2.4.2
Texture segmentation is to segment an image into regions according to the textures of the regions. In the supervised texture segmentation, it is assumed that all the parameters for the textures, and for the noise if it is present, are specified [Hansen and Elliott 1982 ; Elliott et al. 1984 ; Geman and Geman 1984 ; Besag 1986 ; Derin and Cole 1986 ; Cohen and Cooper 1987 ; Derin and Elliott 1987]; and the segmentation is to partition the image in terms of the textures whose distribution functions have been completely specified.
Unsupervised segmentation not only does the partition but also needs to estimate the involved parameters [Lakshmanan and Derin 1989 ; Manjunath and Chellappa 1991 ; Chen and Fan 1992 ; Hu and Fahmy 1987 ; Won and Derin 1992]. Obviously, this is more practical in applications and is also more challenging. There is a chicken-and-egg problem. The estimation should be performed by using realizations of a single class of MRFs, i.e. a single type of textures ( cf. Chapter 6), albeit noisy or noise-free. This requires that the segmentation be done. However, the segmentation depends on the parameters of the underlying textures. A strategy for solving this is to use an iterative algorithm alternating between segmentation and estimation. While the more advanced topic of unsupervised segmentation will be discussed in Section 6.2.2, we focus here on the supervised MRF segmentation with all parameters known.
Texture segmentation, as other labeling problems, is usually performed
in an optimization sense, such as MAP. A main step is to formulate the
posterior energy. In MAP texture segmentation with the hierarchical
texture model, the higher level MRF determines the prior probability
, where the segmentation f partitions 
into regions each of
which is assigned a texture type from the label set 
; the lower
level field contributes to the likelihood function 
where
d is the image data composed of multiple textures.
Let the label set be 
and f represent a
segmentation in which 
is the indicator of the texture type
for pixel i. Denote the set of all sites labeled I by
Then
and
The likelihood energy function can be expressed as
where 
is the potential function for the data d
on c labeled as type I.
Suppose that type-I texture is modeled as an MLL with parameters
Then according to (1.53), the single-site clique potentials are
where 
and according to (1.52), the multi-site
clique potentials are
The above is for the cliques in the interior of 
. At and
near region boundaries, a clique c may ride across two or more
's. In this case, the following rule may be used to determine
the type of texture for generating the data: If c sits mostly in a
particular 
, then choose model parameters 
; if
it sits equally in all the involved 
's, choose an I at
random from the involved labels. When the grey levels for type I
texture image data are known as
where 
is the
number of grey levels for type I texture, more constraints are imposed
on the texture segmentation and better results can be expected. When the
grey levels are also subject to noise, then the constraints become
inexact.
After 
is also defined for the region process, the posterior
energy can be obtained as
Minimizing (2.58) with respect to f is more complicated than the minimization for the restoration and reconstruction formulated in the previous sections because neither term on the RHS can be decomposed into independent sub-terms. In [Derin and Elliott 1987], some assumptions of independence are made to simplify the formulation and a recursive dynamic programming algorithm (see Section 8.2.4) is used to find a sub-optimal solution.
