A New Algorithm to Represent Texture Images

In recent times the spatial autoregressive models have been extensively used to represent images. In this paper we propose an algorithm to represent and reproduce texture images based on the estimation of spatial autoregressive processes. The image intensity is locally modeled by a first spatial autoregressive model with support in a strongly causal prediction region on the plane. A basic criteria to quantify similarity between two images is used to locally select this region among four different possibilities, corresponding to the four strongly causal regions on the plane. Two global image similarity measures are used to evaluate the performance of our proposal.


I. INTRODUCTION
The goal of this work is to introduce a new algorithm to represent and reproduce texture images that uses and improves other recent proposals concerning this topic.
Most of the images of interest, for example, the images of cultivated fields and concentration of population are naturally rich in texture, level of gray, etc.The same thing happens to the images of geographical regions that allow the making of maps and, in general, almost all the images of the earth.During the past decades, image representation and image texture recovery have been two important and challenging topics.In this sense the spatial autoregressive model (AR-2D model) has been extensively used to represent images ( [3], [14]) due to its two main properties.First, simulation experiments have shown that this model is adequate to represent a diversity of real scenarios ( [4]).Second, the AR-2D model does not require a large number of parameters to represent different real scenarios (parsimony) ( [4]).In particular, the first-order AR-2D model is able to represent a wide range of texture images, as is shown in Figure 1; the image (a) have been generated by a first-order AR-2D model with three parameters, while images (b), (c), (d) and (e) have been generated by a model of the same type with two parameters.Theoretical properties of the first-order AR-2D model were studied by Basu and Reinsel ([2]).They derived the correlation structure of the model and the maximum likelihood estimators of the parameters.Also, the spatial autoregressive models have benefited other topics in image processing like image segmentation.An approach to perform image segmentation based on the estimation of AR-2D processes has been recently suggested by Ojeda et al. ([15]).
First an image is modeled using a spatial autoregressive model for the image intensity.Then the residual autoregressive image is computed.This resulting image possesses interesting texture features.The borders and edges are highlighted, suggesting that the algorithm can be used for border detection.Later, a new scheme was proposed to enhance the segmentation yielded by the previous algorithm (Vallejos et al., 2012, [18]).It is based on the identifying of the best prediction window, and generalizes the previous algorithm to different prediction windows associated with unilateral processes on the plane.An analysis of the association between the original and fitted images shows how the algorithm works in practice.In all experiments carried out in [18], the first step of the segmentation algorithm was implemented using the same strongly causal prediction window.Consequently the support of the the local autoregressive models used to represent the images was always the same.Our proposal is a methodology that allows to identify locally the strongly causal prediction window (and consequently the support of the first-order AR-2D processes) associated with the better local representation of the image.To analyze the performance of our method, we quantified the similarity between the original and fitted images by two image measures.The study shows that the new method is capable to enhance the capacity of the AR-2D models to reproduce and represent images.
The rest of the paper is organized as follows: Section II presents a brief description of most used schemes of neighborhoods in .Section III provides an overview of the spatial ARMA models.Section IV presents the recent algorithm developed by Ojeda et al. ([15]) and the improvement due to Vallejos et al. ([18]) in the context of image segmentation.In Section V we explain our proposal to reproduce and represent texture images based on the estimation of spatial autoregressive processes.Section VI shows the results of our study and provides an analysis of the performance of our methodology using two similarity image measures.Conclusions and future scopes will appear in sections VII and VIII respectively.

II. NEIGHBOROODS IN
In time series, there is a natural neighbor structure induced by the existing total order of (the set of all past values of ; is the set of all integers that are less than t).However, for points on the plane, for instance , there are several different notions of neighborhood.www.ijacsa.thesai.orgIn general, definitions of neighborhood of a point on the plane are motivated by the physical acquisition system of the data like in the case of images that have been captured by satellites.A description of the most commonly used neighbor structures in statistical image processing can be found in [13].In this paper we define the structure of neighborhood based on strongly causal regions.

For all
we distinguish the following strongly causal regions: Considering the region , a strongly causal prediction window containing two elements is is shown in Figure 2 (a).Similarly, the strongly causal prediction windows , , and , with two elements each, can be defined considering , and respectively (Figure 2, (b)-(d)).

III. SPATIAL ARMA MODELS
Spatial ARMA processes have also been studied in the context of random fields indexed over , , where is endowed with the usual partial order; that is, for , in , iff for , .Let and , where ; and .A random field is said to be a spatial ARMA with parameters if it is weakly stationary and satisfies the equation (1) where and denotes, respectively the autoregressive and moving average parameters with ; and denotes a sequence of independent and identically distributed centered random variables with variance Notice that if , the sum over is supposed to be zero and the process is called spatial moving average MA random field, Similarly if the process is called spatial autoregressive AR random field, and it is defined as: ( The ARMA random field is called causal if it has the following unilateral representation (4) with .Similarly to the time series case, there are conditions for the (AR or MA) polynomials to have stationarity and invertibility respectively.Let and , where and .A sufficient condition for the random field to be causal is that the AR polynomial has no zeros in the closure of the open disc in .
Applications of spatial ARMA processes and the study of spatial unilateral first order ARMA model have been developed in [10], [7], [2].Other extensions of the theory developed for time series to spatial ARMA models can be found in [11], [6], [1], [17], [9], [5], [8].As an example, consider a particular case of model (3) when and .This model is called a first-order autoregressive process.In this case, and the model is of the form: Note that , implies that for all , is the strongly causal prediction window of the intensity www.ijacsa.thesai.orgIV.APPROXIMATION OF IMAGES BY USING 2D UNILATERAL AR PROCESSES In this section, we describe an algorithm to approximate an image by first-order AR-2D processes whit two parameters using blocks.This algorithm was originally defined by [14] and later adapted by [15].Following, we briefly describe this methodology.
Suppose that a real image is available.This fitted image is constructed by dividing the original image into squared subimages (e.g., ) and then fitting a first-order AR 2D model to each sub-image (i.e., block).Then, a sub-image is generated from each local fitted model and the final fitted image is yielded by putting together all generated sub-images.Let , , be the original image, and let , , where, for all , , ; and is the mean of .Let (e.g. ).For simplicity we shall consider from now on that the images to be processed ( and ) [18]) suggested new variants of this algorithm specially to address the problem of determining the most convenient (in terms of the quality of the segmentation) prediction window of unilateral AR-2D processes.In effect, they generalize the Algorithm 1 to different prediction windows associated with unilateral processes on the plane.Three variants of the Algorithm 1 (called Algorithm 2, Algorithm 3 and Algorithm 4) were implemented in the strongly causal prediction regions , and , using the prediction windows , and , respectively.In each block , and for , we denote the output corresponding to step 2-Algorithm , as .Similarity, denote the output corresponding to step 3 -Algorithm .The computation of the distance between each filtered image and the original was be done by using and image quality measures ( [18], [19], [16]).We described briefly this measures: Let two weakly stationary processes, and , , the index ( [19]) is where is the mean of , is the standard deviation of , and is the covariance between and (and similarly for and ).The quantity models the linear correlation between and , measures the similarity between the sample means (luminance) of and , and measures the similarity related to the contrast (i,j) (i+1,j) (i+1,j+1) www.ijacsa.thesai.org between the images.Coefficient is defined as a function of the correlation coefficient; hence, it is able to capture only the linear association between and .The index was suggested by [16] and it is defined as: where is the sample codispersion coefficient in the direction .This index, can quantify the similarity between images that are generated using a local approximation of AR-2D processes with different window sizes.Moreover, captures different levels of spatial similarity between two images by considering different directions in two-dimensional space.Note that values of and close to point out a low similarity level between two images; instead, values of or close to 1, indicate a high similarity level (direct or inverse, respectively).

V. A NEW ALGORITHM TO REPRESENT TEXTURE IMAGES
In order to represent and reproduce texture images, we suggest a methodology that we have called Algorithm 5.As Algorithms 1-4, our proposal is also based on the idea that is advantageous approximate an image by first-order AR-2D processes using blocks, just that we introduce the possibility to select in each block the causal prediction window of the model.The goal is to get , a new and better representation of the original image , that improves the results achieved by the algorithms 1-4.Essentially in each block, our approach selects one among the algorithms 1, 2, 3 or 4, and uses it to approximate the image in the block.Previously to defined , a approximated image of is provided by the algorithm.This selection is based on the mean square error (MSE).This basic similarity measure for images is computed from the difference of intensity, of pixel to pixel, between two images.More formally, if and , with , are two weakly stationary processes, More details about the MSE can be consult in [19].In the following we describe the new algorithm.

Algorithm 5:
1) In each block , and for , compute the least squares estimators ; of the and corresponding to the block , in the strongly causal prediction region , using the prediction windows (Apply step 1-Algorithm , with ).

3) In each block
, and for , compute .Choose the approximated image generated by the lowest mean square error.Denote it as .4) Let be defined in the block by with and .

5) The approximated image of the original image is:
with and .Note how Algorithm 5 (step 3), allows to locally identify (based on the MSE) the strongly causal prediction window associated with the more adequate local representation of the original image.

VI. EXPERIMENTS AND RESULTS
In this section we developed some examples in different scenarios to explore the performance of Algorithm 5. Six real images were used, Elaine, Lenna, Peppers, Threads, Rind and Aerial (Figure 3 (a)-(f)), all taken from the USC-SIPI image database [12].We applied the Algorithms 1 to 5, for each original image, considering a block of size , and we obtained five representations of the original image.Figure 4 shows the image representations obtained by the five methods, from the original image Elaine.
To gain insight on the quality of each image representation, the images produced by the five algorithms were compared with the original image using the similarity index ( [19]) and ( [16]), described in section IV.The results are shown in TABLE I.In all cases the highest values of the image quality measures were obtained for the image representation produced by the Algorithm 5.In practice, this means that the residual image (difference between the original and the approximated) is more compatible with a null image, when the Algorithm 5 is used.
Visually, in the residual images produced from the Algorithm 5, it is more difficult to detect the patterns of the respective original images, in comparison with the residual images generated from the others methods.As an example, Figure 5 shows the residual images obtained by applying the five algorithms to the original image Elaine, and Figure 6 shows the histograms of these images.Note that the residual image produced from Algorithm 5 (Figure 5  The results presented in this section are not restrictive to the images treated in this paper.There is a large set of images for which the experiments developed in this article can be replicated.www.ijacsa.thesai.orgVII.CONCLUSION This paper proposes a new algorithm to represent and reproduce texture images based on the estimation of spatial autoregressive processes.Our proposal, as other methods, suggests approximating an image by first-order AR-2D processes using blocks, just incorporating the option to select in each block the causal prediction window of the model.This selection is based on the mean square error, a quantitative performance metric in the field of signal processing ( [20]).The method, called Algorithm 5, can generate a wide range of textures using four different local approximations of AR-2D processes, corresponding to the four strongly causal regions on the plane.Using two image quality index that are extensively used in image similarity analysis, we carried out experiments that support our algorithm ( [19], [20], [16]).Specifically, a set of images belonging to the image database ([12]) were processed and provided satisfactory results in order to reproduce and represent image textures.In addition the paper performs a review about the main characteristics and applications of the spatial autoregressive and moving average models.In the light of the examples presented in this article, we suggest in practice to use the Algorithm 5 as a replacement of other similar methods based on only one strongly causal region on the plane, to produce a more adequate representation of the texture images.

VIII. FUTURE RESEARCH
The performance of the algorithm under different kinds of contamination in images is an interesting open problem to be addressed in the future.The effect of considering noncausal and semi-causal prediction windows in the spatial AR model, with different window sizes, is also an open issue to be addressed in future research.

Fig. 2 Prediction
Fig.2 (e)) does not highlight the original borders and boundaries because the original image and the image representation are too similar.The histogram of the image (Figure 6 (e)) confirms this fact.
are arranged in such a way that the number of columns and the number of rows are multiples of ;

TABLE I .
PERFORMANCE OF THE ALGORITHMS 1, 2, 3, 4, AND 5 BY Q AND CQ INDEX.