Texture representation and retrieval using the causal autoregressive model

Abstract

In this paper we propose to revisit the well-known autoregressive model (AR) as a texture representation model. We consider the AR model with causal neighborhoods. First, we will define the AR model and discuss briefly the parameters estimation process. Then, we will present the synthesis algorithm and we will show some experimental results. A perceptual interpretation of the AR estimated parameters will be then proposed and discussed. In particular, a computational measure to estimate the degree of randomness/regularity of textures is proposed. The set of the estimated parameters will be then applied in content-based image retrieval (CBIR) to model texture content and experimental results are shown. Benchmarking, using the precision/recall measures conducted on the well-known Brodatz database, shows interesting results.

Introduction

Among statistical models that have been used to model texture, random fields models such as the autoregressive (AR) model have been among the most successful. Such models are generally characterized by a set of parameters that must be estimated using different methods such as the least squares error (LSE) or the maximum likelihood (ML) methods. These estimated parameters are considered to represent the content of the texture. Another important property of the autoregressive model is its forecasting ability, that is, with this model, the grey-level value of a pixel is predicted using the grey-level values of pixels in its neighborhood.

The autoregressive (AR) model has been used in different works related to texture classification and segmentation [11], [14], [17], [19], texture synthesis [18] and more rarely to texture retrieval [12]. Sondge [18] studied different models, among them the AR model, in order to synthesize textures. He considered the simultaneous model (SAR) as well as the separable model which is a particular case of the SAR model. Kashyap and Chellappa [11] proposed a circular autoregressive model (CAR) in order to take into account rotation invariance. Mao and Jain [14] proposed a multiresolution AR model (MRSAR) which consists in the use of multiple resolutions in order to capture different scales of the texture. They also proposed a rotation-invariant simultaneous autoregressive model (RISAR) to take into account rotation invariance. In the framework of image retrieval, Liu and Picard [12] used the MRSAR proposed by Mao and Jain for texture retrieval. They benchmarked the MRSAR model against the Wold model and their results show that the MRSAR gives interesting results even if the Wold model is shown to give slightly better results.

Given their nature as random fields, such models, generally, work better for random textures than regular textures. A common problem with the use of these models is the choice of a neighborhood in which pixels are considered as related to each other. A priori, it is better to choose a small neighborhood for a fine texture and a large neighborhood for a coarse texture. However, in practice there is another problem which arises with large neighborhood. In fact, when the neighborhood is large, and due to the averaging effect phenomenon [14], the discrimination power of the estimated parameters tends to decrease. This phenomenon is related to curse of dimensionality problem as reported by Mao and Jain [14]. So, such model works better for fine textures. This has also some link with the randomness of textures. In fact, coarse textures are generally perceived as regular while fine textures are perceived as random, and as it has been already mentioned, the autoregressive model works better, by essence, with random textures.

Another problem associated with such models is the fact that their efficiency is altered with large neighborhoods. In fact, the size of the set of estimated parameters becomes much larger as the neighborhood becomes larger. Also, the more the size of the set of estimated parameters is large, the more the computational cost to extract those parameters is high.

In this paper, we propose to reconsider the SAR model in its causal version without considering multiple resolutions. Causality is not a natural constraint in a bidimensional (2D) space. However, this causality allows to simplify both the model and the synthesis process. With causal neighborhoods, diminishing the parameters number increases the efficiency of parameters extraction. We consider both non-symmetric half-plane (NSHP) and quarter-plane (QP) neighborhoods Fig. 1. Also, we consider small neighborhoods which allows to diminish both the number of parameters and the computational cost necessary to extract parameters. We show that, when such a model (causal SAR with small neighborhood) is applied in content-based image retrieval, the loss in terms of search effectiveness (relevance) is acceptable since there is an important gain in terms of search efficiency. Furthermore, we propose a new perceptual interpretation of the estimated parameters. Such a perceptual interpretation is easily comprehensible by users and allows them to better formulate their queries and better understand results returned by the model.

The rest of the paper is organized as follows: In Section 2, we define the model, discuss briefly the parameters estimation scheme and present the main steps in the synthesis process; In Section 3, we present an evaluation methodology, and experimental results, allowing to depict the ability of the causal SAR model to capture texture content as well as its efficiency; In Section 4, we propose a perceptual interpretation of the estimated parameters and a computational measure of the randomness of textures is proposed based on the estimated parameters; In Section 5, the causal model, with both QP and NSHP neighborhoods, is applied in content-based image retrieval [1], [2], [6] and compared to the MRSAR version used by Liu and Picard [12] in terms of search relevance as well as efficiency; And finally, in Section 5, we give a conclusion and a brief description of future investigations related to this work.