A New Wavelet-Based Texture Descriptor forImage Retrieval

E. de Ves1, A. Ruedin2, D. Acevedo2, X. Benavent3, and L. Seijas2

1 Computer Science Department, University of Valencia,2 Departamento de Computación, Facultad de Ciencias Exactas y Naturales,

Universidad de Buenos Aires,3 Robotics Institute, University of Valencia{Esther.Deves,Xaro.Benavent}@uv.es

Abstract. This paper presents a novel texture descriptor based on thewavelet transform. First, we will consider vertical and horizontal coef-ficients at the same position as the components of a bivariate randomvector. The magnitud and angle of these vectors are computed and itshistograms are analyzed. This empirical magnitud histogram is modelledby using a gamma distribution (pdf). As a result, the feature extractionstep consists of estimating the gamma parameters using the maximalikelihood estimator and computing the circular histograms of angles.The similarity measurement step is done by means of the well-knownKullback-Leibler divergence. Finally, retrieval experiments are done us-ing the Brodatz texture collection obtaining a good performance of thisnew texture descriptor. We compare two wavelet transforms, with andwithout downsampling, and show the advantage of the second one, whichis translation invariant, for the construction of our texture descriptor.

Keywords: Texture descriptor, Wavelet Transform, Image retrieval.

1 Introduction

The increasing amount of information available in today’s world raises the needto retrieve relevant data efficiently. Unlike text-based retrieval, where key wordsare successfully used to index documents, content-based image retrieval posesup-front the fundamental questions of how to extract useful image features andhow to use them for intuitive retrieval [11] [5]. Interest in content-based imageretrieval (CBIR) systems has been growing in the last few years. This interest hasbeen motivated by the increasing number of image databases which need effectiveand efficient techniques for retrieving multimedia information. Attempts havebeen made to develop general purpose image retrieval systems based on multiplefeatures (e.g. color, shape and texture), which describe the image content [10].

The new visual information retrieval systems extract visual image features usu-ally related to color, texture and shape from each image stored in the database,and use this representation to compare images by means of a similarity measure.

W.G. Kropatsch, M. Kampel, and A. Hanbury (Eds.): CAIP 2007, LNCS 4673, pp. 895–902, 2007.c© Springer-Verlag Berlin Heidelberg 2007

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The features extracted from an image can be classified into low and high levelfeatures, which are normally obtained by combining low level features with a rea-sonable predefined model. The low level features are obtained by preprocessingeach image in the database. Among these characteristics we can mention thosewith chromatic information, those related to textures present in the image andthose related to the shape of objects in the image. Generally speaking, the struc-ture of a CBIR system is composed of two main modules: the feature extractionmodule and the similarity measurement module. In the latter a distance betweenthe query image and each image in the database is computed, by making use ofthe extracted features. Each image obtains a relevance score related to the queryimage in order to rank the database.

Texture analysis plays an important role in many image processing tasks.There are a great number of references for texture analysis and particularly tex-ture classification. A central point in texture analysis is the definition of goodfeatures to characterize textures, that can be useful for content-based retrieval.Gabor features for texture classification followed by a linear discriminant analysiswas used in [2]. Ayala proposes in [1] a new descriptors for binary and gray-scaletextures based on defined spatial size distributions (SSD). In [7] a functionaldescriptor of the multivariate random closed set defined from the texture is pro-posed as features to describe grayscale textures. Randen presents an interestingcomparative study of different filtering approaches where the features used fortexture classification are obtained from signal processing techniques like Lawand Gabor filters, wavelet transforms or the discrete cosine transform [12].

Wavelets have been successfully applied to different image processing appli-cations. When transformed, an image is represented as the sum of its details atdifferent scales and orientations, plus a coarse approximation of the image. Thisnaturally led to consider wavelets as a possible tool for texture classification.

Some traditional approaches use the energy of the wavelet coefficients in eachsubband as texture descriptors, under the assumption that the energy of thedistribution in the frequency domain identifies textures. A natural extension ofthe mentioned method consists in shaping every texture by means of marginaldensities of the wavelet coefficients in every subband. This is justified by psy-chological studies that suggest that two homogeneous textures are difficult todistinguish if they produce similar marginal distributions as response to a bankof filters ([8]). A number of authors have observed that wavelet subband coeffi-cients have highly non-Gaussian statistics [3], [14]. Numerous tests show that thenormalized histograms of wavelet coefficients in each detail subband can be wellapproximated with a Generalized Gaussian Distribution (GGD). This model hasbeen used with the Kullback-Leibler divergence as a similarity measure amongdistributions [6].

In this work, by applying a wavelet transform and considering at each scalethe pair (horizontal detail, vertical detail) associated with each position, we ex-tract information on on the importance (contrast) and orientation of the edgesin the image. We present a novel texture descriptor, used for content-based re-trieval, based on the wavelet transform. The basic idea consists in modeling the

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distributions of moduli of wavelet detail coefficients in each decomposition level,and computing their empirical angle histogram at each level (assuming verticaland horizontal coefficients at the same position are the components of a bivari-ate random vector). Our tests indicate that the gamma density function (pdf),described by two parameters, is a reasonable model for the moduli coefficientshistogram. Thus, we propose to characterize each texture in the database by in-formation extracted both from the moduli and the orientation of the coefficients.

Section 2 introduces an illustrative example of our work, and explains our ap-proach in detail. In section 3 we present experimental results which evaluate theperformance of our texture descriptors. Finally, in section 4 we have concludingremarks.

2 Modelling Coefficients Distribution of WaveletTransform

2.1 Analyzing the Joint Histograms of Wavelet Coefficients

For our tests we have chosen the orthogonal Daubechies 4 wavelet. In the tradi-tional wavelet transform ([4] [9]), coefficients are calculated via convolutions with2 filters (lowpass and highpass). This is followed by downsampling operations,which prevent the wavelet from being invariant to translations. For comparison,in our tests we have included another wavelet transform that is translation invari-ant, proposed by Mallat [9]. It is calculated with the so–called à trous algorithm,via convolutions with 2 filters, but has no downsampling operations, so that itgives a redundant representation of an image. To capture lower frequencies ateach step of the transform, the filters are upsampled.

Daubechies 4 filters are lowpass [ h3 h2 h1 h0] = 14√

2[3+e, 1−e, 3−e, 1+e],

with e =√

3, and highpass [−h0 h1 − h2 h3].The filters for Mallat’s translation invariant transform correspond to a a

biorthogonal wavelet: lowpass√

2[ a b b a ] and highpass [ c −c ], with a = 0.125,b = 0.375 and c =

√2/2.

A previous step before proposing a new model for the joint histogram ofwavelets coefficients for each detail level, is to study this distribution functionfor some simple images.

The image in figure 1(a), chosen for the analysis, corresponds to a naturalimage from the Brodatz collection. It represents a texture with a privilegedorientation. Both mentioned wavelet transforms have been applied to this imageup to three levels of decomposition. In each level, four subbands are obtained:the approximation subband, the horizontal, vertical and diagonal details. In ourstudy, we shall only use the vertical and horizontal detail subbands. It is worthmentioning that for the traditional wavelet transform, the subbands for the firstlevel are a fourth (in size) of the image; when there is no downsampling step,they have the same size as the original image.

In figure 1(a) we have an original texture, to which we have applied 3 levelsof the biorthogonal wavelet transform without downsampling. In figure 1(b) are

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Level 1 Level 2 Level 3

Fig. 1. Empirical distributions for image (a). First row (b): joint distribution of Mal-lat’s à trous wavelet coefficients, second row (c): joint distribution of Daubechies 4detail coefficients, third row (d): magnitude histogram of Mallat’s à trous coefficients,fourth row (e): magnitude histogram of Daubechies 4 transform coefficients, (f): circularhistogram of Mallat’s à trous coefficients.

A New Wavelet-Based Texture Descriptor for Image Retrieval 899

plotted the joint empirical distributions of the detail coefficients for 3 levels,assuming that horizontal and vertical detail coefficients at the same positionare the components of a bivariate random vector. From these joint distributionsit can be inferred that a correlation exists between the horizontal and verticalwavelet subbands. This correlation is noticeable in the second and third levels ofthe transform, whereas the finer detail coefficients do not give much information:they are dominated by noise. The magnitude and angle of the random vectorsamples can be computed and analyzed. The empirical magnitude histogram(figure 1(d)) and the circular histogram (figure 1(f)) are shown for this wavelet.The circular histogram clearly indicates a privileged orientation of the edges inthe texture. We have similar results for images with clearly oriented edges.

The same study has been done applying the traditional Daubechies 4 wavelettransform to the same image. The joint empirical distributions of detail coef-ficients are shown in figure 1(c). It is evident that in this case there is no no-ticeable correlation between horizontal and vertical subbands. The histogramsbehave differently.

2.2 The Proposed Model for Wavelet-Coefficients Distributions

The previous section shows the importance of treating horizontal and verticaldetails jointly. However, it is a very difficult task to find a model to which theempirical joint histogram will fit reasonably well for any kind of texture. Thereare some papers in this approach, as [13], where the distribution of the subbandcoefficients is modeled using a joint alpha-stable sub-Gaussian distribution.

Our approach is different. We do want to make use of the existing relationbetween the vertical and horizontal coefficients at a certain position, but we alsowant a model which is simple to fit and capable of characterizing different typesof textures.

We want a model for the moduli histograms. Observe that for both wavelettransforms, the shape of the empirical distribution changes for each level, insuch a way that the peak of the histogram is shifted to the right. This behaviormay be modelled by means of the gamma distribution (pdf). This distributionis defined by two parameters: k and θ.

f(x; k, θ) = xk−1 e−(x/θ)

θkΓ (k)(1)

where k > 0 is a shape parameter, and θ > 0 is related to the scale of thedistribution (if θ is large, then the distribution will be more spread out). Thegamma distribution is related to many other distributions. The chi-square andexponential distributions, which are children of the gamma distribution, are one-parameter distributions that fix one of the two gamma parameters.

The idea is to fit our empirical moduli histograms to a gamma distributionusing the maxima-likelihood estimator. A goodness-of-fit Kolmogorov-Smirnovtest has been applied to the different samples in order to justify the use of thismodel, giving very high p-values (larger than α = 0.05) for most of the samples

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analyzed, for both wavelet transforms considered. It seems that the gamma dis-tribution may be a reasonable model for the moduli of wavelet coefficients.

As seen in the previous section, the circular histogram gives valuable informa-tion about privileged orientations in textures. Thus, these histograms can also beused to describe textures. An image with a privileged orientation will present abimodal circular histogram with 180 degrees separated statistical modes whereasa random texture, without privileged orientations, will present a uniform circularhistogram.

Thus, we propose to characterize each texture in the database by:

– Information from the moduli: parameters kn, θn, of the gamma pdf for n =1 . . . 3, where n is the wavelet decomposition level.

– Information from the angles given by the empirical circular histogram. An-gles are quantized by dividing interval [0 , 2 π] into 40 bins. Pn(r) is theobserved frequency of the angles in the rth bin for level n.

Retrieval experiments in section 3, have been performed by using the gammadistribution parameters and the circular histograms.

3 Experimental Results

The objetive of this section is to test the new wavelet-based texture descriptor.Images from the well-known Brodatz image database have been used for the ex-periments; this image database is composed of 105 images representing differentkinds of textures.

Thirty six images from this collection were selected and each original imageof size 512 × 512 was partitioned into sixteen 128 × 128 subimages by randomlychoosing the left corner, thus creating a test database of N = 36 × 16. Thetexture descriptor was computed for each image in the test database.

In our experiments, a simulated query image was anyone in our test database.The relevant images for each query were defined as the other 15 subimages fromthe same original Brodatz image. We evaluated the performance in terms of theaverage rate of retrieval of relevant images. We have done the same experimentthree times: with the Daubechies 4 filters and the traditional wavelet transform,with the Daubechies 4 filters without downsampling, and Mallat’s biorthogonalwavelet without downsampling.

The Kullback-Leibler divergence (KLD), commonly used to compare 2 dis-tributions, was the chosen similarity measure between query image Iq and eachimage in the database {Ij , j = 1 . . .N}: we retrieved the top 16 closest imagesto the query, and counted how many corresponded to the same texture as thequery image. The score used to measure the similarity between image Ij and Iq

was computed as

S(j, q) = S1(j, q) + S2(j, q), where S�(j, q) =3∑

n=1

S�(j, q, n), � = 1, 2, (2)

A New Wavelet-Based Texture Descriptor for Image Retrieval 901

S1(j, q, n) =∫

f(x; kjn, θj

n) logf(x; kj

n, θjn)

f(x; kqn, θq

n)dx, S2(j, q, n) =

∑

r

P jn(r) log

P jn(r)

P qn(r)

.

(3)The term S1(j, q, n) is the KLD (or cross entropy) of the moduli pdf at level n

for images j and q. The term S2(j, q, n) is the KLD of the empirical distributionof the quantized angles at level n for images j and q. The final score was thesum of both scores (modulus and angle KLD scores) at the first 3 levels.

Table 1. Average retrieval rate in the top 16

Wavelet Transform Extracted Features

Wavelet filters Algorithm S1 S1 + S2

Daubechies 4 with downsampling 0.75 0.84Daubechies 4 no downsampling 0.75 0.89

Mallat’s biorthogonal no downsampling 0.79 0.89

Table 1 shows the percentage of relevant images retrieved in the top 16matches. The maximun and minimun percentage rate were 89% and 75% re-spectively. The main points that we can observe from these results are that thedownsampling operation of the wavelet transform and the features used are verysignificant in the retrieval performance. Omitting the downsampling steps in thewavelet transform seems very meaningful for the results. This is because theresulting transform is translation invariant.

Our results in the first column of the table, which do not take into accountthe angle information, and only base the results on the moduli information, aresimilar to the ones obtained by modelling the marginal histograms (vertical andhorizontal independently) with the GGD distribution. Moreover, in our proposeddescriptor we need only half the number of parameters.

The inclusion of the orientation information in the features improves the av-erage percentage rate in 10% independently of the wavelet considered and thepresence– or absence– of the downsampling operation. It means that the orien-tation information is a valuable feature to discriminate textures.

4 Conclusions

We have presented a novel wavelet-based texture descriptor for visual informa-tion retrieval. The basic idea to characterize each texture in the image databaseis to use the information from moduli and orientations of wavelet coefficients,assuming that vertical and horizontal coefficients at the same position are thecomponents of a bivariate random vector. The Kullback-Leibler divergence hasbeen chosen as measure of similarity. The good performance of this new descrip-tor is revealed in retrieval experiments using the Brodatz database.

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Acknowledgement. This work has been partially supported by grants GV04-177 (for research stays), MCYT TIN2006-10134, UBACYT X166 and BID 1728/OC-AR-PICT 26001.

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