I Introduction
Fractals have played an important role in many areas with applications related to computer vision and pattern recognition
Scarlat et al. (2010); Han et al. (2008); Chappard et al. (2003); Wool (2008); TianGang et al. (2007); Lorthois and Cassot (2010); Rossatto et al. (2011); Florindo et al. . This broad use of fractal geometry is explained by the flexibility of fractals in representing structures usually found in the nature. In such objects, we observe a level of details at different scales which are described in a straightforward manner by fractals, rather than through classical Euclidean geometry.Most of fractalbased techniques are derived from fractal dimension concept. Although this concept was defined originally only for real real fractal objects, it contains some properties which turn it into a very interesting descriptor for any object of real world. Indeed, the fractal dimension measures how the complexity (level of details) of an object varies along scales. Such definition corresponds to an effective and flexible means of quantifying the spatial occupation of an object as well as physical and visual important aspects which characterize an object, like luminance and roughness, for instance.
Among these fractal techniques, we may cite Multifractals Harte (2001); Lashermes et al. (2008); Lovejoy et al. (2000), Multiscale Fractal Dimension Manoel et al. (2002); da F. Costa and Cesar, Jr. (2000); Plotze et al. (2005) and Fractal Descriptors Bruno et al. (2008); Backes et al. (2009); Florindo et al. (2010, 2012); Backes et al. (2012); Florindo and Bruno (2011). Here we are focused on the later approach which demonstrated the best results in such kind of application Florindo and Bruno (2012). The main idea of fractal descriptors theory is to provide descriptors of an object represented in a digital image from the relation between the values of fractal dimension taken under different observation scales. These values provide a valuable information about the complexity of the object in the sense that they capture the degree of details at each scale. In this way, fractal descriptors are capable of quantifying important physical characteristics of the structure, like fractal dimension, but presenting a sensibly richer information than a unique number (fractal dimension).
Here, we proposed a novel fractal descriptors based on probability fractal dimension. We used the whole powerlaw curve of the dimension and applied a spacescale transform to emphasize the multiscale aspect of the features. Finally, we test the proposed method over two wellknown datasets, that is, Brodatz and Vistex, comparing the results with another fractal descriptors approach showed in Backes et al. (2009) and other conventional texture analysis methods. The results demonstrated the power of probability descriptors, achieving a more precise classification than other classical techniques.
Ii Fractal Theory
In the last decades we observe a growing number of works applying fractal geometry concepts in the solution of a wide range of problems Scarlat et al. (2010); Han et al. (2008); Chappard et al. (2003); Wool (2008); TianGang et al. (2007); Lorthois and Cassot (2010). This interest in fractals is mainly motivated by the fact that conventional Euclidean geometry has severe limitations in providing accurate measures of real world objects.
ii.1 Fractal Dimension
The first definition of fractal dimension provided in Mandelbrot (1968) is the Hausdorff dimension. In this definition, we consider a fractal object as being a set of points immersed in a topological space. Thus, we can use results from Measure Theory to define a measure over this object. This is the Hausdorff measure expressed through:
where denotes the diameter of , that is, the maximum possible distance between any elements of :
Here, we say that a countable collection of sets , with , is a cover of if .
Notice that also depends on a parameter which expresses the scale under which the measure is taken. We can eliminate such dependence by applying a limit over , defining in this way the dimensional Hausdorff measure:
If we plot as a function of we observe a similar behavior in any fractal object analyzed. The value of is for any and it is for any , where is always a nonnegative real value. is the Hausdorff fractal dimension of . In a more formal way we may write:
In most practical situations, Hausdorff dimension uses to be complicate or even impossible to calculate. Thus, assuming that any fractal object is intrinsically selfsimilar, we may derive a simplified version, also called similarity dimension or capacity dimension:
where is number of rules with linear length used to cover the object.
In practice, the above expression may be generalized by considering as any kind of selfsimilarity measure and as any scale parameter. This generalization gives rise to a lot of estimation methods for fractal dimension, with a broad application to the analysis of objects which are not real fractals (mathematically defined) but which present some degree of selfsimilarity in specific intervals. An example of such method is the probability dimension, used in this work and described in the following section.
ii.2 Probability Dimension
The probability dimension, also known as information dimension is derived from the information function. This function can be defined in any situation where we have an object populating a physical space. We must divide this space into a grid of squares with sidelength and calculate the probability of points of the object pertaining to some square of the grid. The probability function is given by:
where is maximum possible number of points of the object inside a unique square.
The dimension itself is given through:
When this dimension is estimated over a graylevel digital image , an usual approach is to map it onto a threedimensional surface through:
(1) 
In this case, we construct a threedimensional grid of 3D cubes also with sidelength . The probability is therefore calculated as being the number of grid cubes containing points of the surface divided by the maximum number of points inside a grid cube.
Iii Fractal Descriptors
The main idea of fractal descriptors is to extract values (descriptors) from the relation common to most methods of estimating fractal dimension. Actually, any fractal dimension method derived from Hausdorff dimension concept obey a powerlaw relation which may be explicit in the following:
where is a measure depending on the dimension method and is the scale under this measure is taken.
Therefore, fractal descriptors are provided from the the function :
In order to simplify the notation we name the independent variable as . Thus, and our fractal descriptors function is denoted . For the probability dimension used in this work, we have:
The values of function may be used directly as descriptors of the analyzed image or may be postprocessed by some kind of operation aiming at emphasizing some specifically aspects of that function. Here, we apply a multiscale transform to the function. In this way, we obtain a twodimensional function , in which the variable is related to and is related to the scale on which the function is observed. A usual means of obtain is through a derivative process:
where is the wellknown Gaussian function and is the smoothing parameter:
Given the finite domain of function , we also must restrict the response of Gaussian filter to a finite interval :
where is a real value which should satisfy:
being the tolerance error. Usually, we may have:
where is a real constant with values commonly varying between and .
Iv Proposed Method
The idea here proposed is to obtain fractal descriptors from textures based on the probability fractal dimension. Thus, such descriptors are computed from the curve in the Equation II.2. Therefore, we apply a multiscale transform to .
The multiscale process is achieved by deviating numerically and convoluting with a Gaussian filter, as described in the previous section:
where is the Gaussian function descritized over the interval .
As multiscale transform maps a onedimensional signal onto a twodimensional function, it is a process which generates intrinsic redundancies. We may find different approaches to eliminate such redundancies keeping only the relevant information da F. Costa and Cesar, Jr. (2000). Here, we adopt a simple method named finetuning smoothing in which is projected under a specific value of the Gaussian parameter. Here we tested values of varying between and and used that values which provided the best performance in the experiments.
Finally, we selected a particular region from to compose the descriptors. Empirically, we observed that the initial points in this curve are relevant to a good performance in our application. Then, we established a threshold after which all points in the convolution curve are disregarded. Thus, the values in the curve are taken as the proposed descriptors.
V Experiments
In order to verify the efficiency of the proposed technique, we applied the probability descriptors to the classification of two benchmark datasets and compared to the performance of other wellknown and stateoftheart methods for texture analysis.
The first classification was accomplished over the Brodatz dataset, a classical set of natural gray level textures photographed and put together in a book Brodatz (1966). This dataset is composed by 111 classes with 10 textures with dimension 200200 in each class.
The second data set is the Vistex, a set of color textures extracted from natural scenes Singh and Sharma (2001). Here, we employ a version of the dataset in which we have 7 classes, each one with a variable number of images of 256256 pixels and converted them to graylevel images.
We compare probability descriptors to other 4 other techniques, that is, Gabor wavelets Manjunath and Ma (1996), Cooccurrence matrix Haralick (1979), Gray Level Difference Method (GLDM) Weszka et al. (1976), a multifractal approach described in Parrinello and Vaughan (2002) and BouligandMinkowski fractal descriptors Backes et al. (2009).
Vi Results
The Table 1 shows the correctness rate in the classification of Brodatz dataset using the compared descriptors. The proposed method obtained the best result with a 12% advantage. For this result we used and threshold
. A particular important aspect in Brodatz data set is the reduced number of descriptors of the proposed approach. This point may be specially important in large data basis when the computational performance is more relevant. Furthermore, the small number of features avoid the curse of dimensionality, which prejudices the reliability of the global result.
Method  Correctness Rate (%)  Number of descriptors 

Gabor  81.3  20 
Cooccurrence  53.5  84 
GLDM  52.2  20 
Multifractal  35.1  101 
BouligandMinkowski  47.6  85 
Proposed method  91.2  17 
On the other hand, the Table 2 shows the results for the Vistex textures. In this case, we obtained the best result by using and . Again, the proposed approach provided the greater correctness with a 2% advantage. Again, we have a good result in a data set which presents a lot of challenges once it is aimed at color analysis while the proposed approach is graylevel based. This aspect turns significant even a tiny classification enhancing.
Method  Correctness Rate (%)  Number of descriptors 

Gabor  85.7  20 
Cooccurrence  70.1  120 
GLDM  43.5  20 
Multifractal  20.1  101 
BouligandMinkowski  68.8  85 
Proposed method  87.7  80 
Finally, the Figures 4 and 5 show the confusion matrices of the methods with best performances. In such figures, we have the predicted classes in the rows and the actual ones in the columns. The number of classes in each configuration is given by the intensity of graylevel in each point (brighter points correspond to large number of classes). In this kind of representation, a good descriptor must produce a matrix with a diagonal as brighter and continuous as possible and the minimum of brighter points outside the diagonal. In this sense, we see that, in Brodatz data, the probability descriptors presented exactly these characteristics, with almost no “hole” in the diagonal and with a lower density of brighter points outside.
In Vistex case, the matrices are not so distinguishable visually, even due to the similarity in the results. Thus, a perspective which we may use is analyze directly the number of corrected samples in each class. Based on this aspect, we observe in the legend bar that the proposed technique has its matrix normalized on a greater number of samples. This implies that although with a similar aspect, our approach presented a higher number of samples classified correctly in each class.
A general analysis of the results demonstrates that the proposed method overcame the compared ones in both data sets, using a reasonable amount of descriptors. Such results was expected from fractal theory once we have a lot of works in the literature showing the efficiency of fractal geometry in the analysis of natural textures. Actually, fractal geometry presents a flexibility in the modeling of objects which cannot be well represented by Euclidean rules. The fractal dimension is a powerful metric for the complex patterns and spatial arrangement usually found in the nature. Fractal descriptors enhance such ability providing a way of capturing multiscale variations and nuances which could not be measured through conventional tools. More specifically, the probability descriptors here proposed aggregates a statistical approach to fractal analysis, composing a framework which supports a precise and reliable discrimination technique, as confirmed in the above results.
Vii Conclusion
The present work proposed a novel method to extract descriptors based on fractal theory for texture analysis application.
Here we obtained such descriptors by applying a multiscale transform over the power law relation of fractal dimension estimated by the probability method.
We tested the efficiency of the novel technique in the classification of wellknown benchmark texture dataset and compared its performance to that of other classical texture analysis methods. The results demonstrated that probability fractal descriptors are a powerful tool to model such textures. The method provides a rich way of representing even the most complex structures in texture images, being a reliable approach to solve a large class of problems involving the analysis of texture images.
Acknowledgments
J.B.F. acknowledges support from CNPq. O.M.B. acknowledges support from CNPq (Grant #308449/20100 and #473893/20100) and FAPESP (Grant # 2011/015231).
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