I Introduction
Fractals have played an important role in many areas with applications related to computer vision and pattern recognition
SMS10 ; HWZ08 ; CDHLAB03 ; W08 ; TWZ07 ; LC10 , owing to their flexibility in representing structures usually found in nature. In such objects, we observe different levels of detail at different scales, which are described in a straightforward manner by fractals, rather than through classical Euclidean geometry.Most fractalbased techniques are based on the concept of fractal dimension. Altough this concept was originally defined only for mathematical fractal objects, it contains some properties that make it a very interesting descriptor for any object in the real world. Indeed, fractal dimension measures how the complexity (level of detail) of an object varies with scale, an effective and flexible means of quantifying how much space an object ocupies, as well as important physical and visual properties of the object, such as luminance and roughness.
Fractal techniques include the use of Multifractals H01 ; LRAJ08 ; LGS00 , Multiscale Fractal Dimension MCSM02 ; CC00 and Fractal Descriptors BPFC08 ; BCB09 ; PPFVOB05 ; FBCB12 . Here we are focus on the last approach, which has demonstrated the best results in texture classification FB13 . The main idea of fractal descriptors theory is to provide descriptors of an object represented in a digital image from the relation among fractal dimensions taken at different observation scales, thus these values provide a valuable information on the complexity of the object, in the sense that they capture the degree of detail at each scale. In this way, fractal descriptors are capable of quantifying important physical characteristics of the structure, as the fractal dimension, but presenting a richer information than can be provided by a single number (fractal dimension).
Although fractal descriptors have demonstrated to be a promising technique, we observe that they are defined mostly on wellknown methods to estimate the fractal dimension. Here, we propose fractal descriptors based on a less known definition of fractal dimension: the probability dimension. This is a statistical approach, which measures the distribution of pixel intensities along the image. In this way, such descriptors can express how the statistical arrangement of pixels in the image changes with the scale and how much such correlation approximates a fractal behavior. In this sense, our descriptor also measure the selfsimilarity and complexity of the image but upon a statistical viewpoint. This is a rich and not explored perspective, which is studied in depth in this work.
We use the whole powerlaw curve of the dimension and apply a timescale transform to emphasize the multiscale aspect of the features. Finally, we test the proposed method over two wellknown datasets, that is, Brodatz and Outex, comparing the results with another fractal descriptor approach showed in BCB09 and other conventional texture analysis methods. The results demonstrat that probability descriptors achieve a more precise classification than other classical techniques.
Ii Fractal Theory
In recent years, fractal geometry concepts have been applied to the solution of a wide range of problems SMS10 ; HWZ08 ; CDHLAB03 ; W08 ; TWZ07 ; LC10 , mainly because conventional Euclidean geometry has severe limitations in providing accurate measures of realworld objects.
ii.1 Fractal Dimension
The first definition of fractal dimension provided in M68 , is the Hausdorff dimension. In this definition, a fractal object is a set of points immersed in a topological space. Thus one can use results from Measure Theory to define a measure over this object. This is the Hausdorff measure expressed by
(1) 
where denotes the diameter of , that is, the maximum possible distance among any elements of :
(2) 
Here, a countable collection of sets , with , is a cover of if .
Notice that also depends on a parameter , which expresses the scale at which the measure is taken. We can eliminate such dependence by applying a limit over , defining in this way the dimensional Hausdorff measure:
(3) 
The plot of as a function of shows a similar behavior in any fractal object analyzed. The value of is for any and it is for any , where always is a nonnegative real value. is the Hausdorff fractal dimension of . More formally,
(4) 
In most practical situations, the Hausdorff dimension is difficult or even impossible to calculate. Thus assuming that any fractal object is intrinsically selfsimilar, the literature shows a simplified version, also known as the similarity dimension or capacity dimension:
(5) 
where is the number of rules with linear length used to cover the object.
In practice, the above expression may be generalized by considering to be any kind of selfsimilarity measure and to be any scale parameter. This generalization has given rise many methods for estimating fractal dimension, with widespread applications to the analysis of objects that are not real fractals (mathematically defined) but that present some degree of selfsimilarity in specific intervals. An example of such a 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 the information dimension, is derived from the information function. This function is defined for any situation in which we have an object occupying a physical space. We can divide this space into a grid of squares with sidelength and compute the probability of points of the object pertaining to some square of the grid. The probability function is given by
(6) 
where is the maximum possible number of points of the object inside a unique square. Here we use a generalization of teh above expression defined in the multifractal theory PV02 :
(7) 
where is any real number.
The dimension itself is given as
(8) 
When this dimension is estimated over a graylevel digital image , a common approach is to map it onto a threedimensional surface as
(9) 
In this case, we construct a threedimensional grid of 3D cubes also with sidelength . The probability is therefore given by the number of grid cubes containing points on the surface divided by the maximum number of points inside a grid cube.
Iii Fractal Descriptors
Fractal descriptors are values extracted from the relationship common to most methods of estimating fractal dimension. Actually, any fractal dimension method derived from the concept of the Hausdorff dimension obeys a powerlaw relation, which may be expressed as
(10) 
where is a measure depending on the fractal dimension method and is the scale at which this measure is taken.
Therefore Fractal descriptors are provided from the function :
(11) 
We call the independent variable to simplify the notation. Thus and our fractal descriptor function is denoted . For the probability dimension used in this work, we have
(12) 
The values of may be directly used as decriptors of the analyzed image or may be postprocessed by some kind of operation aimed at emphasizing some specifical aspects of that function. Here, we apply a multiscale transform to and obtain a bidimensional function , in which the variable is related to and is related to the scale at which the function is observed. A common means of obtaining is through a wavelet transform:
(13) 
where is a wavelet basis function and is the scale parameterGM84 . Figure 2 shows an example where two textures with the same dimension, but visually distinct, provide different descriptors.
Iv Proposed Method
This work proposes to obtain fractal descriptors from textures by using the probability fractal dimension, computing them from the curve in Eq. 8. Empirically, we obtained as the best value of in the Equation 7. Therefore we apply a multiscale transform to .
The multiscale process employs a wavelet transform of , as described in the previous section:
(14) 
As the multiscale transform maps a onedimensional signal onto a bidimensional function, it is a process that generates intrinsic redundancies. There are different approaches to elliminating such redundancies and keeping only the relevant information CC00 . Here, we adopt a simple method, finetuning smoothing, in which is projected onto a specific value of the Gaussian parameter. We tested values of ranging between and and used the values that provided the best performance in the training experiments.
Finally, we selected a specific region from to compose the descriptors. Empirically, we observed that the initial points in this curve provided better performance in our application. Then, we established a threshold after which all points in the convolution curve are disregarded and 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 our probability descriptors to the classification of two benchmark datasets and compared our results to the performance of other wellknown and stateoftheart methods for texture analysis.
The first classification task used the Brodatz dataset, a classic set of natural graylevel textures photographed and assembled in an architecture book B66 . This dataset is composed by 111 classes with 10 textures in each class. Each image has a pixel dimension of 200200.
The second data set was Outex, a set of color textures extracted from natural scenes OMPVKH02 . Here, we used the first 20 classes, each one having 20 images with a 128128 pixel dimension, and converted them to graylevel images.
We compared our probability descriptors to six other techniques, namely, Local Binary Patterns (LBP) PHZA11 , Gaborwavelets MM96 , GrayLevel Difference Method (GLDM) WDR76 , a multifractal approach described in PV02 and BouligandMinkowski fractal descriptors BCB09 ; FB13 .
Therefore we applied a Principal Component Analysis (PCA)
DH00over the data to elliminate or at least attenuate the correlation among the features. Finally we classified each descriptor by a Kfold process, with
, using the Support Vector Machine (SVM) method
V99 and compared the results.Vi Results
Table 1 shows the correctness rate in the classification of the Brodatz dataset using the compared descriptors. The proposed method obtained the best result, outperforming the powerful BouligandMinkowski fractal descriptors and taking substantial advantage over other stateofthe art techniques such as Gabor and LBP. For this result we used and a threshold
. A particularly important aspect of our method with this data set is the reduced number of descriptors needed to provide a precise classification. This point is especially important in large databases, for which computational performance is more relevant. Furthermore, the small number of features avoids the curse of dimensionality, which impairs the reliability of the global result.
Table 1 shows the results for the Outex textures. In this case, we obtained the best result by using and . Again, the proposed approach provided the greatest success rate, despite the challenge of applying a grayscalebased method to color analysis. In fact, Outex textures exhibit nuances which are better expressed in the color information, such as the changes in the lighting perspective and the images from different classes presenting similarities in the intensity distribution, though distiguished by color. Based on this result, our method demonstrates that although it does not use any color information, it is powerful also for color image analysis.
Figure 4
shows how the success rate varies according to the number of descriptors used in both datasets. The graphs show a wellknown property of KarhunenLoève transform. The most expressive information is concentrated in the initial descriptors, so that the success curves show a quick growing and then tend to stabilize at a constant rate. The larger size and the native grayscale format of Bradatz data leads to a clearer advantage of probability descriptors in that database. In Outex, the first descriptors, corresponding to the PCA components with higher variance, do not have as much significance for the classification purpose. However, the sum of all of them provide the best result. This is a specific property of fractal descriptors, as can be observed in the BouligandMinkowski descriptors for the Brodatz data as well. Fractal descriptors are tightly correlated among themselves, thus we do not have a large significance carried only in a few descriptors.
Finally, Figures 5 and 6 show the confusion matrices of the methods with the best performances. In this kind of representation, a good descriptor must produce a matrix with a diagonal as lighter and continuous as possible and the minimum of dark points outside the diagonal.
As can be seen, in Brodatz data, the probability descriptors clearly presented these characteristics, with almost no “gap” in the diagonal and with a few dark points outside. Both gaps and gray points indicate the confusion of the classifier, that is, elements classified incorrectly in some way. This confusion is caused mostly by the high similarity interclass and low similarity intraclass. A precise descriptor, like the proposed, avoids such confusion by providing measures capable of faithfully representing the most complex structures.
In the case of Vistex, the diagonal gaps are not so clear, given the small number of classes. Thus the advantage of the proposed method can be seen in the reduced number of gray squares outside the diagonal. Such squares correspond to the confused classes. We observe that, particularly, the last classes have some discrimination difficulties. The elements of those classes are often assigned to other classes as depicted in the matrices. However, even in these cases, the proposed descriptors showed the expected robustness, assigning the elements correctly.
An overall analysis of the results demonstrates that the proposed method outperformed the compared ones in both datasets, using a small number of descriptors. Such results were expected from fractal theory given its wide applicability to the analysis of natural textures. Actually, fractal geometry presents a remarkable flexibility in the modeling of objects that cannot be well represented by Euclidean rules. The fractal dimension is a powerful metric for the complex patterns and spatial arrangements usually found in nature. Fractal descriptors provide a way of capturing multiscale variations and nuances that could not be measured by conventional methods. More specifically, the probability descriptors proposed here combine a statistical approach with fractal analysis, comprising a framework that supports a precise and reliable discrimination technique, as confirmed in the above results.
Vii Conclusion
We have proposed a novel method for extracting descriptors by applying a multiscale transform over the powerlaw relation of the fractal dimension estimated by the probability method.
We tested the efficiency of the proposed technique in the classification of two wellknown benchmark texture datasets and compared its performance to that of other classical texture analysis methods. The results demonstrated that probability fractal descriptors are a powerful tool for modeling such textures. The proposed method achieved a high success rate in the classification of the benchmark data sets, using fewer than 10 descriptors in this task. These results demonstrate that the proposed method is capable of combining precision, low computational cost and robustness.
As a consequence, our method offers a reliable approach to solve a large class of problems involving the analysis of texture images.
Acknowledgments
J. B. Florindo gratefully acknowledges the financial support of FAPESP Proc. 2012/191433. O. M. Bruno gratefully acknowledges the financial support of CNPq (National Council for Scientific and Technological Development, Brazil) (Grant #308449/20100 and #473893/20100) and FAPESP (The State of São Paulo Research Foundation) (Grant # 2011/015231).
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