Lucas C Ribas
email: lucasribas@usp.br

Antoine Manzanera
email: antoine.manzanera@ensta-paristech.fr

Odemir Bruno
email: bruno@ifsc.usp.br

A Fractal-Based Approach to Network Characterization Applied to Texture Analysis

Keywords: Complex Networks, Fractal Dimension, Texture analysis

This work proposes a new method for texture analysis that combines fractal descriptors and complex network modeling. At first, the texture image is modeled as a network. Then, the network is converted into a surface where the Cartesian coordinates and the vertex degree is mapped into a 3D point in the surface. Then, we calculate a description vector of this surface using a method inspired by the Bouligand-Minkowski technique for estimating the fractal dimension of a surface. Specifically, the descriptor corresponds to the evolution of the volume occupied by the dilated surface, when the radius of the spherical structuring element increases. The feature vector is given by the concatenation of the volumes of the dilated surface for different radius values. Our proposal is an enhancement of the classic complex networks descriptors, where only the statistical information was considered. Our method was validated on four texture datasets and the results reveal that our method leads to highly discriminative textural features.

Introduction

Texture is a visual pattern related to the surface of a material or an object and it is considered as a key feature to image interpretation. Texture classification is used in many fields such as material science [START_REF] Zimer | Investigation of aisi 1040 steel corrosion in h2s solution containing chloride ions by digital image processing coupled with electrochemical techniques[END_REF], industrial inspection [START_REF] Malamas | A survey on industrial vision systems, applications and tools[END_REF], geology [START_REF] Wenk | Preferred Orientation in Deformed Metal and Rocks: An Introduction to Modern Texture Analysis[END_REF], etc. There are many classical texture characterization methods in the literature that can be divided into four different categories: statistical-based (e.g., gray-level co-occurrence matrices (GLCM) [START_REF] Haralick | Textural features for image classification[END_REF] and local binary patterns (LBP) [32]), spectral methods (e.g. Gabor filters [START_REF] Jain | Unsupervised texture segmentation using Gabor filters[END_REF] and wavelet transform [START_REF] De Ves | A statistical model for magnitudes and angles of wavelet frame coefficients and its application to texture retrieval[END_REF]), structural methods (e.g. morphological decomposition [START_REF] Lam | Rotated texture classification by improved iterative morphological decomposition[END_REF]) and model-based methods (e.g. Fractal models [START_REF] Backes | Color texture analysis based on fractal descriptors[END_REF][START_REF] Backes | Plant leaf identification based on volumetric fractal dimension[END_REF][START_REF] Ribas | Fractal dimension of maximum response filters applied to texture analysis[END_REF], complex networks [START_REF] Ribas | Fusion of complex networks and randomized neural networks for texture analysis[END_REF] and stochastic models [START_REF] Panjwani | Markov random field models for unsupervised segmentation of textured color images[END_REF]) Currently, methods based on complex networks theory have attracted significant attention of the computer vision community, due to the promising results and their capacity to represent the relationships among structural properties of texture [START_REF] Florindo | Fractal measures of complex networks applied to texture analysis[END_REF][START_REF] Ribas | Distance transform network for shape analysis[END_REF][START_REF]Dynamic texture analysis with diffusion in networks[END_REF]. The complex network techniques represent the texture image as a network and extract measures capable of characterizing precisely the structural patterns of texture modeled. In this sense, the network modeling is an important step whose aim is to represent different aspects like spatial and pixel intensity distribution, texture irregularities, auto-similarity scales, etc. [START_REF] Florindo | Fractal measures of complex networks applied to texture analysis[END_REF]. Now a challenging problem is to find a relevant set of measures to extract from the network to characterize it. Previous works [START_REF] Backes | Texture analysis and classification: A complex network-based approach[END_REF][START_REF] Chalumeau | Texture discrimination using hierarchical complex networks[END_REF] use simple metrics based on degree distribution, however, it is noticeable that more complex measures are needed for a finer characterization [START_REF] Florindo | Fractal measures of complex networks applied to texture analysis[END_REF][START_REF] Costa | Characterization of complex networks: A survey of measurements[END_REF].

To this end, we propose in this paper a new approach for texture analysis that uses fractal descriptors as measures for complex network characterization. The proposal is to use the fractal measures to characterize the network topology instead of using network statistical measures. To achieve this, firstly, the texture image is modeled as a network and this network is converted to a surface. Then, we calculate a feature vector using a method inspired by the Bouligand-Minkowski technique to estimate the fractal dimension of the surface that maps a network. The proposed signature is composed of the influence volume computed of this surface for different radius values. We verify the performance of the proposed approach in the classification of four texture image datasets and the results are compared to other literature methods.

The paper is organized as follows. In Section 2, we describe the proposed approach in detail. Section 3 reports the experimental setup. The results and discussion are presented in Section 4, followed by the conclusion in Section 5.

Proposed Method

Modeling texture as network

According to Backes et al [START_REF] Backes | Texture analysis and classification: A complex network-based approach[END_REF] a texture image I can be modeled as a network G = (V, E) considering each pixel as a vertex. Each pixel i ∈ V is characterized by its Cartesian coordinates x i and y i , and an integer value I(i) ∈ [0, 255] which represents its gray-level. Two vertices (pixels) i and j are connected by a nondirected edge e ij ∈ E if the Euclidean distance between them is smaller or equal than a given value d, according to:

e ij ∈ E ⇐⇒ (x i -x j ) 2 + (y i -y j ) 2 ≤ d (1)
For each edge a weight w(e ij ) is defined by a normalized combination of the Euclidean distance and the difference of intensities between two pixels:

w(e ij ) = (x i -x j ) 2 + (y i -y j ) 2 + d 2 |I(i)-I(j)| 255 2d 2 (2)
This weight function includes information about the pixel surroundings and the normalization of the difference of pixel intensity in the interval [0, d 2 ] aims to balance the relative importance between geometric and color information in the texture representation [START_REF] Backes | Texture analysis and classification: A complex network-based approach[END_REF]. Each network vertex i has a degree k i , which is the number of edges connected to i. Note that until now the network vertices all have the same degree. Thus, it is necessary to apply a transformation in order to highlight texture properties. This can be done by applying a threshold t over the network, removing all edges whose weight is higher than a given value t and obtaining a new set of edges E t . In this way, a set of thresholds T, t ∈ T can be applied over the network in order to study the behavior of its properties along the successive transformations.

Fractal Dimension of Network Degrees

A way to describe how irregular is an object is to use the Fractal dimension, which can be estimated in various ways [START_REF] Goncalves | Texture descriptor combining fractal dimension and artificial crawlers[END_REF]. In this paper, we compute the fractal descriptors of the complex network using a method inspired by the Bouligand-Minkowski method, which is one of the most accurate approach to estimate the fractal dimension [START_REF] Backes | Plant leaf identification using color and multi-scale fractal dimension[END_REF][START_REF] Bruno | Fractal dimension applied to plant identification[END_REF]. This method estimates the fractal dimension based on the size of the influence area |S(r)| generated by the dilation of surface S ⊂ R 3 using a radius r [START_REF] Goncalves | Texture descriptor combining fractal dimension and artificial crawlers[END_REF]. The Bouligand-Minkowski fractal dimension D of a surface S varying the radius r is defined as:

D(S) = 3 -lim r→0 log V S (r) log r , (3)
where V S (r) is the influence volume achieved with the dilation process of all points s of the surface S using a sphere of radius r [START_REF] Goncalves | Texture descriptor combining fractal dimension and artificial crawlers[END_REF][START_REF] Backes | Plant leaf identification using color and multi-scale fractal dimension[END_REF]. V S (r) can be calculated by the Euclidean Distance Transform (EDT) in order to achieve a fast algorithm [START_REF] Backes | Plant leaf identification using color and multi-scale fractal dimension[END_REF][START_REF] Fabbri | 2d euclidean distance transform algorithms: A comparative survey[END_REF]. The distance transform DT is the function defined as:

DT (x, y, k) = min (x ,y ,k )∈S d ((x, y, k), (x , y , k )) , (4)
for all points (x, y, k) from the cubic grid N 3 [START_REF] Backes | Plant leaf identification using color and multi-scale fractal dimension[END_REF][START_REF] Da | Enhancing fractal descriptors on images by combining boundary and interior of minkowski dilation[END_REF]. This way, the set of possible radii (distances) R is

R = {r : r = x 2 + y 2 + k 2 ; (x, y, k) ∈ N 3 }, (5)
with the values r sorted increasingly to create the set of possible radii

R = {0, 1, √ 2, √ 3, 2 √ 2, ..., r max } [39]. In this approach, the influence volume is com- puted by V S (r) = #{(x, y, k) : DT (x, y, k) = r} [39].
In the proposed method, we apply the fractal dimension theory to characterize the topology of a network that models a texture image. To achieve this, a network modeling texture can be easily mapped onto a 3D surface S ⊂ R 3 , by converting the Cartesian coordinates (x i , y i ) and the degree k i of each vertex i into a 3D point s i = (x i , y i , k i ). In this way, we can characterize the network topology relating the degree and the vertex/pixel position in the image. Figure 1(a) illustrates a network modeling a texture image and (b) the degree of the vertices.

Figures 1(b) and (c) illustrate the mapping of the network onto a surface. Note that the Z axis is the degree of the vertices. The dilation process of this surface is shown in Figures 1(d) and (e). Figure 2(a) shows the degree of vertices of a network modeling a texture image represented by an image. Figures 2(a)-(c) show the dilatation process of the surface that represents this network with each point of the 3D surface dilated by a sphere of radius r. Notice that more collisions happen among the dilated points as we increase the value of radius r. These collisions are directly related to the geometry of the surface S and determine the way the total influence volume V S (r) increases, according to the changes in the network topology. Thus, the pattern produced by the influence volume V S (r) for different radii can be used to describe the network topology and, consequently, the texture patterns.

Feature Vector

In this paper, we propose to use the influence volume V S (r) of the surface (that maps a network) as a feature vector. The influence volume provides a rich shape descriptor and it was used with success to discriminate texture in previous works [START_REF] Goncalves | Texture descriptor combining fractal dimension and artificial crawlers[END_REF][START_REF] Backes | Plant leaf identification using color and multi-scale fractal dimension[END_REF]. Thus, the feature vector is composed of the concatenation of the influence volume V S (r) computed for different values of radius r:

Ψ d t = [V S (1), V S ( √ 2), V S ( √ 3), ..., V S (r max )], (6)
where t is the value of threshold applied in the network, d is the distance used in network modeling and r max is the maximum radius.

Note that the feature vector Ψ d t is built using all possible radius values from r = 1 to r max . In order to analyze the behavior of the network topology from the successive transformation, we combine the feature vector Ψ d t for different values of threshold t, starting from t 0 to t f , to finally get the complete texture signature as follows:

Υ d = [Ψ d t0 , Ψ d t1 , ..., Ψ d t f ]. (7)
3 Results and Discussion

Experimental setup

The proposed signature is computed for each image. After that, the supervised classification is carried out by applying a Canonical Analysis [START_REF] Correa | Musical genres: beating to the rhythms of different drums[END_REF] followed by a Linear Discriminant Analysis (LDA) [START_REF] Fukunaga | Introduction to Statistical Pattern Recognition[END_REF]. This is a simple classifier that emphasizes the features extracted by the methods. We apply the Canonical Analysis here due to the presence of high correlation of the volume features [START_REF] Da | Enhancing fractal descriptors on images by combining boundary and interior of minkowski dilation[END_REF]. Basically, the canonical analysis is a geometric transformation of the feature space to generate new uncorrelated features. Thus, p-canonical variables can be obtained from the original features and the LDA supervised classification is accomplished by using the most significant p-variables [START_REF] Da | Enhancing fractal descriptors on images by combining boundary and interior of minkowski dilation[END_REF]. To separate the training and sets we adopted the leave-one-out scheme. Thus, one sample is used for testing and the remainder for training. This process is repeated for each sample of dataset. The average accuracy of all tests is used as a performance measure.

To evaluate the methods the databases used as benchmark were:

-Brodatz [START_REF] Brodatz | Textures: A photographic album for artists and designers[END_REF]: just as in [START_REF] Backes | Texture analysis and classification: A complex network-based approach[END_REF] We compare the accuracy of our proposed method to other descriptors proposed in the literature. For fair comparison purposes, all methods are compared using the LDA classifier and leave-one-out cross-validation scheme. The compared methods are: Grey-Level Co-occurrence Matrix (GLCM) [START_REF] Haralick | Statistical and structural approaches to texture[END_REF], Gray Level Difference Matrix (GLDM) [START_REF] Kim | Statistical textural features for detection of microcalcifications in digitized mammograms[END_REF], Windowed Fourier transforms [START_REF] Azencott | Texture classification using windowed Fourier filters[END_REF], Gabor Filters [START_REF] Manjunath | Texture features for browsing and retrieval of image data[END_REF], Fractal [START_REF] Backes | Plant leaf identification based on volumetric fractal dimension[END_REF], Fractal Fourier [START_REF] Florindo | Fractal descriptors based on Fourier spectrum applied to texture analysis[END_REF], Local Binary Patterns (LBP) [32], Local Binary Patterns Variance (LBPV) [START_REF] Guo | Rotation invariant texture classification using lbp variance (lbpv) with global matching[END_REF], Complete Local Binary Pattern (CLBP) [START_REF] Guo | A completed modeling of local binary pattern operator for texture classification[END_REF], Local Phase Quantization (LPQ) [START_REF] Ojansivu | Blur insensitive texture classification using local phase quantization[END_REF], Local Configuration Pattern (LCP) [START_REF] Guo | Texture classification using a linear configuration model based descriptor[END_REF], Local Frequency Descriptor (LFD) [START_REF] Maani | Noise robust rotation invariant features for texture classification[END_REF], Binarized Statistical Image Features (BSIF) [START_REF] Kannala | Bsif: Binarized statistical image features[END_REF] and Complex Network Texture Descriptors (CNTD) [START_REF] Backes | Texture analysis and classification: A complex network-based approach[END_REF].

Parameter Evaluation

In this paper, the set of thresholds T and distance for connection d used are identical as the ones used in [START_REF] Backes | Texture analysis and classification: A complex network-based approach[END_REF]. It was defined by an initial threshold t 0 = 0.005, an increment of ∆t = 0.015 and a final threshold t f = 0.530. The maximal distance d for connection of the vertices was set up as d = 3. Figure 3(a) shows the accuracy variation with respect to the number of the p-canonical variables used in the LDA classifier for different datasets and with a fixed value of maximal radius r = 3. Note that, for all datasets, the accuracy increases at a first moment, achieves an optimal accuracy and then stabilizes. Such behavior is expected and is a good indicator for adjusting the dimension of the descriptor, since the high number of features damages the efficiency of the classifier [START_REF] Da | Enhancing fractal descriptors on images by combining boundary and interior of minkowski dilation[END_REF]. From the behavior observed in Figure 3(a), we accomplish the remainder of the experiments using a total of 130 p-canonical variables.

Figure 3(b) presents the accuracy variation for all datasets when the maximal value of radius r max is ranged. The original feature vector before the canonical analysis has the size of 36, 144, 288 ,504, 792, 1 116, 1 512, 1 944, 484, 3 060, 3 672, 4 356, 5 112, 940, 6 804 for the 15 respective integer values of r max between 1 and 15. We observe that, as we increase the maximal value the accuracy also increases. However, it begins to stabilize or decrease from the radius r max = 5 and r max = 6. Based on this behavior, we define the value of radius r max = 6 as the maximum dilation radius. Therefore, we define the following feature vector

Ψ d t = [V S (1), V S ( √ 2), V S ( √ 3), ..., V S ( √ 36)].

Comparison with other methods

In this section, we performed comparisons with literature methods in order to evaluate the performance obtained by the proposed approach. In all experiments were used the LDA classifier with leave-one-out, except for CLBP descriptor (1-Nearest Neighborhood (1-NN) classifier with distance Chi-square), which followed the original paper. For our approach, it was adopted the parameter setup discussed in section 3.2.

The results obtained by all the texture methods in the four datasets are presented in Table 1. We can note that the proposed approach achieved the best results when compared to the other literature methods in the four datasets. In the Brodatz and Vistex datasets, the CLBP and CNDT methods obtained the second best results. On the other hand, the second best accuracy in the Outex and USPTex datasets was obtained by the GLDM and CNDT methods.

We can also verify in Table 1 that our method reached higher accuracy than the CNDT method. The CNDT method is also based on complex networks, and we use in our approach the same network modeling as them. The difference between the CNDT method and our approach is the measure (statistical measures based on the degree histogram) used to characterize the network. Therefore, this suggests that our approach obtained superior performance due to the fractal measure extracted from the network. Our interpretation is that our volumetric fractal descriptors can characterize more richly the network topology, considering the degree and spatial arrangement of the vertices, in contrast with CNDT method that only used degree frequency.

On the other hand, the proposed method also outperformed the Fractal method [START_REF] Backes | Plant leaf identification based on volumetric fractal dimension[END_REF] which also uses the volumetric fractal descriptors to describe the image. However, this approach uses the Cartesian coordinates and the gray intensities to represent the surface. This shows that the proposed approach that combines fractal measures and complex networks improves the ability of discrimination when compared to both approaches isolated.

We can also compare our results with learned descriptors (e.g. using convolutional neural network (CNN)). For comparison purposes, we considered the Outex and USPTex datasets, which are the most challenging. The results using the HardNet++ [START_REF] Mishchuk | Working hard to know your neighbor's margins: Local descriptor learning loss[END_REF], InceptionV3 [START_REF] Szegedy | Rethinking the inception architecture for computer vision[END_REF] and ResNet101 [START_REF] He | Deep residual learning for image recognition[END_REF] methods for feature extraction were considered. On the USPTex dataset, our approach obtained the highest accuracies when compared to the InceptionV3 (92.71%) and HardNet++ (94.20%). On the other hand, the ResNet101 obtained an accuracy of 96.50%. For the Outex dataset, the accuracies were 86.98%, 88.97% and 89.34% for the InceptionV3, ResNet101 and HardNet++, respectively. Our approach overcomes the CNN methods in some cases and in others it obtain close results. Thus, our method is still competitive due to its simplicity.

Conclusion

In this paper, we have proposed a new method based on fractal descriptors and complex network for texture analysis. The method extracts fractal measures from a network that models a texture image. For this, we have proposed to map the network a 3D surface by converting the Cartesian coordinates and the vertex degree as a point in the surface.

We have demonstrated how the texture description can be improved by combining the fractal descriptors and complex networks, instead of using only the fractal descriptor or complex network approaches separated. Experiments on four datasets indicate that our method significantly improved the classification rate with regard to the original complex networks method and fractal descriptors method. The results also showed that our approach overcomes the other compared literature methods. As future work, we believe that different techniques to estimate the fractal dimension of the network can be investigated. In addition, new ways to obtain the feature vector can be studied, such as using different sets of radius values. Another future idea is to extend the proposed approach to boundary shapes analysis. The proposed idea also opens a promising research field for network characterization using fractal measures.

Fig. 1 . 3 Fig. 2 .

 132 Fig. 1. Illustration of the proposed approach inspired by the Bouligand-Minkowski method for estimating the fractal dimension of a network. (a) Network modeling pixels. (b) Degree of the vertices. (c) 3D surface mapping the network in (a), converting each vertex into a 3D point. (d)-(e) Dilation process of the surface.

Fig. 3 .

 3 Fig. 3. Accuracy of our approach for different datasets. (a) Accuracy versus number of p-canonical variables. (b) Accuracy versus the maximal values of the radius.

  , this dataset is composed of 1776 texture images of 128 × 128 pixel size divided into 111 classes, 16 images per class. -Outex [31]: the dataset used in this work is composed of 68 texture classes from TC Outex 00013 with 20 samples each of 128 × 128 pixel size without overlapping. The dataset has a total of 1 360 textures. -USPTex [6]: this dataset is composed of 2 292 samples divided into 191 classes, 12 images per class, and each image has 128 × 128 pixels size. -Vistex [35]: the database Vision Texture has 54 images 512 × 512 which were split into 16 sub-images 128 × 128 pixel size without overlapping, totalizing 864 images.

Table 1 .

 1 Comparison of accuracies of different texture analysis methods in four texture databases.

	Methods	Number of features Outex USPTex Brodatz Vistex
	GLCM [21]	24	80.73 83.63	90.43 92.24
	GLDM [26]	60	86.76 91.92	94.43 97.11
	Gabor Filters [30]	64	81.91 83.19	89.86 93.28
	Fourier [3]	63	81.91 67.70	75.90 79.51
	Fractal [5]	69	80.51 78.22	87.16 91.67
	Fractal Fourier [14]	68	68.38 59.45	71.96 79.75
	LBP [32]	256	81.10 85.42	93.64 97.92
	LBPV [20]	555	75.66 55.13	86.26 88.65
	CLBP [19]	648	85.80 91.13	95.32 98.03
	BSIF [25]	256	77.43 77.48	91.44 88.66
	LCP [18]	81	86.25 91.31	93.47 94.44
	LFD [28]	276	82.57 83.59	90.99 94.68
	LPQ [33]	256	79.41 85.29	92.51 92.48
	CNTD [7]	108	86.76 91.71	95.27 98.03
	Proposed approach	130	87.86 94.41	96.45 98.96

Acknowledgments

Lucas Ribas gratefully acknowledges the financial support grant #2016/23763-8 and #2019/03277-0, São Paulo Research Foundation (FAPESP). Odemir Bruno thanks the financial support of CNPq (Grant #307797/2014-7) and FAPESP (Grant #s 14/08026-1 and 16/18809-9).