## 1. Introduction

During the past decades, image segmentation and edge detection have been two important and challenging topics. The main idea is to produce a partition of an image such that each category or region is homogeneous with respect to some measures. The processed image can be useful for posterior image processing treatments.

Spatial autoregressive moving average (ARMA) processes have been extensively used in several applications in image/signal processing. In particular, these models have been used for image segmentation, edge detection and image filtering. Image restoration algorithms based on robust estimation of a two-dimensional process have been developed (Kashyap & Eom 1988). Also the two-dimensional autoregressive model has been used to perform unsupervised texture segmentation (Cariou & Chehdi, 2008). Generalizations of the previous algorithms using the generalized M estimators to deal with the effect caused by additive contamination was also addressed (Allende et al., 2001). Later on, robust autocovariance (RA) estimators for two dimensional autoregresive (AR-2D) processes were introduced (Ojeda, 2002). Several theoretical contributions have been suggested in the literature, including the asymptotic properties of a nearly unstable sequence of stationary spatial autoregressive processes (Baran et al., 2004). Other contributions and applications of spatial ARMA processes have been considered in many publications (Basu & Reinsel, 1993, Bustos 2009a, Francos & Friendlaner1998, Guyon 1982, Ho 2011, Illig & Truong-Van 2006, Martin1996, Vallejos & Mardesic 2004).

A new approach to perform image segmentation based on the estimation of AR-2D processes has been recently suggested (Ojeda 2010). First an image is locally 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. Experimental results with real images clarify how the algorithm works in practice. A robust version of the algorithm was also proposed, to be used when the original image is contaminated with additive outliers. Applications in the context of image inpainting were also offered.

Another concern that has been pointed out in the context of spatial statistics is the development of coefficients to compare two spatial processes. Coefficients that take into account the spatial association between two processes have been proposed in the literature. (Tjostheim, 1978) suggested a nonparametric coefficient to assess the spatial association between two spatial variables. Later on, (Clifford et al. 1989) proposed an hypothesis testing procedure to study the spatial dependence between two spatial sequences. Rukhin & Vallejos (2008) studied asymptotic properties of the codispersion coefficient first introduced by Matheron(1965). The performance and impact of this coefficient to quantify the spatial association between two images is currently under study Ojeda et al. (2012). An adaptation of this coefficient to time series analysis was studied in Vallejos (2008).

In the context of clustering time series Chouakria & Nagabhushan (2007) proposed a distance measure that is a function of the codispersion coefficient. This measure includes the correlation behavior and the proximity of two time series. They proposed to combine these distances in a multiplicative way, introducing a tuning constant controlling the weight of each quantity in the final product. This makes the measure flexible to model sequences with different behaviors, comparing them in terms of both correlation and dissimilarity between the values of the series.

The structure of this chapter consist in two parts. In the first part we review some theoretical aspects of the spatial ARMA processes. Then the algorithm suggested by Ojeda(2010), its limitations and advantages are briefly described. In order to propose a more efficient algorithm new variants of this algorithm are suggested 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. The computation of the distance between the filtered images and the original one will be done by using the codispersion coefficient and other image quality measures (Wang and Bovik 2002). Examples with real images will highlight the features of the modified algorithm. In the second part, the codispersion coefficient previously used to measure the closeness between images is utilized in a distance measure to perform cluster analysis of time series. The distance measure introduced in Chouakria & Nagabhushan (2007) is generalized in the sense that considers an arbitrary lag

## 2. Image Segmentation Through Estimation of Spatial ARMA Processes

### 2.1. The Spatial ARMA Processes

Spatial ARMA processes have been studied in the context of random fields indexed over

A random field

where

with

Applications of spatial ARMA processes have been developed, including analysis of yield trials in the context of incomplete block designs (Cullis & Glesson 1991, Grondona et al. 1996) and the study of spatial unilateral first-order ARMA model (Basu & Reinsel, 1993). Other theoretical extensions of time series and spatial ARMA models can be found in (Baran et al., 2004, Bustos et al., 2009b, Gaetan & Guyon 2010, Choi 2000, Genton & Koul 2008, Guo 1998, Vallejos and Garccía-Donato 2006).

### 2.2. An Image Segmentation Algorithm

In this section, we describe an image segmentation algorithm that is based on a previous fitting of spatial autoregressive models to an image. This fitted image is constructed by dividing the original image into squared sub-images (e.g.,

Let

where

where

Algorithm 1.

For each block

1. Compute estimators

where

2. Let

where

Then the approximated image

The image segmentation algorithm we describe below is supported by a widely known notion in regression analysis. If a fitted image very well represents the patterns on the original image, then the residual image (i.e., the fitted image minus the observed image) will not contain useful information about the original patterns because the model already explains the features that are present in the original image. On the contrary, if the model does not well represent the patterns that are present in the original image, then the residual image will contain useful information that has not been explained by the model. Thus, to implement an algorithm based on these notions, we must characterize which patterns are present in the residual image when the fitted image is not a good representation of the original one, and we must develop a technique to produce a fitting that is satisfactory in terms of segmentation but not a very good estimation in that the residual image still contains valuable information. (Ojeda et al. 2010) investigated these concerns and, based on several numerical experiments with images, determined that the residual image associated with a good local fitting is in fact poor in terms of structure (i.e., it is very similar to a white noise). However, when the fitted image is poor in terms of estimation, the residual image is useful for highlighting the boundaries and edges of the original image. Moreover, a bad fitting is related to the size of the block (or window) used in Algorithm 1. The best performance is attained for the maximum block size, which would be the size of the original image. The image segmentation algorithm introduced by (Ojeda et al. 2010) can be summarized as follows.

Algorithm 2.

1. Use Algorithm 1 to generate an approximated image

2. Compute the residual autoregressive image given by

Example 1. We present examples with real images to illustrate the performance of Algorithms 1 and 2. These images were taken from the database http://sipi.usc.edu/database. Figure 1(a) shows an original image of size

### 2.3. Improving the Segmentation Algorithm

In all experiments carried out in (Ojeda et al., 2010) and (Quintana et al., 2011), Algorithm 1 was implemented using the same prediction window for the AR-2D process, which contains only two elements belonging to a strongly causal region on the plane. Here, we consider other prediction windows to observe the effect on the performance of Algorithm 2. A description

of the most commonly used prediction windows in statistical image processing is in Bustos et al., (2009a). A brief description of the strongly causal prediction windows is given below.

For all

For a given

In particular, if

The set

Visually, the best segmentation for the aerial image is yielded by the prediction window

To gain insight on image quality measures, the fitted images produced by Algorithm 1 associated with the images shown in Figure 4(a) -(d) were compared aerially with the original image using three coefficients described in (Ojeda et al., 2012). These coefficients are briefly described below.

Consider two weakly stationary processes,

where

For

with

The index

where

where

The correlation coefficient and the coefficients defined in (6), (7) and (8) were computed to compare the fitted images, which were generated with a prediction window with two elements and associated with the images shown in Figure 4(a) -(f), and the original images. The results are shown in Table 1. In all cases, the highest values of the image quality measures are attained for the image fitted using the prediction window

experiment was carried out for the image shown in Figure 2(a). Table 2 summarizes the values of the image quality coefficients for the fitted images generated by Algorithm 2 with prediction windows

generated with prediction window

Algorithm 3.

1. Use Algorithm 1 to generate the approximated images

2. Compute an image quality index between

3. Compute the residual autoregressive image

## 3. Clustering Time series

### 3.1. Measuring Closeness and Association Between Time Series

Let

where

where

Dynamic time warping (DTW) is a variant of the Fréchet distance that considers mapping length as the sum of the spans of all coupled observations. That is,

Dynamic time warping is then defined as

The distances defined above are based on the proximity of the values

Several distance measures that are functions of the correlation between two sequences (

where

### 3.2. The Codispersion Coefficient for Time Series

Consider two weakly stationary processes,

where

### 3.3. Dissimilarity Index for Time Series

This coefficient involves a distance measure and a correlation-type measure that addresses both the correlation behavior and the proximity of two time series. The dissimilarity index depends on similarity behaviors, which should be specified in advance. The suggested dissimilarity index

where

where

Note that (13) is a generalization of the dissimilarity index introduced in Chouakria & Nagabhushan, (2007). The dissimilarity index (13) can capture high-order serial correlations between the sequences because the distance lag

The dependence of (13) on

When the variance of the codispersion coefficient is difficult to compute, resampling methods can be use to estimate the variance of the sample codispersion coefficient (Politis & Romano, 1994, Vallejos, 2008).

In the next section, we present two simulation examples to illustrate the capabilities of the hierarchical methods using the distance measure (13) under the tuning function given by (14). All else being constant, the clusters produced using traditional distances are usually different from those yielded using the distance measure (13).

### 3.4. Simulations

In this example, we simulate observations from six first-order autoregressive models to illustrate the clustering produced by hierarchical methods when the sequences exhibit serial correlation. To generate the series, we consider the following models.

where

with

Two hundred observations were generated from each model for

In Figure 5, we see that the dendrogram obtained using hierarchical methods with the Euclidean distance does not recognize the correlation structure between

To obtain better insight into the classification process using the proposed distance measure (13), we carried out a second simulation study that involves clustering measures based on other distances (but using the same setup). Observations from models 1-6 were generated using Gaussian white noise sequences for the errors, thereby preserving the same correlation structure used in the first study. The goal was to explore the ability of the distance measure (13) to group strongly correlated series first. A total of 1000 runs were considered for this

experiment, and 200 observations were generated in each run. We used measure (13) under the tuning function (14) for

Note from Table 3 that the traditional distance measures failed to group the correlated sequences, with the exception of the Minkowski distance, which correctly grouped the correlated series 99% of the time. The hierarchical algorithm that uses the distance measure (13) has a higher percentage of well-clustered correlated sequences than the same algorithm using the traditional distance measures described in Section 2 (see Table 4). The percentage of correct clusters increased in all cases with the distance measure (13), suggesting that hierarchical algorithms can be improved by including coefficients of association that consider high-order cross-correlation.

### 3.5. The NDVI Data Set

In this section, we consider time series from four different locations in Argentina. The data set consists of 15 monthly NDVI series measured during a period of 19 years (i.e., January 1982-December 2000). The observed values correspond to a transformation to the interval

We can observe a variety of different patterns in Figure 6. In particular, the data collected during the period 1994-1995 show irregular behavior. Additionally, the original data lack some information (less than one percent) for all series over the period 1999-2000. An imputation technique based on moving averages, which takes into account past and future values of the series, was used to replace missing values. The series were grouped by geographical region and then plotted (Figure 7). Similar patterns are observed for the series across each group.

An exploratory data analysis was carried out for each of the 15 series. There exists significant autocorrelation of order of at least one in all series. Seasonal components are present in most of partial autocorrelations. Because there is no large departure from the weakly stationary assumptions (i.e., constant means and variances), all series can be modeled using the Box-Jenkins approach. Specifically, seasonal ARIMA models can be fitted to each single series with a small number of parameters (i.e.,

### 3.6. Clustering

Using the NVDI data set described in Section 3.5, the distance measure

## 4. Concluding Remarks and Future Work

This chapter described two problems. The first problem involved image segmentation, while the second problem involved clustering time series. For the first problem, a new algorithm was proposed that enhances the segmentation yielded by a previous algorithm (Ojeda et al., 2010). Identifying the best prediction window improves segmentation based on the estimation of AR-2D processes 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 relies on the selection of a suitable image quality measure. Using three image quality coefficients that are commonly used in image segmentation, we carried out experiments that support our algorithm. Specifically, a set of images belonging to the image database (http://sipi.usc.edu/database/) were processed and provided satisfactory results (not shown here) in terms of image segmentation.

This chapter also proposed an extension of the dissimilarity measure first introduced in (Chouakria & Nagabhushan,2007). The simulation experiments performed and the data analysis carried out for relevant ecological series show that the distance lag

Now, further research for the topics presented in this chapter is outlined.

Following the notation used in the Algorithm 3, consider the following residual image.

One interesting open problem involves the characterization of the types of images and distributions associated with the segmentation produced by Algorithms 2 and 3. In addition, the definition and study of linear combinations of residual images produced by distinct prediction windows is also of interests. For example,

where

Regarding the clustering technique problem, the distribution of