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Fractal Descriptors Applied to Texture Analysis

Members

João Batista Florindo

Odemir M. Bruno (Advisor)

Introduction

This research project studies application of fractal descriptors for texture analysis. Marginally, we also developed a novel method to compute the fractal dimension of closed shapes and studied the implications of some statistical methods as an auxiliary tool to the texture analysis through fractal geometry.

The fractal theory is widely used in the literature, chiefly for the modelling of natural objects in image analysis techniques. Usually, these works employ the fractal dimension to characterize and discriminate such objects. The drawback of such approach is that fractal dimension is only a single value and it is not capable of capturing complex important nuances. Then, we propose the application of a multiscale transform to the fractal dimension allowing the obtainment of a curve (signature) which describes more exactly the object.

Here, we verified the performance of the proposed technique by employing different fractal dimension estimation methods, like Bouligand-Minkowski, Fourier, wavelets, etc. Then, we apply the descriptors to tasks of classification and segmentation of texture images in benchmark data sets, like Brodatz and Vistex or from practical applications, like in nanotechnology and botany.

Fractal geometry

Formally developed in Mandelbrot [1], the fractal geometry provides tools and methodologies to study fractal objects. A fractal is a complex structure caracterized by a self-similarity behavior, that is, each part is a quasi-copy of the whole. The Figure 1 illustrates such self-similarity. Observe that a zoom in one region exhibits a scaled and rotated version of the entire object.

The most used metric of a fractal is the fractal dimension. Actually, this measure captures the composition law of the object and as a direct consequence also expresses the self-similarity degree of that structure. Particularly, another possible interpretation for the concept is that it measures the spatial occupation of the object. In this sense, fractal dimension appears as a powerful tool to describe objects from the real world which, despite not being exact mathematical fractals, have some degree of self-similarity and may be modelled through fractal geometry. This is the case of textures analyzed in this work.

The most classical definition of fractal dimension coincides with the Hausdorff-Besicovitch dimension. Although, theoretically, this dimension could be computed for any object, in practice, its application is expensive or even inviable. In such cases, the literature shows an alternative definition known as the similarity dimension. This is represented by the following expression:

where N(u) represents the number of units u needed to cover the object. This expression can be generalized giving rise to a lot of methods used to estimate the fractal dimension even of objects which are not exact fractals as those we are dealing with. In the following, we see a brief description of these approaches.

Bouligand-Minkowski fractal dimension

In this technique [1], the image is representsd in a three-dimensional surface, in such a way that each pixel with coordinate (x,y) in the original image is mapped onto a point with coordinate (x,y,z), where z is the intensity of the pixel. Thus, this surface is dilated with a sphere with a variable radius. The dimension is computed by replacing N(u) by the volume of the dilated surface and u by the dilation radius.

Figure 2. Bouligand-Minkowski dilation process.

Fourier fractal dimension

In this approach [2] we compute the Fourier transform of the image to be analyzed and obtain the fractal dimension from similarity expression, replacing N(u) by the power spectrum and u by the frequency, taken radially. We also developed a variation of this approach to be used in color texture images [6].

Figure 3. Fourier fractal descriptors.

Figure 4. Fourier fractal descriptors applied to color textures.

Wavelets fractal dimension

Wavelets method [3] is based on a specifical type of wavelets transform named Best Basis Selection (BBS). In BBS, each level of the wavelets transform may or not be decomposed depending on a cost function. Here, we used the Shannon entropy as such function. The dimension itself is obtaines by replacing N(u) by the sort energy of the transformed image and u by a sorting index.

Figure 5. Wavelets fractal descriptors.

Fractal descriptors

Altough fractal dimension may be a valuable tool in the solution of many problems, the fact that it is only a single value limits seriously its efficiency in the description of more complex objects. For instance, we may find objects with the same fractal dimension but with a completely distinct aspect, as illustrated in the Figure ???. As a solution to solve this gap, we propose the fractal descriptors method [4].

Figure 6. Fractal descriptors.

The essential idea of fractal descriptors consists in using the following function u, instead of simple fractal dimension:

This function can be used directly or after some statistical operations. For instance, it may be summitted to a scale-space or still to a Karhunen-Loeve transform or in functional data space [5]. The Figure ??? shows the discrimination power of fractal descriptors in a set of natural textures.

Figure 7. Discrimination power of fractal descriptors.

Practical Applications

Among the applications, we have successfully tested fractal descriptors in a data set of titanium oxide films prepared under different experimental conditions [8]. Another application still in development stage is the segmentation of satellite images (Figure 8). In a paralel study on fractal theory we also developed a novel method for calculate the fractal dimension of closed contours [7].

Figure 8. Fractal descriptors applied to satellite images segmentation.

Results

KNN classifier in Vistex (partial) data set.
Method Correctness rate (%)
Cromaticity moments 68.8312
Color Gabor 92.5325
Color Fourier fractal descriptors 95.1299
Bayesian classifier in UMD data set.
Method Correctness rate (%)
Standalone fractal dimension 27.8
Co-occurrence 57.1
Gray-level Gabor 66.7
Wavelets fractal descriptors 76.7
LDA classifier in Brodatz data set.
Method Correctness rate (%)
Laws 87.25
Multifractal 48.39
Gray-level Gabor 94.57
Bouligand-Minkowski fractal descriptors 99.34

We can observe from the tables that fractal descriptors is a quite promising tool in a lot of problems involving image pattern recognition, discriminations and description of objects represented digitally. In fact, this result was expected given the ability of fractal geometry in representing the spatial occupation and arrangement relation inside an image. This perspective turns possible the measurement of important physical characteristics of the object represented in the image, as, for instance, roughness and luminosity patterns. On the other hand, such aspects are related to the cognitive notion of pattern recognition present in our vision system.

Considering this relevance of fractal dimension, the method proposed here increments the fractal dimension precision by extracting a set of values (descriptors) which capture the fractality of the object under distinct scales. This is strongly important when we study real-world structures. In such objects, the fractality does not present a perfect power-law relation as it occurs with real fractals. Thus, we notice a behavior close to that found in real exact fractals but with some irregularities which are more faithfully captured through fractal descriptors approach.

References

[1] B. B. Mandelbrot, The Fractal Geometry of Nature, Freeman, 1968.

[2] J. C. Russ, Fractal Surfaces, Plenum Press, New York, 1994.

[3] C. L. Jones, H. F. Jelinek, Wavelet packet fractal analysis of neuronal morphology, Methods 24 (4) (2001) 347-358.

[4] O. M. Bruno, R. de Oliveira Plotze, M. Falvo, M. de Castro, Fractal dimension applied to plant identi cation, Inf. Sci 178 (12) (2008) 2722-2733.

[5] J. B. Florindo, M. Castro, O. M. Bruno, Enhancing multiscale fractal descriptors using functional data analysis, International Journal of Bifurcation and Chaos 20 (11) (2010) 3443-3460.

[6] J. B. Florindo, O. M. Bruno, Fractal descriptors in the Fourier domain applied to color texture analysis, Chaos 21 (4) (2011) 043112.

[7] J. B. Florindo, O. M. Bruno, Closed contour fractal dimension estimation by the Fourier transform, Chaos, Solitons and Fractals 44 (2011) 851-861.

[8] J. B. Florindo, M. S. Sikora, E. C. Pereira, O. M. Bruno, Multiscale Fractal Descriptors Applied to Nanoscale Images (to appear), Journal of Superconductivity and Novel Magnetism.