Texture analysis using volume-radius fractal dimension
TTexture analysis using volume-radius fractal dimension
Andr´e. R. Backes ∗ Faculdade de Computa¸c˜ao, Universidade Federal de Uberlˆandia,Av. Jo˜ao Naves de ´Avila, 2121, 38408-100, Uberlˆandia, MG, Brasil
Odemir M. Bruno † Instituto de F´ısica de S˜ao Carlos (IFSC), Universidade de S˜ao Paulo,Av. Trabalhador S˜ao Carlense, 400, 13560-970, S˜ao Carlos, SP, Brasil (Dated: August 22, 2018)Texture plays an important role in computer vision. It is one of the most important visualattributes used in image analysis, once it provides information about pixel organization at differentregions of the image. This paper presents a novel approach for texture characterization, based oncomplexity analysis. The proposed approach expands the idea of the Mass-radius fractal dimension,a method originally developed for shape analysis, to a set of coordinates in 3D-space that representsthe texture under analysis in a signature able to characterize efficiently different texture classes interms of complexity. An experiment using images from the Brodatz album illustrates the methodperformance.
I. INTRODUCTION
Texture is a visual attribute that performs an impor-tant role in computer vision, image analysis and patternrecognition. There are a lot of applications using tex-tures in different areas of knowledge, ranging from med-ical images [1], passing by remote sensing [2], analysis ofgeological images [3], etc.The full definition of texture is a complex task. In-deed, there is no formal definition in the literature thatis capable of explaining it completely. This occurs, due tothe nature of the texture that can be modeled in differentways. Texture can be formed by simple repetitions of setof pixels or simple patterns, but it can also be formed bycomplex arrangements. These arrangements can be con-stituted by tiles of natural patterns such as leaves, rocks,clouds or even for more abstract patterns. In fact, eventhe absence of patters can characterize a texture (e.g., aregion formed by noise in an image). If, on one hand, tex-ture is very difficult to be formally defined, on the otherhand, the importance of the attribute and its applicationhas been motivated the development of many algorithmsand methods. Over the years, many approaches havebeen proposed to describe texture patterns: second-orderstatistics [4, 5], spectral analysis [6–12], wavelet packets[13, 14] and fractal dimension [15–17].This paper presents a novel approach for texture char-acterization, based on fractal analysis. The proposedapproach expands the methodology of the Mass-radiusfractal dimension. The Mass-radius fractal dimensionwas originally developed to deal with binary images andshape analysis. The proposed method considers the pix-els of an image as a set of coordinates in 3D-space, wherethe z-axis is the pixel intensity. In this way, it estimates ∗ [email protected] † [email protected] the fractal dimension of the surface of the image and,consequently, it is capable of dealing with textures. Be-sides the extension of the Mass-radius fractal dimensionmethod, the paper approach uses a signature, obtainedby a vector calculated by the fractal dimension, to char-acterize textures. The method is described in detail andan experiment using images from Brodatz album showsthe method performance. The proposed method is com-pared with popular texture ones. II. PROPOSED APPROACH
In this work, a novel approach for texture analysis isproposed. The approach is based on the mass-radiusmethod for shape complexity analysis [18–20]. Thismethod consists of covering the shape with circles of ra-dius r and to compute the amount of shape that is inter-cepted by the circle as the radius increases.Consider an image texture as a set of coordinates S .Each texture pixel is represented as a triple s = ( y, x, z ), s ∈ S , where y and x are the Cartesian coordinates ofthe pixel at the original image and z is the gray-levelassociated to the pixel ( y, x ). Note that now an imageis represented as a set of points in a 3D-space. Thus,the circle employed in the original method is replacedby a sphere of radius r and the amount of points s ∈ S intercepted by the sphere is computed.One important step in the method is to select the totalnumber of spheres that will be employed to sample thetexture complexity. Each sphere is centered at a specificpoint s i ∈ S , u = 1 , , . . . , N , randomly chosen. So, thenumber of points intercepted by a sphere of radius r , V i ( r ), is defined as: V i ( r ) = |{ s i ∈ S |∃ s ∈ S : | s − s i | ≤ r }| , (1)where s i is a point in S which dists r or less from s . For N spheres, we consider the occupied volume V ( r ) as a r X i v : . [ c s . C V ] D ec V ( r ) = 1 N (cid:88) i =1: N V i ( r ) (2)From occupied volume V ( r ), the fractal dimension D is estimated as D = lim r → log V ( r )log r . (3) III. TEXTURE SIGNATURE
From log-log curve computed from proposed approach,the fractal dimension can be easily estimated by applyinglinear regression over the curve log r × log V ( r ), where theresulting line presents angular coefficient α and D = α isthe estimated fractal dimension.However, a single non-integer value may not be suitableto represent all complexity and self-similarity present inthe image. In fact, if we analyze the computed log-logcurve we may note that it presents considerable informa-tion along the scales that are lost during the process oflinear regression.Thus, we propose to compute the linear regression atdifferent sections of the log-log curve. Each linear regres-sion is computed for a section of the curve composed by M points in sequence and one single point is not allowedto belong to two different curve sections (Figure 1).As a result, a vector (cid:126)ϕ = { α , α , . . . , α k , } capable ofdescribing the complexity changes at different portionsof the log-log curve is yielded, where k is the number ofline segments computed, thus providing a more efficienttexture characterization. Figure 1. Example of texture signature computed for M = 10,resulting in k = 8 line segments. Log-log curve computed for r = 10. IV. EXPERIMENTS
The proposed approach was evaluated considering im-ages collected from Brodatz album [21]. These imageswere selected once they are widely employed by literatureas benchmark for texture analysis methods in computervision and image processing applications. Each imageconsidered has 200 ×
200 pixels of size, with 256 graylevels. The image set used contains 400 images groupedinto 40 Brodatz classes, with 10 samples each. Figure 2presents one example of each texture class considered inthe experiment.
Figure 2. Example of each Brodatz texture class considered.
The proposed signature was computed for each tex-ture and analysis step was carried out applying a LinearDiscriminant Analysis (LDA) [22, 23]. The LDA is a su-pervised method which enables us to find a feature spacewhere the distribution of classes presents good discrim-inative properties. Descriptors are considered ”good”when the variance between classes is larger than the vari-ance within classes in this feature space. Leave-one-outcross-validation scheme was also employed during theanalysis.
V. RESULTS AND DISCUSSION
An important issue from the proposed approach thatclaims for attention refers to the number of spheres N used by the method to sample the texture pattern un-der analysis. Each sphere is centered at random over thetexture and a small number of spheres may produce anunsuitable texture sampling. This may result in an un-derestimated fractal dimension value or even in differentfractal dimensions for different executions of the methodover a same texture sample. Otherwise, after a givennumber of spheres, the texture is over sampled, i.e., norelevant information is add to the log-log curve by eachadditional sphere. Figure 3 shows the fractal dimensionvalue D estimated for a given texture sample accordingto the number of spheres N used during the samplingstep. In this experiment, for each value of N , the frac-tal dimension D was estimated 30 times and its averagecomputed. As a result, we note that fractal dimension isstable for N ≥ Figure 3. The fractal dimension D as a function of the numberof points N considered for texture sampling.Figure 4. Success rate as a function of the slope interval ( M )considered. Best classification (98.50 %) is achieved whenusing M = 10. Another important parameter of the method is thenumber of points M used to compute the texture sig-nature. Figure 4 shows the success rate of the proposedapproach according to the number of point M used tocompute each line segment during the texture signaturemaking process. In the experiments, a maximum radius r = 20 was considered, thus resulting in a log-log curvecontaining 335 points. Each point in the curve corre-sponds to a radius value at interval [0 , radius ] in the dis-creet 3D-space. According to the length of the line seg-ments, different sections of the log-log curve are selected,so emphasizing details at different resolutions. As theline segment increases, more information is used to com-pute a single angular coefficient. We have that differentoscillations in the log-log curve are now represented by the same angular coefficient. These oscillations are dueto the volume of the spheres do not increase equally in alltexture points sampled. Small variations in the texturepattern disturb how the influence volume V ( r ) increases,and this makes V ( r ) very sensitive to structural changeson texture patterns. Thus, an increase in the line seg-ment size tends to decrease the relevance of the detailspresent in that section of the curve. Moreover, longer linesegments produce a smaller set of linear coefficients and,as a consequence, a less discriminative texture signature.In fact, a subtle decrease in success rate is perceived as M increase and the best result (98.50 %) is found when M = 5 is considered.Results yielded by different texture analysis methodsare presented in Table I. The methods considered forcomparison are: Fourier descriptors [11], Co-occurrencematrices [4] and Gabor filters [8, 9, 24]. A brief descrip-tion of the methods is presented as follows: Fourier descriptors : it is a set containing the energy ofthe 99 most meaningful coefficients of the Fourier Trans-form applied over the image. Each coefficient representsthe sum of the spectrum absolute values from a givenradial distance from the center transformation.
Co-occurrence matrices : they represent the joint prob-ability distributions between the gray-levels of pairs ofpixels at a given orientation and distance. Energy andentropy were computed from non-symmetric matrices ob-tained for distances of 1 and 2 pixels with angles of − ◦ ,0 ◦ , 45 ◦ , 90 ◦ , totalizing 16 descriptors. Gabor filters : an input image is convolved by a fam-ily of filter, where each filter is a bi-dimensional gaussianfunction moduled with an oriented sinusoid in a deter-mined frequency and direction. In this paper, 16 filters(4 rotation filter and 4 scale filters), with lower and up-per frequencies equal to 0.01 and 0.3, respectively, wereemployed. Energy from the resulting images was used asits descriptors.The proposed approach performs texture analysis di-rectly over texture pixels, i.e., no transformation is ap-plied over the image pixels. However, its result overcomesthe ones from traditional texture analysis methods, suchas Fourier descriptors and Gabor filters. These meth-ods employ more complexes and sophisticated comput-ing than the proposed approach, and this contributes tovalidate our approach as a feasible texture descriptor.
VI. CONCLUSION
This paper presented a novel approach for texture dis-crimination using complexity analysis. The proposed ap-proach is based on the idea of the mass-radius method,which is used in literature to compute the fractal dimen-sion of shapes. By considering texture as a set of coordi-nates in 3D-space, the original method is easily expandedfrom shape analysis to texture analysis, thus enabling usto estimate the fractal dimension of a texture pattern. Anexperiment using the texture signature computed using
Method Images correctly classified Success rate (%)Co-occurrence matrices 330 82.50Fourier descriptors 351 87.75Gabor Filters 381 95.25Proposed Method 394 98.50Table I. Comparison results for different texture methods. the proposed approach and linear discriminant analysisto classify texture samples extracted from Brodatz albumwas performed. Results show that the method presentsgreat potential to be used in texture identification/clas-sification tasks.
ACKNOWLEDGMENTS