I Introduction
The classification of texture images is an important problem in pattern recognition and consequently forms the subject of many research works in this field. Texture is an important image feature with a strong discriminative capability and is therefore widely used in computer vision. The texture classification problem addressed in this paper is a multiclass classification problem and two different wellknown databases are considered: the Brodatz database, which contains a number of unique textures that each form a class, and the Vistex database, which consists of a number of classes, each with several texture images belonging to it. For the Brodatz database, ten subimages of the same size are taken from each texture to obtain different texture images for the corresponding class. For each texture image of both databases a feature vector, i.e. a vector of characteristics, is obtained by using a novel method introduced in this paper, as well as by using a number of popular methods from literature. This feature vector is then employed to classify the different texture images using Linear Discriminant Analysis (LDA) following a stratified 10fold crossvalidation scheme. Feature vectors for image texture are usually obtained from the analysis of groups of pixels and the way this analysis is performed is used to classify the different texture analysis methods. Five main categories can be distinguished: structural
Goyal et al. (1995); Serra (1983); Zhang and Tan (2002), statistical Haralick (1979), modelbased Backes et al. (2009); Backes and Bruno (2008); Cohen et al. (1991), spectral Chen (1982); Tang and Stewart (1995), and agentbased methods Backes et al. (2010a, b); Lai et al. (2006).This paper proposes a novel method to analyze the structural elements of textures by means of a cellular automaton (CA) inspired by the pitting corrosion phenomenon, further on referred to as the CorrosionInspired Texture Analysis (CITA) method. The basic mechanisms behind this detrimental reaction which occurs between metals (or alloys) and their environment serve as inspiration to develop a CAbased model. Next, this CAbased model is employed to perform texture analysis by treating the image to be classified as a metal surface. The CITA method, like real corrosion, amplifies existing differences in material and height (in this case grayscale value) so that the biggest contrasts in the original texture image will become more pronounced and smaller contrasts will be nullified. The eroded mass of ‘metal’ by the progression of pitting corrosion at each iteration is used to generate a feature vector that describes the image to be classified. The effectiveness of this strategy is demonstrated on two texture data sets with natural and synthetic textures. In addition, to verify the robustness of the CITA method, its invariance to noise and rotation were tested, obtaining satisfactory results.
This paper is organized as follows. Section II describes the basics behind the pitting corrosion phenomenon, while the definition of a CA as well as further explanation of some parts of this definition form the subject of Section III. The CITA method is described in Section IV and the experimental setup needed to test its efficacy is explained in Section V. Section VI presents the results and discussion of the study. Finally, the paper is concluded in Section VII.
Ii Pitting corrosion
Corrosion is the disintegration of metals (and alloys) into their constituents due to reaction with the environment and is one of the main causes of structural failure in industrial systems, and poses as such an economic problem Roberge (2008). Dealing with corrosion is difficult because of its complex nature and the involvement of many variables. Therefore, modeling and simulation could allow for predicting more accurately the corrosion process in time. CAbased models are excellent candidates for modeling corrosion due to their intrinsic simplicity and therefore, since the beginning of the new millennium, attempts are being made to employ these models in the field of corrosion engineering Contreras et al. (2011); di Caprio et al. (2011); Lishchuk et al. (2011); Malki and Baroux (2005); Pidaparti et al. (2005). Corrosion is present in a wide range of metals and environments, which points to the universality of this phenomenon. The latter suggests that corrosion does not depend on the details of the underlying mechanism, so that it may be modeled adequately using simple models Valor et al. (2007). Moreover, CAbased models are able to capture the stochasticity of the involved electrochemical reactions at the mesoscopic scale Lishchuk et al. (2011).
Pitting corrosion is a very harmful and common form of localized corrosion where all or most of the metal loss occurs concentrated in certain areas. Upon close inspection of the metal surface, pitting can be recognized by the appearance of small holes on the metal surface as shown in Figure 1.The first step in pitting corrosion is the pit initiation which is the result of impurities or irregularities of the metal surface or the environment, making perfectly polished surfaces more resistant to this type of corrosion. From there on, the acidity inside the pit is maintained by the spatial separation of the cathodic and anodic halfreactions, which creates a potential gradient and electromigration of aggressive anions into the pit (see Figure 1). As pit growth progresses, different solution compositions develop inside the cavity and the consequent voltage (IR) drop along the metal/electrolyte interface illustrates that the deeper the pit, the lower the pit growth rate Malki and Baroux (2005); Pickering (1989); Wang et al. (2011).
Iii Cellular automata
CAs are mathematical constructs in which the space, state and time domains are discrete as opposed to partial differential equations (PDE) in which these three domains are continuous
Berec (2002); Wolfram (1983). The ability of CAs to generate a rich spectrum of sometimes complex spatiotemporal patterns from relatively simple underlying transition functions has led to their successful employment in the study of several (a)biological processes Milne and Fu (2003); Picioreanu et al. (1998); Preziosi (2003); Schiff (2008); Van der Weeën et al. (2012); Vasilkoski and Weaver (2000). Models based on CAs can be seen as an alternative to PDEbased models, to provide researchers with a wider range of modeling tools and, in some complex cases, a solution to problems encountered with some of the more classical modeling methods El Yacoubi and El Jai (2002); Toffoli (1984).In this paper, we make use of a homogeneous CA, in which a single transition function, constructed using a combination of knowledge on the pitting corrosion phenomenon and intuition, governs the dynamics of all cells. The following definition of a homogeneous 2D CA is relied upon.
Definition I. (Homogeneous 2D cellular automaton)
A homogeneous 2D cellular automaton can be represented as
where

is a twodimensional grid of cells .

is a finite set of states, with .

The output function yields the state of every cell at the th discrete time step.

The neighborhood function determines the neighboring cells of every cell , including the cell itself.

The transition function yields the state of every cell at the next time step, based on its state and that of its neighboring cells at the current time step.
For reasons of comprehensiveness, some parts of this definition will be elaborated in the remainder of this section.
iii.1 Grid
In this paper, a finite twodimensional grid consisting of squares is used, because it has the most straightforward implementation and provides an easy way of linking the cells of to the pixels of the texture images to be classified (cfr. infra). Furthermore, an indexing of the cells of a 2D CA is introduced, which is shown in Figure 2. For a square grid, it holds that = = .
iii.2 Neighborhood function
Many different neighborhoods can be defined in 2D, the two most important ones being the Moore and the von Neumann neighborhood. The Moore neighborhood of a cell comprises those cells that share at least a vertex with (see Figure 3). The von Neumann neighborhood is a more restricted neighborhood in which only those cells that share an edge with are considered as neighbors (see Figure 3).
iii.3 Discrete states
Every cell has one of the discrete states comprised in the set . The states of the cells of at , i.e. , constitute the initial condition of . In this paper, the initial condition of is determined by the grayscale value of the different pixels of the corresponding texture image (cfr. infra).
iii.4 Transition function
The transition function determines the state of a cell at the ()th time step based on the cell’s current state and the states of its neighboring cells. The transition function employed in this paper is executed in a deterministic and synchronous manner, meaning that is used to evaluate the state of every cell of at every time step and for all cells at the same time Baetens et al. (2012).
Iv CorrosionInspired Texture Analysis
The CITA method proposed in this paper starts by converting the texture image into the initial state of a CA. Thereafter, a CAbased model inspired by the pitting corrosion phenomenon is evaluated for a number of time steps. The cumulative mass of corroded metal after each iteration of the CAbased model is used to construct a feature vector for every texture image. Finally, these vectors of characteristics are used to classify the images via LDA. In the remainder of this section, the CITA method is explained in more detail.
A twodimensional grayscale image is treated as a discrete object and is seen as a grid . The dimensions of are defined by the size of the image, where each pixel of the image is a cell of the CA. The original image is then used to determine an initial state of the cells of the CA by converting the gray level image into a discrete initial state for each cell. Thus, for the initial configuration there are 256 possible states, ranging from 0 to 255. This conversion is described by
(1) 
where is the original image and represents the gray level of the pixel at the th row and th column of the image . In order to introduce the ideas of pitting corrosion, the 2D grid will be regarded as a metal surface and the state of each cell will represent the depth of the local pit in the metal (i.e. along the third dimension), with state 0 meaning that there is no pit and 255 being the largest pit depth of the metal at . It is important to point out that for the maximum pit depth can exceed 255 and from thereon it is possible that the grid can no longer be represented as a grayscale image.
An important consideration is the choice of boundary conditions in order to obtain an appropriate behavior of the CAbased model. The two most popular boundary conditions are the periodic and reflecting boundary conditions. The former tries to simulate an infinite grid, where the new boundaries of the top, bottom, left and right are filled with the values of the opposite side, thus forming a torus in a 3D space. This boundary condition is useful for simulating systems where the physical boundaries do not play an important role. However, throughout this paper, reflecting boundaries will be used as they give rise to better results for the studied databases as was observed from preliminary tests. Firstly, an imaginary row at the top and at the bottom of the grid and an imaginary column at the left and at the right of the grid are added. Then, the reflecting boundary conditions are applied at every time step as follows:
(2) 
with the size of the original image with pixels.
The updated state of each cell of at time depends on the analysis of the states of the cells in the neighborhood of at time . In this paper, the Moore neighborhood (see Figure 3) is employed. Furthermore, the CITA method makes use of a transition function inspired by pitting corrosion. In a first step, is calculated for every cell as the difference between the state of this cell and the lowest state value within its Moore neighborhood (see Eq. (3)):
(3) 
where is the set of states of the cells in the Moore neighborhood of .
Bearing in mind the principles of pitting corrosion, a local ‘impurity’ or minimum height difference is needed at a certain location in order to initiate or propagate pitting corrosion. For this purpose, a surface roughness parameter is introduced. All differences lower than this parameter are considered insignificant, i.e. not real impurities, in order to account for the fact that not even a polished metal surface is perfectly smooth. This means that differences lower than will not give rise to (further) pitting. On the other hand, the larger the difference grows, the lower the pit growth rate will be due to the IR drop, until finally the pit growth rate becomes zero. In this paper, it is assumed that if the difference is greater than 254, the greatest possible difference at , the corresponding pit growth rate is zero. This means that only the state of those cells with a difference greater than or equal to and smaller than 255 are evaluated.
Figures 44 illustrate the selection process to determine whether a cell will be evaluated or not. In this example, is set to five meaning that cells whose state differs less than five with the lowest state in its neighborhood are considered to belong to the surface and will not have their state changed. Figure 4 shows the cells belonging to a square tessellation with their initial state. Figure 4 depicts the difference for each of these cells calculated according to Eq. (3). Finally, in Figure 4 the gray cells indicate the cells that are evaluated in that time step, because their is greater than or equal to five and smaller than 255.
Under these assumptions, the transition function establishes the state of a cell at the ()th time step according to
(4) 
where is the pitting power. This parameter represents the metalspecific resistance to corrosion under given environmental conditions, where stands for completely resistant metal. Further, is a function that employs and to determine the level of corrosion to be applied. In this paper, is defined as
(5) 
From Eq. (5), it can be seen that the function gives, depending on the value of , a noninteger output, meaning that the employed model structure is actually a continuous CA or Coupled Map Lattice rather than a CA Levine and Reynolds (1992). However, in order to keep working with a CAbased model and to not overcomplicate the model, the choice was made to limit the output of to integer values (see Eq. (6)).
(6) 
where in denotes the floor of .
The output of the CAbased model at every time step is the cumulative mass of corroded product. In each iteration, after updating the state of the cells, the mass that suffered corrosion in that iteration is added to the eroded total mass from the previous iteration. Finally, this cumulative corroded mass is expressed relative to the number of pixels of the texture images such that texture images with different sizes can be compared using the CITA method.
Figure 5 shows some examples of initial images and the simulated result after 90 iterations of the pittingcorrosioninspired CAbased model. In the first column the original images are shown, the second column shows the simulated output of the CAbased model in grayscale, while the third column displays the same results as the second column, but scaled in a color map where blue indicates the lowest and red the highest resulting values. After simulation, some structural details from the original image can still be retrieved in the simulated output. Regions with similar state values are mostly considered by the model as belonging to the same local surface and therefore tend to keep the same state value throughout the simulation.
In a final step, the time series of cumulative mass of corroded metal, relative to the total number of pixels, is used as the feature vector for each texture image and classification is performed using LDA following a stratified 10fold crossvalidation scheme. LDA is traditionally used in texture analysis to find a linear combination of attributes resulting in a good separation of the classes. The proposed method is summarized in Figure 6. Figure 7 shows the feature vectors for four texture images from the Brodatz database. It can clearly be seen from this figure that the feature vectors from textures belonging to the same class are very similar on the one hand and that these vectors are different from the feature vectors from images belonging to different classes on the other hand.
V Experimental setup
To validate the CITA method, it is employed for the classification of the images of two classical texture databases, the Brodatz and Vistex databases, and the results obtained by applying the CITA method are compared with the results obtained with several established methods from literature. The remainder of this section includes a short description of the employed databases and the methods from literature used for comparison as well as an optimization of the parameters of the CITA method, i.e. , and the number of iterations performed in order to obtain the most favorable results. To ensure the CITA method is not sensitive to the parameter configuration, Usptex, a different database than the databases used for the validation of the method is used to perform the parameter optimization.
v.1 Databases
Two important databases, widely used in the literature and each with its own peculiarities, are employed for testing the different methods for pattern classification: the Brodatz and Vistex databases and a third database, Usptex, is used to perform the parameter optimization.
v.1.1 The Brodatz database Brodatz (1966)
contains 111 unique natural textures (and therefore also 111 classes) with image size of pixels and 256 gray levels. From each image ten subimages with size of pixels were obtained, resulting in an image database containing 1110 images. Figure 8 shows the complete Brodatz database and Figures 9 and 9 show for two of the original Brodatz images the ten selected subimages.
v.1.2 The Vistex database VisTex (2009)
contains 864 images belonging to 54 texture classes. Each texture class contains 16 texture samples of pixels, each extracted from a particular texture pattern without overlapping (see Figure 10). The true color RGB images are converted to grayscale intensity images, because the CITA method in its present form works only on grayscale images.
v.1.3 The Usptex database Backes et al. (2012)
contains 191 color images that each form a texture class (see Figure 11). Each image has a size of pixels from which 12 subimages with a size of 128 128 pixels are extracted without overlapping, so that a total of 2292 images is obtained. The images are again converted to grayscale images.
v.2 Established methods for texture analysis
v.2.1 Fourier descriptors Azencott et al. (1997); Gonzalez and Woods (2006)
consider attributes in terms of spectral density considering the texture as a Gaussian random field. The Fourier transform was calculated for each image, where the spectrum was divided into 64 sectors with eight radial distances and eight angles. The sum of the absolute spectrum values for each sector is calculated, resulting in 64 descriptors per image.
v.2.2 Gray Level Cooccurrence Matrix  GLCM Haralick (1979)
is based on the spatial gray level dependence matrices. Haralick descriptors (Contrast, Correlation, Energy and Homogeneity) were computed from resulting cooccurrence matrices with angles of 0, 45, 90 and 135, distances equal to one or two pixels and 64 gray levels in order to obtain a set of 32 descriptors for each image.
v.2.3 Gray Level Difference Matrix  GLDM Weszka et al. (1976); Kim and Park (1999)
calculates the absolute gray level difference between two pixels with distance . Here, 60 descriptors were obtained using
1, 3 and 5 and the attributes contrast, angular second moment, entropy, mean, and inverse difference moment from the estimated probability density function.
v.2.4 Gabor filter Manjunath and Ma (1996); Daugman and Downing (1998); Idrissa and Acheroy (2002)
is a bidimensional Gaussian function modulated with an oriented sinusoid in a determined frequency and direction. To perform the tests, 64 filters were used, composed of eight rotation filters and eight scale filters with lowest frequency equal to 0.01 and highest frequency equal to 0.4.
v.2.5 Local Binary Pattern Variance  LBPV Guo et al. (2010)
v.3 Parameter Evaluation
The parameterization of the CITA method is performed using the Usptex database, a different database than the ones used for validation. This is done to ensure that the CITA method is not susceptible to the parameters and therefore the same configuration can be used for classification of textures from different databases.
v.3.1 Number of iterations
To describe each image, the cumulative mass of corroded metal after each iteration of the CAbased corrosion model is used. These values constitute the vector of characteristics that is used to discriminate each of the images. However, finding a single number of iterations that gives rise to the smallest, most informative feature vector for all images of both databases is a nontrivial task due to the variety of the type of texture images and also because this number is dependent on the values of and . In order not to overcomplicate the problem, the choice is made to look for a single optimal number of iterations for both databases that overall gives the best result for all the texture images in both databases. This optimal number of iterations is nevertheless still kept dependent on and .
v.3.2 Surface roughness
One of the parameters that defines the pitting corrosion is the surface roughness . According to the proposed corrosionbased method, pixels having a difference lower than (Eqs. (3) and (4)) do not suffer from the action of the corrosion process, considering that they are part of the local surface. However, if the neighborhood has a difference greater than the permitted threshold surface, the center pixel will pass through a corrosion process having its value eroded according to Eqs. (4) and (5). Figure LABEL:fig:parametersusptex shows the success rate surface, i.e. the percentage of correctly classified texture images, for the Usptex database for varying from 0 to 10 and and varying from 0.01 to 0.08. The figure shows that higher values of lead to a lower success rate. However, when equals 0 the obtained success rate is smaller than for equal to 1 for almost all values of . Thus, a value of 1 for is chosen as optimal value.
v.3.3 Pitting power
Another model parameter with a physical meaning is the pitting power . This parameter is important because it determines the level of corrosion according to the material being eroded. However, as we are not dealing with real metal surfaces, this parameter is not known for the image texture analysis. Figure LABEL:fig:parametersusptex is now studied for ranging from 0.01 to 0.08. It can be seen that the highest success rate is obtained with and equal to 0.05 and 1, respectively. For values of below 0.05 the rate tends to be reduced while for values above 0.05 the rate also tends to decrease. When looking at combined high values of and the success rates drop sharply. Figure LABEL:fig:iterationsusptex shows the number of iterations to obtain the highest success rate for each of the parameter combinations. The graph shows that the optimal number of iterations necessary for equal to 0.05 and equal 1 is relatively low in comparison to the other results.
Vi Results and Discussions
This section reports on the performance of the CITA method for texture analysis. Results for the classification with the proposed method are compared to the traditional texture analysis methods in literature described in Section V to evaluate the performance of the method, where the classification is performed using LDA following a stratified 10fold crossvalidation scheme. Three sets of experiments were performed: firstly on the original test databases and afterwards on modified versions of the test databases to test noise and rotation invariance. All tests were performed using the optimized values for and found in the previous section and the corresponding optimal number of iterations for that specific combination of and , which are shown in Table 1.
Parameter  Value 

Number of iterations  158 
1  
0.05 
vi.1 Unmodified databases
Table 2 lists the results for the different texture analysis methods for the unmodified databases. As can be seen, the CITA method achieves an excellent success rate, which outperforms all methods for the Brodatz database and shares the best result for the Vistex database together with the GLDM method. To further test the robustness of the CITA method, firstly, noise is applied to the images and, secondly, a rotation of the images is performed to verify whether the performance of the CITA method persists under these circumstances.
Success rate (RMSE)  
Method  Brodatz  Vistex 
Fourier descriptors  94 ( 2.2)  94 ( 1.8) 
GLCM  94 ( 3.1)  94 ( 3.3) 
GLDM  98 ( 0.9)  97 ( 1.7) 
Gabor filter  92 ( 3.5)  92 ( 1.6) 
LBPV  88 ( 3.2)  82 ( 3.7) 
CITA Method  99 ( 1.5)  97 ( 1.6) 
vi.2 Noise Invariance
In order to demonstrate the tolerance of the proposed method to noise, experiments were performed on modified versions of the Brodatz and Vistex databases with addition of noise in the form of ‘Salt & Pepper’ noise. By applying this type of noise to an image, black and white pixels are randomly added to the image matrix with an intensity which may vary from 0 to 1 and represents the share of the image affected by the noise. The robustness of the CITA method to the addition of noise is demonstrated by performing the texture classification on six modified databases generated from both the Brodatz and Vistex databases. The six different databases were generated in both cases by adding ‘Salt & Pepper’ noise with intensities , 0.05, 0.07, 0.1, 0.5 and 0.7. For all different cases the CITA method is compared with the established methods described in Section V in order to get an idea of how the CITA method, in comparison with the other methods, deals with deformation of texture. Figure 13 shows samples of the modified Vistex databases where noise was added to the images and where each column shows examples of an intensity of noise.
The success rates for classifying the perturbated images from the modified Brodatz and Vistex databases using the different methods, are given in Tables 3 and 4, respectively. These results demonstrate the good performance of our method even with the addition of various intensities of noise. For all databases generated from the Brodatz database, the CITA method has a higher success rate compared to traditional methods in literature. It is important to note that even with increasing noise levels, the CITA method yields high success rates, while for all other methods the success rate declines. For the databases generated from the Vistex database our method gives rise to the second best success rate, preceded by the GLDM method, but still demonstrating its robustness to noise.
Success rate (RMSE)  
Method  = 0.01  = 0.05  = 0.07  = 0.1  = 0.5  = 0.7 
Fourier  93 ( 2.5)  93 ( 3.1)  91 ( 3.3)  90 ( 3.3)  82 ( 4.8)  67 ( 6.0) 
GLCM  94 ( 3.2)  94 ( 2.6)  95 ( 2.6)  94 ( 2.9)  87 ( 4.6)  76 ( 5.9) 
GLDM  98 ( 1.4)  98 ( 1.5)  98 ( 1.4)  98 ( 1.7)  95 ( 2.0)  91 ( 3.4) 
Gabor filter  91 ( 3.5)  90 ( 4.3)  90 ( 4.4)  90 ( 4.8)  82 ( 5.5)  67 ( 3.6) 
LBPV  88 ( 3.0)  87 ( 3.6)  87 ( 4.3)  87 ( 4.5)  66 ( 5.0)  46 ( 6.1) 
CITA method  99 ( 1.6)  98 ( 1.3)  98 ( 1.3)  98 ( 1.3)  97 ( 2.1)  97 ( 1.9) 
Success rate (RMSE)  
Method  = 0.01  = 0.05  = 0.07  = 0.1  = 0.5  = 0.7 
Fourier  93 ( 2.3)  91 ( 2.3)  89 ( 1.5)  89 ( 2.0)  67 ( 3.5)  35 ( 5.4) 
GLCM  94 ( 2.8)  95 ( 1.6)  95 ( 2.7)  94 ( 1.7)  84 ( 3.6)  66 ( 4.1) 
GLDM  98 ( 1.1)  97 ( 1.4)  97 ( 1.5)  97 ( 1.3)  94 ( 1.3)  86 ( 2.9) 
Gabor filter  91 ( 2.1)  89 ( 1.7)  89 ( 3.0)  86 ( 3.1)  56 ( 5.5)  34 ( 3.6) 
LBPV  83 ( 3.6)  83 ( 2.2)  82 ( 3.8)  81 ( 3.9)  54 ( 2.9)  39 ( 7.3) 
CITA method  96 ( 1.7)  94 ( 1.3)  94 ( 1.5)  94 ( 1.4)  94 ( 4.4)  81 ( 4.2) 
The results in Tables 3 and 4 are obtained with addition of noise to both the training as well as the test data. Further, experiments using nonperturbated texture images for training and images with addition of noise for testing were performed. The results are shown in Tables 5 and 6. The success rate of the CITA method is comparable to the other methods analyzed in both cases. It is never the worst method seen over the different intensities and two databases, but neither is it clearly the best method.
Success rate (RMSE)  

Method  = 0.01  = 0.05  = 0.07  = 0.1  = 0.5  = 0.7 
Fourier  90 ( 2.4)  62 ( 3.4)  50 ( 3.8)  36 ( 2.4)  4 ( 0.9)  1 ( 1.2) 
GLCM  49 ( 2.0)  6 ( 0.9)  4 ( 0.4)  3 ( 0.6)  1 ( 0.0)  1 ( 0.0) 
GLDM  84 ( 5.0)  42 ( 3.0)  16 ( 1.5)  9 ( 0.6)  1 ( 0.0)  1 ( 0.0) 
Gabor filter  73 ( 4.7)  57 ( 1.8)  32 ( 1.5)  26 ( 2.1)  7 ( 1.1)  4 ( 0.8) 
LBPV  57 ( 3.2)  14 ( 1.2)  10 ( 1.3)  9 ( 0.7)  1 ( 0.0)  1 ( 0.0) 
CITA method  97 ( 2.6)  37 ( 3.2)  20 ( 1.6)  11 ( 1.3)  6 ( 1.8)  4 ( 0.7) 
Success rate (RMSE)  
Method  = 0.01  = 0.05  = 0.07  = 0.1  = 0.5  = 0.7 
Fourier  58 ( 4.1)  10 ( 2.0)  8 ( 2.0)  6 ( 2.0)  3 ( 1.1)  2.1 ( 1.0) 
GLCM  64 ( 3.5)  33 ( 1.4)  5 ( 0.5)  2 ( 0.6)  2 ( 0.6)  2 ( 0.6) 
GLDM  90 ( 2.5)  13 ( 1.4)  13 ( 1.1)  9 ( 1.2)  2 ( 0.5)  2.3 ( 1.0) 
Gabor filter  58 ( 2.6)  14 ( 3.2)  10 ( 2.1)  5. ( 1.5)  2 ( 0.5)  2 ( 0.6) 
LBPV  52 ( 2.6)  14 ( 2.5)  7 ( 1.5)  5.6 ( 1.5)  2 ( 0.6)  2 ( 0.6) 
CITA method  83 ( 3.2)  23 ( 4.1)  17 ( 2.5)  11 ( 1.6)  6 ( 3.1)  5 ( 2.8) 
vi.3 Rotation Invariance
The proposed CITA method is intrinsically rotation invariant, so good results are expected when test are performed with modified databases with rotated images. To demonstrate the rotation invariance of the CITA method, additional versions of both the Brodatz and Vistex databases are created. Each image from the databases is rotated with the following angles: 0, 45, 90, 135, 180, 225 and 270 and in this way, seven images are obtained from each original database image. Therefore, the new database with rotated Brodatz images has 70 images per class with 111 classes in total and the new database with rotated Vistex images has 112 images per class with 54 classes in total. Figure 14 shows for some texture images from the Brodatz database the seven rotated images obtained under the different rotation angles, with all images on the same row originating from the same original image.
Table 7 shows the success rates for the classification of the texture images of the Brodatz and Vistex databases with rotated images. In this case, the success rates are obtained with LDA following a stratified 10fold crossvalidation scheme for the complete rotated Brodatz and Vistex databases, where each database consists of all rotated texture images of all the original images. The success rates achieved with the CITA method are better than the success rates obtained with any of the other methods and are comparable to the results obtained on the unmodified databases. These experimental results indicate that our method has a good generalization ability. Hence, the method described here has proven to be performant also for rotated texture classification.
Success rate (RMSE)  
Method  Rotated Brodatz  Rotated Vistex 
Fourier descriptors  83 ( 1.0)  77 ( 1.3) 
GLCM  71 ( 0.5)  74 ( 1.2) 
GLDM  84 ( 0.5)  83 ( 0.7) 
Gabor filter  78 ( 0.9)  71 ( 1.3) 
LBPV  63 ( 1.0)  62 ( 1.6) 
CITA method  98 ( 0.4)  97 ( 0.3) 
Vii Conclusions
In this paper, a new method for texture analysis and classification was described. Combining concepts from corrosion engineering, cellular automata and pattern recognition a texture descriptor was generated, able to characterize an image according to the iterations of a CAbased model drawing inspiration from the pitting corrosion phenomenon. The developed CITA method was used to classify the texture images of two wellknown databases: Brodatz and Vistex. The method was applied to images of the original databases and the robustness of the method under addition of noise and rotation was investigated. For this purpose, several new databases were created, starting from the original databases. Six new databases were obtained by adding ‘Salt & Pepper’ noise with different intensities to each of the images of the test databases and another new database was obtained by rotating the images of the databases under seven angles. In all cases, the CITA method obtained good results compared to the methods from literature, showing a good generalization ability and proving to be performant for texture classification.
The results presented in this paper demonstrate the potential of the CITA method. Therefore, future work should focus on further refining the method as well as expanding it so that it is applicable for more types of texture images. This can be done by integrating measures of corrosion frequency via a histogram of eroded pixels and measuring the velocity of corrosion by calculating the difference between initial and final values divided by the number of iterations. Further, the CITA method has to be expanded so that it can deal with RGB color images as well as with dynamic textures, i.e. sequences of images that together form a texture.
Acknowledgments
Núbia Rosa da Silva acknowledges support from FAPESP (The State of São Paulo Research Foundation). Pieter Van der Weeën was sponsored by the Fund for Scientific Research in Flanders (FWO). Odemir Martinez Bruno gratefully acknowledges the financial support of CNPq (National Council for Scientific and Technological Development, Brazil) (Grant Nos. 308449/20100 and 473893/20100) and FAPESP (Grant No. 2011/015231).
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