Structural Identification of a Generic MGRF Model

A generic MGRF model of spatially homogeneous textures is specified by the explicit spatial geometry (the characteristic neighbourhood) and quantitative strengths of statistical dependencies (clique potentials). Conventional model identification involves estimating both the characteristic neighbourhood and the Gibbs potential of each selected family, from a given texture. The process, in particular potential refinement via stochastic approximation, involves exponential time complexity [38].

In order to simplify the identification, a structural approach is proposed which focuses on identifying the geometric structure of texels (short for TEXture ELement [43]) and placement rules of their spatial arrangement. In the resulting texture description, a texture is constructed by a group of texels that repeats many times over the image plane by certain regular or stochastic spatial arrangement. This description is in line with structural approach to texture analysis [43], which suggests texels and their spatial organisation constitute a ``two-layered structure'' in a texture, specifying local and global properties, respectively,

In the proposed method, each texel is a micro geometric element, consisting of a group of image pixels (not necessarily continuous) with a certain spatial configuration. Each individual texel is distinguished by both the geometric structure and the combination of image signals over the structure. Usually, a texel involves a rather simple spatial configuration of only a small number of pixels. For example, Figure 6.4 shows a simple texel with hexagonal structure.

Figure 6.1: A simple hexagonal texel of six-pixels.

The structural identification of a generic MGRF model in Eq. (6.2.5) for a given training image ${g}^{\circ}$ involves the following steps:

1. Construct the MBIM using the approximate relative partial energy given by Eq. (6.3.4).
2. Select a geometric structure of the texels by taking account of the higher-energy clique families in the MBIM.
3. Derive the placement rules for these texels.

Figure 6.2: Brodatz textures [8] and their MBIMs. The textures and their MBIMs are of size $128\times 128$ and $125\times 125$ respectively. On the left-hand side are stochastic textures, and on the right-hand side are regular textures.

D4		D6
D12		D20
D24		D52
D66		D76
D105		D77

Figure 6.3: Stochastic textures D29 [8] and Bark0009 [82], their MBIMs and recovered texels.

D29	Bark0009
D29 MBIM	Bark0009 MBIM
D29 Texels	Bark0009 Texels

Figure 6.4: Regular textures D34 and D101 [8], their MBIMs and recovered texels.

D34	D101
D34 MBIM	D101 MBIM
D34 Texel	D101 Texel