I Introduction
In the last decades, Fractal Geometry has demonstrated to be a worthy tool to develop robust and precise methods of image analysis SMS10 ; HWZ08 ; CDHLAB03 ; W08 ; TWZ07 ; LC10 . Such methods have been successfully applied to a number of problems comprising the analysis of digital images in several areas, such as Physics POPMBD97 ; BOAM10 , Medicine KSASKC12 ; GQPJ11 , Engineering ACNS09 ; DOSSS09 , etc.
Among such fractalbased approaches, methods like multifractals XJF09 , multiscale fractal dimension MCSM02 , local fractal dimension K99 , etc. have outperformed other classical and stateoftheart image analysis methods in many situations.
More recently, a novel fractalbased imaging method named fractal descriptors has been proposed in BCB09 ; FB11
. Roughly speaking, this approach extracts features of the image by computing the fractal dimension at different scales of observation and taking all these values over a predefined range. Although this approach has demonstrated to be a promissing solution for image analysis problems, it was defined and studied only for a limited number of wellknown techniques to estimate the fractal dimension. Among the studied possibilities, BouligandMinkowski has provided remarkable results both for the analysis of natural and synthetic images
BCB09 . These descriptors are obtained from the interior measures (volumes) of the object of interest dilated by spheres with a predefined range of radius values.Despite the great results achieved, BouligandMinkowski takes into account only the interior (volume) of the dilated structure. It is wellknown from Geometry that the boundary of a structure encloses information as rich as the interior and, for instance, in a threedimensional space, two objects with the same volume may have different areas. In this way, this work proposes to enhance the BouligandMinkowski descriptors by including information from the boundary, that is, the area in a threedimensional space. The combination is accomplished by means of a simple concatenation of measures, followed by a dimensionality reduction through the cannonical analysis CorreaSC10 . The performance of the proposal is assessed over databases of texture images and the results are compared to other classical and stateoftheart methods in the literature.
Ii Fractal Geometry
A number of works characterizing and analyzing images using fractal geometry have been reported in the literature SMS10 ; HWZ08 ; CDHLAB03 ; W08 ; TWZ07 ; LC10 . Generally, these works model objects and scenarios from the real world as an approximation of mathematical or statistical fractals and extract fractal properties of the element. The most used of these properties is the fractal dimension.
The formal definition of the fractal dimension, also called HausdorffBesicovitch dimension, is obtained from the Hausdorff measure. Let be a geometrical set of points in an dimensional topological space. Its Hausdorff measure is calculated by
(1) 
where is a cover of ; that is, there exists a countable collection of sets , with , such that and denotes the diameter of , that is, the maximum possible distance between two any elements of :
(2) 
As the parameter is a superior limit for the diameter of the balls covering the fractal object, it can be considered a scale metric and should be removed from the dimension definition once this is scaleindependent. In this way a limit to is applied over giving rise to the measure
(3) 
As it can be demonstrated in Measure Theory, has a particular behavior that arises for any set of points , that is, the value of is always for any and for any , where is a nonnegative real value. The point of discontinuity is the HausdorffBesicovitch fractal dimension of
(4) 
Although the above definition is the most exact and generalist method to calculate the dimension, by using an infinitesimal covering, it is necessary to know the analytical expression of the object being measured. However this is not possible when the realworld element approximated by a fractal is represented in a discrete and finite space as the digital images discussed in this work. To address these situations, several approximation methods have been proposed in the literature F86 ; FB11b . Such methods aim to compute a measure of selfsimilarity and complexity of the object by generalizing the definition of the Euclidean dimension. Thus the object is measured by a rule unit with the same topological dimension of the object. The length of this unit is ranged along an interval to compute the number of units necessary to cover the object. In this context, the dimension is given by
(5) 
ii.1 BouligandMinkowski
One of the most commonly used techniques to estimate the dimension based on Equation 5 is the BouligandMinkowski F86 . Similarly to the HausdorffBesicovitch definition, it is also derived from a measure, in this case, the BouligandMinkowski measure of a set
(6) 
where is a parameter and is the volume of the edge of () morphologically dilated by a structuring element , symmetrical with respect to the origin and with radius .
The BouligandMinkowski fractal dimension is given by
(7) 
In practice, the dimension is computed by a neighborhood strategy. Each point of the object is replaced by a structuring element , with radius , and the number of points within the union of such elements is used to estimate the volume . Thus the dimension is provided by
(8) 
For an object represented in a digital image, an efficient and precise method to compute the BouligandMinkowski dimension is the Euclidean Distance Transform FCTB08 , by Saito’s Algorithm SaitoT94 . Starting from a texture image , it is mapped into a 3D structure (surface) , such that each point with coordinate and pixel intensity is converted into the point with coordinate in . To simplify the idea, the object is supposed to be in . The distance transform is given by
(9) 
for all , where is the Euclidean Distance between and .
The set of possible distances (Euclidean) is given by
(10) 
In the following, these values of are sorted increasingly
(11) 
Thus the dilation volume is computed by
(12) 
and the dimension is given by
(13) 
Numerically, this limit uses to be given by the slope of a straight line fit to the curve . Figure 1 illustrates the dilation process.
ii.2 Fractal Descriptors
Equation 5 can be generalized by replacing by any selfsimilarity measure , where is a scale parameter.
This generalization results in the proposal and study of several methods to approximate the dimension F86 ; FB11b , each one providing results more or less close to the theoretical value, depending on the specific application. However, the outcome still is only a single real value to describe all the complexity of an object. Moreover, when this object is not a mathematical fractal, its dimension is highly scaledependent and a global dimension may be of little or no usefulness.
To address these points and make possible a more complete fractalbased analysis of realworld structures, the fractal descriptors were proposed in BCB09 ; FB11 . Basically, instead of computing only the fractal dimension, the fractal descriptors are composed by all the values of dimension at each scale along a range of observation. Considering that is a scale parameter, such set of descriptors can be obtained from the selfsimilarity curve:
(14) 
Using the BouligandMinkowski method described previously, in thres dimensions, these descriptors are given by the logarithm of the dilation volumes
(15) 
Iii Boundary Measure versus Interior Measure
On the above discussion, the Equation 15 and the previous works BCB09 ; FB11 talk about the use of interior information (volume in three dimensional space) to obtain the fractal descriptors. This work proposes to analise the difference between the information comprised within the boundary and the interior of a region. In two dimensions, this analysis consists of studying measures of perimeter and area of a flat shape. In the threedimensional case, the discussion concerns area measures of surfaces and measures of volumes.
The main idea depicted here is that two different objects, with a dilatated radius , can have the same interior measure, but different boundary measures.
To support this theory, the dilation of circles is analysed in two dimensions. Figure 2 illustrates three different circle arrangements, with three different center points for each case and the same radius. It is an exemple that the interior areas of three circles, including the intersections, are the same but the the boundary measures of the regions, i.e. the perimeters, are different.
In Image Processing, the perimeter of a flat region can be estimated by
(16) 
where and
are the number of even and odd codes, respectively, in chaincode representation
CC00 ; C96 . Equation 16 simply performs a count of the number of pixels on the boundary of a region and estimates the arc length of its contour.In the same way, the area measure is estimated by the count of pixels that represent the object . Therefore, the area is obtained by
(17) 
where denotes any pixel on the image.
After applying Equations 17 and 16 for the three arrangements on Figure 2, similar area measures and different perimeters are obtained. Table 1 shows these results to the arrangements of the Figure 2.
Arrangements  Center Points  Area  Perimeter 

Left  (6,6); (8,6); (15,10)  152  49.799 
Center  (6,6); (6,12); (10,19)  152  44.627 
Right  (6,6); (12,9); (14,7)  152  46.627 
These results support our theory, we can verify that the two different images (or pixels arrangement) can have the same dilated areas (to a given dilatated radius ), but different perimeters. In this situation only the perimeter is useful to discriminate the images. In this way we can use the bondary measure as an complementary feature to compose our fractal descriptor.
Iv Applications on Images
As showed by the above discussion, the area and perimeter of a dilated object express complementary information. In this way the most natural approach is to use both features together. The synergy achieved by this approach can bring a better discrimination power to the fractal descriptors. To do that, a simple concatenation of both feature vectors is made in order to obtain a more rich fractal descriptors.
In order to verify our theory an application of texture analysis is carried out using the novel proposed fractal descriptor. Such analisys is performed over a supervised classification task, though the proposed method can be also used to perform a CBIR (Contentbased image retrieval), segmentation or other kinds of image analysis.
For this application, the both signatures (area and volume) are computed for each image, concatened and the supervised classification is carried out by applying a Canonical Analysis CorreaSC10 followed by a Linear Discriminant Analysis (LDA) (also called Fisher linear discriminant) Fukunaga90 . The 10fold crossvalidation scheme is used in all experiments and over benchmark databases. The canonical analysis is, basically, a geometric transformation of the feature space in order to generate new uncorrellated features based on linear combinations. The idea of this method is to find a new projection of the data where the class separation is maximized. From original features, cannonical variables can be obtained. However, a reduction in the number of variables to be evaluated is usually desired. Therefore, a LDA supervised classification is accomplished by using the most significant cannonical variables.
iv.1 Texture
Three texture sets are used in the experiments: Brodatz, Vistex and Outex.

Brodatz texture database is derived from the Brodatz Album Brodatz66 and has become the standard for evaluating texture algorithms, with hundreds of studies having been applied to this set of images. This database is composed by texture samples grouped into classes. Each image is pixels with 256 graylevels.

The Vision texture database (VisTex) Vistex09 is maintained by the Vision and Modelling group at the MIT Media Lab. The full database contains images representative of realworld textures under practical conditions (lighting, perspective, etc). In this work, we use the original Vistex images with resolution . Each image was split into nonoverlapping subimages with dimension . These images are avaliable on de Vistex site as the test suite Contrib_TC_00006 OjalaMPVKH02 .

The suite Outex_TC_00013 is provided by the Outex texture database OjalaMPVKH02 . This database includes a collection of natural scenes acquired under strictly controlled conditions. The test suite provides a meaningful framework for the empirical evaluation of a candidate texture analysis algorithm. A database of color texture images () was constructed by splitting each one of the original texture image () into nonoverlapping subimages.
After the concatenation and Canonical Analysis the success rate may vary depending on the number of the
cannonical variables used in LDA classifier. Figure
3 illustrates the behavior of the success rates when the number ofcannonical variables is ranged. We observe that, for Vistex data set, the rate increases at a first moment, achieves an optimal rate and thus stabilizes with a very small decrease when we consider more descriptors. This behavior was expected since the high dimensionality of the feature vectors damage the efficiency of the classifier. For all considered fractal descriptors this same behavior can be observed.
We also can verify that the volumetric information is a little bit more effective than area counterpart. However, we see that the synergy between such two kinds of infomation yields a improvement on the perfomace of the texture analysis.
Based on the behavior observed in Figure 3, we setup an experiment with a total of 40 cannonical variables. Table 2 shows the achieved results. For all data sets, the best sucess rate is provided by the combination of volumetric and surface features, the average performance improvement is close to 2%. This result corroborates our theory that the area can be useful as a complementary information in the studied fractal descriptor. It is important to emphasize that combining different features does not increase the number of descriptors used in the classifier.
Success rate (%) and stardart deviation (%)  

Methods  Brodatz  VisTex  Outex 
Volumetric  88.15(0.26)  91.28(0.67)  80.66(0.43) 
Area  87.93(0.29)  89.98(0.52)  80.91(0.37) 
Volumetric + Area  89.29(0.21)  93.09(0.31)  82.57(0.33) 
V Conclusion
This work proposed a new way of computing fractal descriptors from graylevel fractal descriptors. The method combines information from the interior (volume) and boundary (area) of a surface representation of the image dilated by spheres with variable radii (Minkowski dilation).
The results of applying the proposal to the classification of benchmark data sets showed that by using a reasonable number of descriptors, the volume achieved higher correctness rates than the area used in the BouligandMinkowski approach. However the performance is enhanced when combining area and volume information.
In fact, the area enriched the descriptors in that it provides a different and complementary viewpoint of the texture. Thus different images can have the same volume for a specific dilation, but its area may be different, contributing for a more robust discrimination.
Acknowledgments
O. M. Bruno gratefully acknowledges the financial support of CNPq (National Council for Scientific and Technological Development, Brazil) (Grant #308449/20100 and Grant #484312/20138) and FAPESP (The State of São Paulo Research Foundation) (Grant # 11/015231). M. W. S. Oliveira is thankful to CAPES (Coordination for the Improvement of Higher Education Personnel) for financial support. J. B. Florindo gratefully acknowledges the financial support of FAPESP Proc. 2013/222053 and 2012/191433. D. Casanova is grateful to FAPESP Proc. 13/149842 for financial support.
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