 # Texture Analysis And Characterization Using Probability Fractal Descriptors

A gray-level image texture descriptors based on fractal dimension estimation is proposed in this work. The proposed method estimates the fractal dimension using probability (Voss) method. The descriptors are computed applying a multiscale transform to the fractal dimension curves of the texture image. The proposed texture descriptor method is evaluated in a classification task of well known benchmark texture datasets. The results show the great performance of the proposed method as a tool for texture images analysis and characterization.

## Authors

##### This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.

## 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); Tian-Gang 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 fractal-based 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 power-law curve of the dimension and applied a space-scale transform to emphasize the multiscale aspect of the features. Finally, we test the proposed method over two well-known 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); Tian-Gang 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:

 Hsδ(X)=inf∞∑i=1|Ui|s: {Ui is a % δ-cover of X},

where denotes the diameter of , that is, the maximum possible distance between any elements of :

 |X|=sup{|x−y|:x,y∈X}.

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:

 Hs(X)=limδ→0Hsδ(X).

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 non-negative real value. is the Hausdorff fractal dimension of . In a more formal way we may write:

 D(X)={s}|inf{s:Hs(X)=0}=sup{Hs(X)=∞}.

In most practical situations, Hausdorff dimension uses to be complicate or even impossible to calculate. Thus, assuming that any fractal object is intrinsically self-similar, we may derive a simplified version, also called similarity dimension or capacity dimension:

 D=−log(N)log(r),

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 self-similarity 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 self-similarity 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 side-length and calculate the probability of points of the object pertaining to some square of the grid. The probability function is given by:

 NP(δ)=N∑m=11mpm(δ),

where is maximum possible number of points of the object inside a unique square.

The dimension itself is given through:

 D=−limδ→0lnNPlnδ.

When this dimension is estimated over a gray-level digital image , an usual approach is to map it onto a three-dimensional surface through:

 S={i,j,I(i,j)|(i,j)∈[1:M]×[1:N]}. (1)

In this case, we construct a three-dimensional grid of 3D cubes also with side-length . 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. Figure 1: A diagram of probability dimension estimation. Above, the 2D version used for shapes analysis. Below, the 3D version used for gray-level images.

## 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 power-law relation which may be explicit in the following:

 D=−log(M)log(ϵ),

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 :

 u:log(ϵ)→log(M).

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:

 u(t)=−log(NV(δ))log(δ).

The values of function may be used directly as descriptors of the analyzed image or may be post-processed 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 two-dimensional 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:

 U(b,a)=+∞∑n=−∞u(t−n)G(n,a),

where is the well-known Gaussian function and is the smoothing parameter:

 G(n,a)=1√2πaexp(−n2/2t).

Given the finite domain of function , we also must restrict the response of Gaussian filter to a finite interval :

 U(b,a)=+M∑n=−Mu(t−n)G(n,a),

where is a real value which should satisfy:

 2∫∞t=MG(t,a)dt=2∫∞v=M/√aG(v,a)dv

being the tolerance error. Usually, we may have:

 M=Cσ+1=C√(a)+1,

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:

 U(b,a)=dudt∗Ga,

where is the Gaussian function descritized over the interval .

As multiscale transform maps a one-dimensional signal onto a two-dimensional 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 fine-tuning 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 well-known and state-of-the-art 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 gray-level images.

We compare probability descriptors to other 4 other techniques, that is, Gabor wavelets Manjunath and Ma (1996), Co-occurrence matrix Haralick (1979), Gray Level Difference Method (GLDM) Weszka et al. (1976), a multifractal approach described in Parrinello and Vaughan (2002) and Bouligand-Minkowski fractal descriptors Backes et al. (2009).

Finally, we classify each descriptor by a hold-out process (half of data to train and the remaining to test) using K-Nearest Neighbor (KNN), with

and compare the results.

## 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.

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 gray-level based. This aspect turns significant even a tiny classification enhancing.

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 gray-level 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. Figure 4: Confusion matrices in Brodatz dataset. a) Gabor. b) Co-occurrence. c) GLDM. d) Proposed method. Figure 5: Confusion matrices in Vistex dataset. a) Gabor. b) Co-occurrence. c) Bouligand-Minkowski. d) Proposed method.

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 well-known 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 #308449/2010-0 and #473893/2010-0) and FAPESP (Grant # 2011/01523-1).

## References

• Scarlat et al. (2010) E. I. Scarlat, M. Mihailescu,  and A. Sobetkii, “Spatial frequency and fractal complexity in single-to-triple beam holograms,” Journal of Optoelectronics and Advanced Materials, 12, 105–109 (2010).
• Han et al. (2008) D. Han, M. Wang,  and J. Zhou, “Fractal analysis of self-mixing speckle signal in velocity sensing,” Optics Express, 16, 3204–3211 (2008).
• Chappard et al. (2003) D. Chappard, I. Degasne, G. Hure, E. Legrand, M. Audran,  and M. Basle, “Image analysis measurements of roughness by texture and fractal analysis correlate with contact profilometry,” Biomaterials, 24, 1399–1407 (2003).
• Wool (2008) R. P. Wool, “Twinkling Fractal Theory of the Glass Transition,” Journal of Polymer Science Part B - Polymer Physics, 46, 2765–2778 (2008), Annual Meeting of the American-Physical-Society, New Orleans, LA, MAR 10, 2008.
• Tian-Gang et al. (2007) L. Tian-Gang, S. Wang,  and N. Zhao, “Fractal Research of Pathological Tissue Images,” Computerized Medical Imaging and Graphics, 31, 665–671 (2007).
• Lorthois and Cassot (2010) S. Lorthois and F. Cassot, “Fractal analysis of vascular networks: Insights from morphogenesis,” Journal of Theoretical Biology, 262, 614–633 (2010).
• Rossatto et al. (2011) D. Rossatto, D. Casanova, R. Kolb,  and O. M. Bruno, “Fractal analysis of leaf-texture properties as a tool for taxonomic and identification purposes: a case study with species from neotropical melastomataceae (miconieae tribe),” Plant Systematics and Evolution, 291, 103–116 (2011).
• (8) J. B. Florindo, M. S. Sikora, E. C. Pereira,  and O. M. Bruno, “Multiscale fractal descriptors applied to nanoscale images,” arXiv:1201.3410 [physics.data-an] .
• Harte (2001) D. Harte, Multifractals: theory and applications (Chapman and Hall/CRC, 2001).
• Lashermes et al. (2008) B. Lashermes, S. G. Roux, P. Abry,  and S. Jaffard, “Comprehensive multifractal analysis of turbulent velocity using the wavelet leaders,” European Physical Journal B, 61, 201–215 (2008).
• Lovejoy et al. (2000) S. Lovejoy, P. Garrido,  and D. Schertzer, “Multifractal absolute galactic luminosity distributions and the multifractal Hubble 3/2 law,” Physica A - Statistical Mechanics and its Applications, 287, 49–82 (2000).
• Manoel et al. (2002) E. T. M. Manoel, L. da Fontoura Costa, J. Streicher,  and G. B. Müller, “Multiscale fractal characterization of three-dimensional gene expression data,” in SIBGRAPI (IEEE Computer Society, 2002) pp. 269–274, ISBN 0-7695-1846-X.
• da F. Costa and Cesar, Jr. (2000) L. da F. Costa and R. M. Cesar, Jr., Shape Analysis and Classification: Theory and Practice (CRC Press, 2000).
• Plotze et al. (2005) R. O. Plotze, J. G. Padua, M. Falvo, M. L. C. Vieira, G. C. X. Oliveira,  and O. M. Bruno, “Leaf shape analysis by the multiscale minkowski fractal dimension, a new morphometric method: a study in passiflora l. (passifloraceae),” Canadian Journal of Botany-Revue Canadienne de Botanique, 83, 287–301 (2005).
• Bruno et al. (2008) O. M. Bruno, R. de Oliveira Plotze, M. Falvo,  and M. de Castro, “Fractal dimension applied to plant identification,” Information Sciences, 178, 2722–2733 (2008).
• Backes et al. (2009)

A. R. Backes, D. Casanova,  and O. M. Bruno, “Plant leaf identification based on volumetric fractal dimension,” International Journal of Pattern Recognition and Artificial Intelligence (IJPRAI),

23, 1145–1160 (2009).
• Florindo et al. (2010) J. B. Florindo, M. De Castro,  and O. M. Bruno, “Enhancing Multiscale Fractal Descriptors Using Functional Data Analysis,” International Journal of Bifurcation and Chaos, 20, 3443–3460 (2010).
• Florindo et al. (2012) J. Florindo, A. Backes, M. de Castro,  and O. Bruno, “A comparative study on multiscale fractal dimension descriptors,” Pattern Recognition Letters, 33, 798 – 806 (2012), ISSN 0167-8655.
• Backes et al. (2012) A. R. Backes, D. Casanova,  and O. M. Bruno, “Color texture analysis based on fractal descriptors,” Pattern Recognition, 45, 1984 – 1992 (2012), ISSN 0031-3203.
• Florindo and Bruno (2011) J. B. Florindo and O. M. Bruno, “Fractal descriptors in the fourier domain applied to color texture analysis,” Chaos, 21, 043112–043122 (2011).
• Florindo and Bruno (2012) J. B. Florindo and O. M. Bruno, “Fractal descriptors based on fourier spectrum applied to texture analysis (in press),” Physica A (2012).
• Mandelbrot (1968) B. B. Mandelbrot, The Fractal Geometry of Nature (Freeman, 1968).
• Brodatz (1966) P. Brodatz, Textures: A photographic album for artists and designers (Dover Publications, New York, 1966).
• Singh and Sharma (2001) S. Singh and M. Sharma, “Texture analysis experiments with meastex and vistex benchmarks,” in Advances in Pattern Recognition - ICAPR 2001, Second International Conference Rio de Janeiro, Brazil, March 11-14, 2001, Proceedings, Lecture Notes in Computer Science, Vol. 2013, edited by S. Singh, N. A. Murshed,  and W. G. Kropatsch (Springer, 2001) pp. 417–424.
• Manjunath and Ma (1996) B. Manjunath and W. Ma, “Texture features for browsing and retrieval of image data,” IEEE Transactions on Pattern Analysis and Machine Intelligence, 18, 837–842 (1996), ISSN 0162-8828.
• Haralick (1979) R. M. Haralick, “Statistical and structural approaches to texture,” Proceedings of the IEEE, 67, 786–804 (1979).
• Weszka et al. (1976) J. Weszka, C. Dyer,  and A. Rosenfeld, “A comparative study of texture measures for terrain classification,” SMC, 6, 269–286 (1976).
• Parrinello and Vaughan (2002)

T. Parrinello and R. A. Vaughan, “Multifractal analysis and feature extraction in satellite imagery,” International Journal of Remote Sensing,

23, 1799–1825 (2002).