I Introduction
Texture is a visual attribute that performs an important role in computer vision, image analysis and pattern recognition. There are a lot of applications using textures in different areas of knowledge, ranging from medical images
wu_1992 , passing by remote sensing yang_1998 , analysis of geological images Heidelbach_2000 , etc.The full definition of texture is a complex task. Indeed, there is no formal definition in the literature that is capable of explaining it completely. This occurs, due to the nature of the texture that can be modeled in different ways. Texture can be formed by simple repetitions of set of pixels or simple patterns, but it can also be formed by complex arrangements. These arrangements can be constituted by tiles of natural patterns such as leaves, rocks, clouds or even for more abstract patterns. In fact, even the absence of patters can characterize a texture (e.g., a region formed by noise in an image). If, on one hand, texture is very difficult to be formally defined, on the other hand, the importance of the attribute and its application has been motivated the development of many algorithms and methods. Over the years, many approaches have been proposed to describe texture patterns: secondorder statistics haralick1979 ; journals/pr/MurinoOP98 , spectral analysis journals/paa/ShenB06 ; journals/pr/BianconiF07 ; jain1991 ; daugman95gabor ; journals/pami/ManjunathM96 ; journals/pami/AzencottWY97 ; bb28041 , wavelet packets journals/eswa/SengurTI07 ; unser95 and fractal dimension BackesB08 ; journals/ijprai/ChenB99 ; books/tricot .
This paper presents a novel approach for texture characterization, based on fractal analysis. The proposed approach expands the methodology of the Massradius fractal dimension. The Massradius fractal dimension was originally developed to deal with binary images and shape analysis. The proposed method considers the pixels of an image as a set of coordinates in 3Dspace, where the zaxis is the pixel intensity. In this way, it estimates the fractal dimension of the surface of the image and, consequently, it is capable of dealing with textures. Besides the extension of the Massradius fractal dimension method, the paper approach uses a signature, obtained by a vector calculated by the fractal dimension, to characterize textures. The method is described in detail and an experiment using images from Brodatz album shows the method performance. The proposed method is compared with popular texture ones.
Ii Proposed Approach
In this work, a novel approach for texture analysis is proposed. The approach is based on the massradius method for shape complexity analysis Fernandez2001309 ; Caserta1995133 ; journals/bioinformatics/LandiniR93 . This method consists of covering the shape with circles of radius and to compute the amount of shape that is intercepted by the circle as the radius increases.
Consider an image texture as a set of coordinates . Each texture pixel is represented as a triple , , where and are the Cartesian coordinates of the pixel at the original image and is the graylevel associated to the pixel . Note that now an image is represented as a set of points in a 3Dspace. Thus, the circle employed in the original method is replaced by a sphere of radius and the amount of points intercepted by the sphere is computed.
One important step in the method is to select the total number of spheres that will be employed to sample the texture complexity. Each sphere is centered at a specific point , , randomly chosen. So, the number of points intercepted by a sphere of radius , , is defined as:
(1) 
where is a point in which dists or less from . For spheres, we consider the occupied volume as
(2) 
From occupied volume , the fractal dimension is estimated as
(3) 
Iii Texture signature
From loglog curve computed from proposed approach, the fractal dimension can be easily estimated by applying linear regression over the curve
, where the resulting line presents angular coefficient and is the estimated fractal dimension.However, a single noninteger value may not be suitable to represent all complexity and selfsimilarity present in the image. In fact, if we analyze the computed loglog curve we may note that it presents considerable information along the scales that are lost during the process of linear regression.
Thus, we propose to compute the linear regression at different sections of the loglog curve. Each linear regression is computed for a section of the curve composed by points in sequence and one single point is not allowed to belong to two different curve sections (Figure 1).
As a result, a vector capable of describing the complexity changes at different portions of the loglog curve is yielded, where is the number of line segments computed, thus providing a more efficient texture characterization.
Iv Experiments
The proposed approach was evaluated considering images collected from Brodatz album brodatz1966 . These images were selected once they are widely employed by literature as benchmark for texture analysis methods in computer vision and image processing applications. Each image considered has 200 200 pixels of size, with 256 gray levels. The image set used contains 400 images grouped into 40 Brodatz classes, with 10 samples each. Figure 2 presents one example of each texture class considered in the experiment.
The proposed signature was computed for each texture and analysis step was carried out applying a Linear Discriminant Analysis (LDA) everitt2001 ; fukunaga1990
. The LDA is a supervised method which enables us to find a feature space where the distribution of classes presents good discriminative properties. Descriptors are considered ”good” when the variance between classes is larger than the variance within classes in this feature space. Leaveoneout crossvalidation scheme was also employed during the analysis.
V Results and Discussion
An important issue from the proposed approach that claims for attention refers to the number of spheres used by the method to sample the texture pattern under analysis. Each sphere is centered at random over the texture and a small number of spheres may produce an unsuitable texture sampling. This may result in an underestimated fractal dimension value or even in different fractal dimensions for different executions of the method over a same texture sample. Otherwise, after a given number of spheres, the texture is over sampled, i.e., no relevant information is add to the loglog curve by each additional sphere. Figure 3 shows the fractal dimension value estimated for a given texture sample according to the number of spheres used during the sampling step. In this experiment, for each value of , the fractal dimension was estimated 30 times and its average computed. As a result, we note that fractal dimension is stable for , which corresponds to select 10% of the texture pixels during its sampling step.
Another important parameter of the method is the number of points used to compute the texture signature. Figure 4 shows the success rate of the proposed approach according to the number of point used to compute each line segment during the texture signature making process. In the experiments, a maximum radius was considered, thus resulting in a loglog curve containing points. Each point in the curve corresponds to a radius value at interval in the discreet 3Dspace. According to the length of the line segments, different sections of the loglog curve are selected, so emphasizing details at different resolutions. As the line segment increases, more information is used to compute a single angular coefficient. We have that different oscillations in the loglog curve are now represented by the same angular coefficient. These oscillations are due to the volume of the spheres do not increase equally in all texture points sampled. Small variations in the texture pattern disturb how the influence volume increases, and this makes very sensitive to structural changes on texture patterns. Thus, an increase in the line segment size tends to decrease the relevance of the details present in that section of the curve. Moreover, longer line segments 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 increase and the best result (98.50 %) is found when is considered.
Method  Images correctly classified 
Success rate (%) 

Cooccurrence matrices  330  82.50 
Fourier descriptors  351  87.75 
Gabor Filters  381  95.25 
Proposed Method  394  98.50 
Results yielded by different texture analysis methods are presented in Table 1. The methods considered for comparison are: Fourier descriptors journals/pami/AzencottWY97 , Cooccurrence matrices haralick1979 and Gabor filters jain1991 ; daugman95gabor ; Idrissa2002 . A brief description of the methods is presented as follows:
Fourier descriptors
: it is a set containing the energy of the 99 most meaningful coefficients of the Fourier Transform applied over the image. Each coefficient represents the sum of the spectrum absolute values from a given radial distance from the center transformation.
Cooccurrence matrices
: they represent the joint probability distributions between the graylevels of pairs of pixels at a given orientation and distance. Energy and entropy were computed from nonsymmetric matrices obtained for distances of 1 and 2 pixels with angles of
, , , , totalizing 16 descriptors.Gabor filters: an input image is convolved by a family of filter, where each filter is a bidimensional gaussian function moduled with an oriented sinusoid in a determined frequency and direction. In this paper, 16 filters (4 rotation filter and 4 scale filters), with lower and upper frequencies equal to 0.01 and 0.3, respectively, were employed. Energy from the resulting images was used as its descriptors.
The proposed approach performs texture analysis directly over texture pixels, i.e., no transformation is applied over the image pixels. However, its result overcomes the ones from traditional texture analysis methods, such as Fourier descriptors and Gabor filters. These methods employ more complexes and sophisticated computing than the proposed approach, and this contributes to validate our approach as a feasible texture descriptor.
Vi Conclusion
This paper presented a novel approach for texture discrimination using complexity analysis. The proposed approach is based on the idea of the massradius method, which is used in literature to compute the fractal dimension of shapes. By considering texture as a set of coordinates in 3Dspace, the original method is easily expanded from shape analysis to texture analysis, thus enabling us to estimate the fractal dimension of a texture pattern. An experiment using the texture signature computed using the proposed approach and linear discriminant analysis to classify texture samples extracted from Brodatz album was performed. Results show that the method presents great potential to be used in texture identification/classification tasks.
Acknowledgments
A.R.B. acknowledges support from FAPESP (2006/543679). O.M.B. acknowledges support from CNPq (306628/20074 and 484474/20073).
References
 (1) C. M. Wu, Y. C. Chen, K. S. Hsieh, Texture features for classification of ultrasonic liver images, IEEE Transactions on Medical Imaging 11 (1992) 141–152.
 (2) C. Q. Zhu, X. M. Yang, Study of remotesensing image texture analysis and classification using wavelet, International Journal of Remote Sensing 19 (16) (1998) 3197–3203.
 (3) F. Heidelbach, K. Kunze, H. R. Wenk, Texture analysis of a recrystallized quartzite using electron diffraction in the scanning electron microscope, Journal of Structural Geology (2000) 91–104.
 (4) R. M. Haralick, Statistical and structural approaches to texture, Proc. IEEE 67 (5) (1979) 786–804.
 (5) V. Murino, C. Ottonello, S. Pagnan, Noisy texture classification: A higherorder statistics approach, Pattern Recognition 31 (4) (1998) 383–393.

(6)
L. Shen, L. Bai, A review on gabor wavelets for face recognition, Pattern Anal. Appl 9 (23) (2006) 273–292.
 (7) F. Bianconi, A. Fernández, Evaluation of the effects of gabor filter parameters on texture classification, Pattern Recognition 40 (12) (2007) 3325–3335.
 (8) A. K. Jain, F. Farrokhnia, Unsupervised texture segmentation using Gabor filters, Pattern Recognition 24 (12) (1991) 1167–1186.

(9)
J. Daugman, C. Downing, Gabor wavelets for statistical pattern recognition, in: M. A. Arbib (Ed.), The Handbook of Brain Theory and Neural Networks, MIT Press, Cambridge, Massachusetts, 1995, pp. 414–419.
 (10) B. S. Manjunath, W.Y. Ma, Texture features for browsing and retrieval of image data, IEEE Trans. Pattern Anal. Mach. Intell 18 (8) (1996) 837–842.
 (11) R. Azencott, J.P. Wang, L. Younes, Texture classification using windowed fourier filters, IEEE Trans. Pattern Anal. Mach. Intell 19 (2) (1997) 148–153.
 (12) R. K. Bajcsy, Computer identification of visual surfaces, Computer Graphics Image Processing 2 (1973) 118–130.
 (13) A. Sengür, I. Türkoglu, M. C. Ince, Wavelet packet neural networks for texture classification, Expert Syst. Appl 32 (2) (2007) 527–533.
 (14) M. Unser, Texture classification and segmentation using wavelet frames, IEEE Trans. Image Processing 4 (11) (1995) 1549–1560.
 (15) A. R. Backes, O. M. Bruno, A new approach to estimate fractal dimension of texture images, in: ICISP, Vol. 5099 of Lecture Notes in Computer Science, Springer, 2008, pp. 136–143.
 (16) Y. Q. Chen, G. Bi, On texture classification using fractal dimension, IJPRAI 13 (6) (1999) 929–943.
 (17) C. Tricot, Curves and Fractal Dimension, SpringerVerlag, 1995.
 (18) E. Fern ndez, H. F. Jelinek, Use of fractal theory in neuroscience: Methods, advantages, and potential problems.

(19)
F. Caserta, W. D. Eldred, E. Fernandez, R. E. Hausman, L. R. Stanford, S., V. Bulderev, S. Schwarzer, H. E. Stanley, Determination of fractal dimension of physiologically characterized neurons in two and three dimensions, Journal of Neuroscience Methods 56 (2) (1995) 133 – 144.
 (20) G. Landini, J. W. Rippin, Notes on the implementation of the massradius method of fractal dimension estimation, Computer Applications in the Biosciences 9 (5) (1993) 547–550.
 (21) P. Brodatz, Textures: A photographic album for artists and designers, Dover Publications, New York, 1966.

(22)
B. S. Everitt, G. Dunn, Applied Multivariate Analysis, 2nd Edition, Arnold, 2001.
 (23) K. Fukunaga, Introduction to Statistical Pattern Recognition, 2nd Edition, Academic Press, 1990.
 (24) M. Idrissa, M. Acheroy, Texture classification using gabor filters, Pattern Recognition Letters 23 (9) (2002) 1095–1102.
Comments
There are no comments yet.