2.4.1
An MRF model of texture can be specified by the joint probability
. The probability determines how likely a texture pattern f is
to occur. Several MRF models, such as
auto-binomial [Cross and Jain 1983], auto-normal
(or GMRF) [Chellappa 1985 ; Cohen and Cooper 1987], and
multi-level logistic (MLL) model [Derin et al. 1985 ; Derin and Elliott 1987], have been
used for modeling textures. A particular MRF model tends to favor the
corresponding class of textures, by associating them with larger
probability than others. Here, ``a particular MRF model'' is a
particular probability function 
which is specified by its form
and the parameters in it.
In the MLL model, for example, the probability of a texture pattern f is defined based on the MLL clique potential function (1.52). In the second order (8-neighbor) neighborhood system, there are ten types of cliques as illustrated in Fig.1.2, with the parameters associated with them shown in Fig.1.4. When only the pair-wise clique potentials are non-zero, equation (1.52) reduces to (1.54), rewritten below
where 
and 
is a set of pair-site
cliques, and 
is a parameter associated with the type-c
pair-site cliques. After MLL is chosen, to specify an MRF model is to
specify the 
's. When the model is anisotropic, i.e.
with all
, the model tends to generate texture-like patterns;
otherwise it generates blob-like regions. The probability 
can be
calculated from 
's using the corresponding Gibbs
distribution.
A textured pattern corresponding to a realization of a Gibbs
distribution 
can be generated by
sampling the distribution. Two often used sampling algorithms are the
Metropolis sampler [Metropolis and etal 1953] and the Gibbs sampler
[Geman and Geman 1984] ( cf.
Section 9.11).
Figs.2.7 and 2.8
list two sampling algorithms [Chen 1988]. Algorithm 1 is based
on the Metropolis sampler
[Hammersley and Handscomb 1964] whereas Algorithm 2 is based on the Gibbs sampler
of [Geman and Geman 1984]. A sampling algorithm
generates a texture f with probability 
.
Figure 2.7: Generating a texture using Metropolis sampler.
Figure 2.8: Generating a texture using Gibbs sampler.
The differences between the two sampling algorithm is in step (2).
Algorithm 1 needs only to evaluate one exponential function because the
update is based on the ratio 
in step (2.2). Algorithm 2
needs to compute M exponential functions and when the exponents are
very large, this computation can be inaccurate. The following
conclusions on the two algorithms are made by [Chen 1988]: (1)
N=50 iterations are enough for both algorithms, (2) Algorithm 2 tends
to generate a realization with a lower energy than Algorithm 1 if N is
fixed and (3) Algorithm 1 is faster than Algorithm 2.
Figure 2.9: Textured images (
) generated by using MLL
models.
Upper-left: Number of pixel levels M=3 and parameters
.
Upper-right: M=4, 
.
Lower-left: M=4, 
.
Lower-right: M=4, 
.
Fig.2.9 shows four texture images generated
using the MLL model (1.54) and Algorithm 1. It can be seen
that when all the 
parameters are the same such that the model is
isotropic, the formed regions are blob-like; when it is anisotropic, the
generated pattern looks textured. So, blob regions are modeled as a
special type of textures whose MRF parameters tend to be isotropic.
The above method for modeling a single texture provides the basis for
modeling images composed of multiple textured regions. In the so-called
hierarchical model
[Derin and Cole 1986 ; Derin and Elliott 1987 ; Won and Derin 1992],
blob-like regions are modeled by a high level MRF which is an isotropic
MLL; these regions are filled in by patterns generated according to MRFs
at the lower level. A filling-in may be either additive noise or a type
of texture, characterized by a Gibbs distribution. Let f be a
realization of the region process with probability 
. Let d be
the textured image data with probability 
. A texture label
indicates which of the M possible textures in the texture label
set 
that pixel i belongs to.
The MLL model is often used for the higher level region process. Given f, the filling-in's are generated using the following observation model
where 
is a random function that implements the mechanism of a
noise or texture generator. To generate noisy regions, the observation
model is simply 
, where 
denotes the true grey level for region type 
, as in
(2.2). When the noise is also Gaussian, it
follows a special Gibbs distribution, that in the zero-th order
neighborhood system (where the radius of the neighborhood is zero). When
the noise is an identical, independent and zero-mean distribution, the
likelihood potential function is 
where 
is the variance.
For textured regions, 
implements anisotropic MRFs. Because
of contextual dependence of textures, the value for 
depends on the
neighboring values 
. Given 
and 
, its
conditional probability is 
where all the
sites in 
are assumed to belong to the same texture type,
, and hence some treatment may be needed at and near boundaries.
Note that the texture configuration is d rather than f; f is the
region configuration. The configuration d for a particular texture can
be generated using an appropriate MRF model (such as MLL used before for
generating a single texture). That is to say, d is generated according
to 
; see further discussions in
Section 2.4.2. Fig.2.10
shows an image of textured regions generated by using the hierarchical
Gibbs model. There, the region process is the same as that for the
upper-left of Fig.2.9 and the texture
filling-in's correspond to the other three realizations.
Figure 2.10: Multiple texture regions generated using the hierarchical
Gibbs model of textures.