer quantization of orientation will be needed. The re-striction to four orien tationsis made for computational efficiency.

The above choice of the radial frequencies, guarantees thatthe passband of the filter with the highest radial frequency,viz.

(N J4 )f i cycledimage-width, falls inside the image array. Foran image with 256 columns, for example, a total of 28 filterscan be used - orientations and 7 frequencies. Note that filterswith very low radial frequencies (e.g.,1f i nd 2f i cycled image-width) can oftenbe left out, because they capture spatial variationsthat are too large to correspond to texture.To assure that the filters


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Figure 3: Th e filter set in the spatial-frequency domain(256 x 256). There are a total of 28 Gabo r filters. Onlythe half-peak support of the filters areshown. The origin is at(row,col) = (128,128).

do not respond to regions with constant intensity, we have set theMTF of each filter at ( u , o ) = (0,O) o zero. As a result eachfiltered image has a zero mean.

The set of filters used in our algorithm results in nearly uni-form coverage of the frequency domain (Figure3). This filter

set constitutes an approximate basis for a wavelet transform, withthe Gabor filter as the wavelet. The wavelet transform is closel)related to the window F ourier transform. Howev er, unlike win-dow Fourier transforms where the window is fixed, in a wavelettransform the window size is allowed to change according to fre-quency [17]. Intuitively, a wavelet transform canbe interpretedas a band-pass filtering operation on the inpu t image. The Gaborfunction is an admissible wavelet, however, it does not result in anorthogonal decomposition. This means that a wavelet transformbased on Gabor w aveletsis redundant. A decomposition obtainedby our filter set is nearly orthogonal, as the amount of overlapbetween the filters (in the spatial-frequency domain)is small.

Figure 4 shows examples of filtered images for an imagecontaining straw matting@55) and wood grain (D68) texturesfrom the photographic albumof textures by Brodatz[2]. The abil-ity of the filters to exploit differences in spatial-frequency (size)

and orientation in the tw o texturesis evident in these images. T hedifferences in the strength of the responses in regions with dif-ferent textures is the key to the multi-channel approach to textureanalysis. To maximize visibility, each filtered image has beenscaled to full contrast.

2.2 Filter Selection

Using only a subset of the filtered images can reduce the com-putational burden at later stages, because this directly translatesinto a reduction in the number of texture features. Lets(z, ) bethe reconstruction of the input image obtained by adding all thefiltered images. Let .^(z,y) be th e partial reconstruction of theimage obtained by adding a given subset of filtered images. T hee m r involved in us ingS(z ,y ) instead of s(z, y ) ca n be measuredby

SS E = C S^(X,Y) - x 1 Y)12Z, Y

The fraction of intensity variations ins(z, ) that is explained byS^z,y) ca n be measured by the coefficient of determination

SSER 2 = 1 - -

SSTOT

Figure 4 Examplesof filtered imagesfor 055-068 texturepair(128x256). (a) Input image. @e) Filtered images correspondingto Gaborfilters uned to 16 fi h-w and to 0,450, O , nd135O.respectively.

whereSSTOT = C s ( ~ , y ) ~.

Z,Y

Note that s(z,y) has a mean of zero, since the mean gray valueof each filtered image is zero.

For computational efficiency, we determine the best subset01 .he filtered imag es (filters) by the follow ing subo ptimal sequen-tial forward selection procedure:

1. Select the filtered image that best approximatess(z, y), i.e.

2. Select the next filtered image that together with previously

3. Repeat Step 2 until R2 2 0.95.

results in the highest value ofR 2 .

selected filtered image(s) best approximates(z, ).

A minimum value of 0.95 fo r R 2 means that we will use only asmany filtered images as necessary to account for at least95% of

the intensity variations ins(z,

y) .Le t r;(z,y) be the z th filtered image and R, (u ,v ) e its Dis-crete Fourier Transform. The amountof overlap between theMTFs of the Gabor filters in our filter set is small. Therefore,the total energy E in S ( I , y ) can be approximated by

n

E R ~ E ; ,i= 1

whereEi = [ ~ i ( z ? y ) ] ~ ( R , ( U , V ) ~ ~ .

Z, Y u,u

Now, it is easily verified that for any subsetA of filtered images

An approximate filter selection wo uld then consists of comp utingE; fo r z = 1, , . These energies can be computed in the Fourierdomain, hence avoiding unnecessary inverse Fourier transforms.We then sort the filters (channels) based on their energy and pickas many filters as needed to achieveR2 2 0.95. Computationally,this procedure is much more efficient than the sequential forwardselection procedure described before.

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3 Computing Feature Images

We use the following procedure to compute features from eachfiltered image. First, each filtered image is subjected to a non-linear transformation. Specifically, we use the following boundednonlinearity

1 -Q (4= anh((Yt) = 7 (3 )

where (Y is a constant. This nonlinearity bears certain similari-

ties to the sigmoidal activation function used in artificial neuralnetworks. In our experiments, we have used an empirical valueof (Y = 0.25 which results in a rapidly saturating, threshold-liketransformation. As a result, the application of the nonlinearitytransforms the sinusoidal modulations in the filtered images tosquare modulations and, therefore, can be interpreted asa blobdetector. However, the detected blobsare not binary, and unlikethe blobs detected by Voorhees& Poggio [23] they are not nec-essarily isolated from each other. Also, since each filtered imagehas a zero mean and the nonlinearity in (3) is odd-symmetric, bothdark and light blobsare detected.

Instead of identifying individual blobs and measuring their at-tributes, we simply comp ute the average absolute deviation (AAD)from the mean in small overlapping windows. This is similar tothe texture energy measure that w as first proposed by Laws [15].Formally, the feature imag eek(z, y) corresponding to filtered im-

ag e ~( i , ) is given by

where $( ) is the nonlinear function in (3) andW,, is an M x Mwindow centered at the pixel with coordinates(z, y ).

The size, M , of the averaging window in(4) s an importantparameter. More reliable measurementof texture features calls forlarger window sizes. O n the other hand, mo re accurate localiza-tion of region boundaries calls for smaller windows. Furthermore,using Gaussian weighted windows, rather than unweighted win-dows, is likely to result in more accurate localization of textureboundaries. For each filtered image we use a Gaussian windowwhose space constant U is proportional to theaverage size of theintensity variations in the image. For a Gabo r filter with radialfrequency u o his average size is given by

T = N,/uo pixels, ( 5 )

where N , is the width (number of columns) of the image.Wefound a n NN 0.52 to be appropriate in mostof our segmentationexperiments.

4 Integrating Feature Images

Having obtained the feature images, the main question is how tointegrate features corresponding to different filters to produce asegmentation. Lets assume that thereare K texture categories,CI, . CK, resent in the image. If our texture featuresare capa-ble of discriminating these categories then the patterns belongingto each category will form a cluster in the feature space whichis compact and isolated from clusters corresponding to other

texture categories. Pattern clustering algorithmsare ideal vehiclesfo r recovering such clusters in the feature space. In our texturesegmentation experiments we use a square-error clustering algo-rithm known as CLUSTER [13]. Prior to clustering each featureis normalized to have a zero mean and a constant variance. Thisnormalization is intende d to avoid the dom ination of features with

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small numerical ranges by those with larger ranges.Clustering a large number of pattems becomes computation-

ally demanding. Therefore, we first cluster a small randomly se-lected subset of pattems into a specified number of clusters. Pat-rems in each cluster are given a generic category label that distin-guishes them from those in other clusters. These labeled panemsare then used as training patterns to classify pattems (pixels) inthe entire image using a minimum distance classifier.

In texture segmentation, neighboring pixelsare very likelyto belong to the same texture category. We propose a simplemethod that incorporates the spatial adjacency information directlyin the clustering process. This is achieved by includ ing the spatialcoordinates of the pixels astwo additional features.

5 Experimental Results

We now apply our texture segmentation algorithm to severalim-ages in order to demo nstrate its performance. These images arecreated by collaging subim ages of natural as well as artificial tex-tures. We start by a total of 20 Gabor filters in each case. Eachfilter is tuned to o ne of the four orientations and to o ne of the fivehighest radial frequencies.For an image with a width of 256 pix-els, for example,4 f i , 8 d ? , 1 6 f i , 3 2 f i , a n d6 4 f i cycleshmage-width radial frequenciesare used. We then use our filter selectionscheme to determine a subset of filtered images that achieves an

(a) (b)

Figure 5: (a) 055-068 texture pair. (b ) Two-category segmen-tation obtained using a totalof 13 Gabor filters.

(b) (c)

Figure 6: (a) A 256 x 256 image containing four different Gaus-sian Markov random field textures. (b ) Four-category segmen-tation obtained using a total of11 Gabor filters. (c) Same as b,but with pixel coordinates usedas additional features.


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R2 value of at least 0.95. The number of randomly selected fea-ture vectors, that are used as input to the clustering program, isproportional to the size of the input image. For a 256x 256 im-age, for example, 4000 pattems are selected at random, which isabout 6% of the total number of pattems. T he same percentage isused in all the following experiments. The segmentation resultsare displayed as gray-level images, where regions belonging todifferent categoriesare shown with different gray levels.

Figure 5 shows the segmentation results for the055-068texture pair. The algorithm successfully discriminatesthe tw otextured regions and detects the boundary between them quite ac-curately. The segm entation with pixel coordina tes included asadditional features was essentially the same for this example.

Figure 6 (a) shows a 256x 256 image containing four dif-ferent Gaussian Markov random field (GMRF) textures generatedusing non-causal finite lattice models[ 5 ] . These four textures cannot be discrimin ated based on their mean gray values. T he four-category segmentation of the image is shown in Figure 6 (b). Thesegmentation improves considerably when pixel coordinatesareused as ad ditional features (Figure 6 (c)).

Figure 7 (a) shows another256 x 256 im age containing naturaltextures D77, D55, D84, D 17, and D 24 from the Brodatz album.The five-category segm entation of this imag e, using13 Gaborfilters and the pixel coordinates, is shown in Figure7 (b).

Figure 8 (a) shows a 512 x 512 image containing sixteennatural textures, also from the Brod atz album. Ou r filter selection

(with a threshold of 0.95 fo rR 2 ) indicated that only 14 filteredimagesare sufficient. H owev er, the resulting segm entation was notvery good. Using all20 filtered imag es (an d the pixel coordinates)we obtained the 16-category segmentation in Figure8 (b). Recallthat the fitting criterion in our filter selection sc heme is com putedglobally over the entire image.A larger threshold forR 2 should,therefore, be used when one or more texture. categories occupy asmall fraction of the image.

Figure 9 shows the segmentation of a number of texture pairimages that have been used in the psychophysical studies of tex-ture perception [13]. T he two textures in the L-and-+ texturepair have identical power spectra. The textures in the even-odd

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+ + + + + L L L L L L + + + + +

+ + + + + L L L L L L + + + + +

+ + + + + L L L L L L + + + + +

+ + + + + L L L L L L + + + + +

+ + + + + L L L L L L + + + + +

+ + + + + + + + + + + + + + + ++ + + + + + + + + + + + + + + ++ + + + + + + + + + + + + + + ++ + + + + + + + + + + + + + + +

+ + + + + + + + + + + + + + + +

(a) (b)

Figure7 : (a) A 25 6 x 25 6 image containing five natural textures(D77, D55, D84, D17, and D24) from the Bmdatz album. @)Five-category segmentation obtained using a totalof 13 Gaborfilters and the pixel coordinates.

(a) (b )

Figure 8: (a) A 51 2 x 51 2 image containing sixteen naturaltextures (D29, D12, D17, D55; D32, D5, D8 4, D68; D77, D24.D9 , D4; D3, D33, D51, D54) from the Brodatz album@) 16-categoly segmentation obtainedusing a total of 20 G abor filtersand the pixel coordinates.

(a) (b ) (c) (d lFigure 9: Segmentationof texture pairs that havebeen used in the psychophysical studiesof texture perception.All images are256 x 256. The number of Gabor filters used varied between8 - 11 . (a) Land-+. (b) even-odd.(c) triangle-arrow (d )S-IO.

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Table 1: Percentage of misclassified pixels.

texture pair have identical third-orders d e ~ tatistics. The tex-tures in the m-arr an d S-IO texture pairs, on the other hand,have identical second-order statistics. The even -odd andtri-arrtextures are two of the counter-examples to the original Juleszconjecture that texture pairs with identical second-order statisticscannot be preattentively discriminated[14]. While the first threetexture pairs in Figure9 are easily discriminated, theS-IOtexturepair is not preattentively discriminable. Ou r algorithm appears toperform a s predicted by preattentive texture discrimination by hu-ma n - he algorithm successfully segments thefirst three texturepairs, but fails todo so for the S-IO texture pair.

The lack of appropriate quantitative measures of the goodnessof a segmen tation make s it very difficult to evaluate and com paretexture segmentation algorithms. One simple criterion that is of-ten used is the percentage of misclassified pixels. Table1 givesthe percentage of misclassified pixels for the segmentation ex-periments reported here. As seen in this table, there is a clearadvantage in using the pixel coordinates (spatial information) asadditional features.

6 Summary and Conclusions

We have presented an unsupervised texture segmentationalgo-rithm that uses a fixed set of Gabor filters.Our choice of Gaborfilters and their parameters were motivated by a goal to constructan approximate basis for a wavelet transform. The use of anon-linear transformation following the linear filtering operations hasbeen suggested as one way to account for inherently nonlinearnature of biological visual systems[9]. We argued that the lo-calized filtering by our Gabor filter set followed by a nonlineartransformation can be interpreted as a multi-scale blob detectionoperation.

One of the limitations of our texture segmentation algorithmis the lack of a criterionfo r choosing the value ofa in the non-linear transformation. Also, the algorithm assumes that differentchannels are independent from each other. However, there is psy-chophysical and phys iological evidence indicating inhibitory inter-actions between different spatial frequency channels[9]. Allowinginhibitory interactions among the channels is shownto have thepotential to reduce the effective dimen sionality of the feature space131.

In our texture segmentation algorithm, we assumed that thenumber of texture categories is given. The pattem clustering tech-nique that is used by ou r segmentation algorithmwill producea clustering with the de sired number of c lusters, even if it doesnot make sense. We believe that an integrated approach that

uses both a region-basedand an edge-based segmentation can beused to resolve the question of determining the number of texturecategories. Mal& 8~ erona [16], for example, have developeda multi-channel filtering technique that produces edge-based seg-mentations. The basic idea isto generate an oversegmented solu-

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tion using our region-based texture segmentano rl algorithm. T hen,based on the evidence f or an edg e provided by the edge-basedsegmentation,one can test the validity of boundaries between re-gions in the oversegmented solution. This integrated approachiscurrently being investigated.

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Unsupervised Texture Segmentation Using Gabor Filters

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