Maximum entropy random fields for texture analysis☆
Introduction
Texture analysis has a long history and texture analysis algorithms range from using random field models to multi-resolution filtering techniques such as the wavelet transform. Generally, a texture is regarded as the region with a set of properties being constant, slowly varying or approximately periodic. It seems natural to model textures as random fields. For example, the Markov random field has been used widely for modeling texture for many years (Chellappa and Chatterjee, 1985; Cohen and Cooper, 1987; Derin and Elliott, 1987). But in practice, for many applications of texture analysis such as texture segmentation and classification, multi-resolution filtering techniques are used much more frequently than random fields for their better performance and efficiency (Chui, 1992; Manjunath and Ma, 1996).
The maximum entropy random field (MERF) presented in this paper is a random field constructed upon multi-resolution filters. In the MERF, the multi-resolution filters are selected for the purpose of constructing the potential function in the joint probability density function of this random field. Furthermore, these selected filters are designed to be suitable for extracting texture features in various orientations and scales.
From the point of view of random fields, the multi-resolution filters in the MERF are used to describe interactions between adjacent pixels in texture images. While according to multi-resolution filtering techniques, a set of filters which are ideal for texture feature extraction have to be tuned to various orientations and scales in frequency domain. To satisfy both requirements, the multi-resolution filters in the MERF are carefully created in order to cover a variety of orientations and scales. And the parameters in these filters (at the same time they are the parameters of the MERF model) are determined in the process of fitting the MERF model to natural textures.
The rest of this paper is arranged as follows: Section 2 explains the creating of multi-resolution filters in the MERF, Section 3 explains how to incorporate multi-resolution filters into random fields with the maximum entropy (ME) method and Section 4 explains the parameter estimation for the MERF using Markov chain Monte Carlo(MCMC) method, Section 5 presents some experimental results of texture synthesis and texture retrieval, Section 6 gives a conclusion.
Section snippets
Multi-resolution filters in the MERF
The multi-resolution filtering techniques can provide a multi-resolution representation of an image. Each of the sub-images, generated by multi-resolution filters, contains information of a specific scale and orientation. In the MERF, the multi-resolution filters are created in two ways. Some of the filters are created by the product of a low-pass 1D filter and a high-pass 1D filter. For example, the filter for the direction θ and the scale s is defined aswhere denotes the
Maximum entropy random fields
From the point of view of random fields, the multi-resolution filters in the MERF are meant to describe contextual constraints or interactions between adjacent pixels in texture images. The MERF is the random field that not only satisfies contextual constraints but also has the maximum entropy, and it is constructed with the ME principle.
Parameter estimation using Markov chain Monte Carlo
In Bayesian analysis, the complexity of the posterior distribution means that direct estimation procedures are rarely possible, so the MCMC methods, particularly the Metropolis–Hastings algorithm (Cowles and Carlin, 1996; Hastings, 1970), and the Gibbs sampler, are widely used (Aykroyd, 1998).
In this section, only the specification of the posterior distribution is given, and the details of the MCMC estimation can be found in (Aykroyd, 1998).
According to Bayesian theory, the probability density
Experimental results
The MERF model is a random field built upon multi-resolution filters. It describes texture images as random fields, and extracts texture features with multi-resolution filters. In this section, the MERF model is tested with the experiments of texture synthesis and texture retrieval.
Conclusion
The MERF model, a combination of random fields and multi-resolution filters is presented as a novel approach for texture analysis. Its application to texture synthesis and texture retrieval is demonstrated. The MERF model makes use of multi-resolution filters to construct its joint probability density function and to extract texture features in various orientations and scales. The parameter estimation method based on MCMC is also presented in this paper to fit the MERF model to natural
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This work is supported in part by the research fund from Nanyang Technological University under grant number RG 31/94 and RG 21/98.