6.2.2
In the previous simultaneous restoration and estimation, it is assumed there is only a single MRF present in the data. Segmentation has to be performed when the data contains realizations of more than one MRF. For the said reasons, there is a need for simultaneous segmentation and estimation. The optimal parameters are estimated during the computation of the optimal segmentation.
Let 
denote the possible texture types and f
represent a labeling or segmentation with 
indicating the
texture type for pixel i. The segmentation f partitions 
into
M types of disjoint regions (In completely unsupervised procedures, the number of texture types, M, is also an unknown parameter to be estimated. See Section 6.3.1)
( cf. Equations (2.51-2.53) The problem may be formulated in terms of the MAP principle as
Assume that 
and 
are independent of each other and
that they are uniformly distributed when no prior knowledge about their
distributions is available. Then the above is reduced to the following
In general, this maximization problem is computationally intractable.
However, it is solvable when 
can be expressed in closed-form
as function 
[Won and Derin 1992].
Assume that the regions are piecewise constant valued and are governed
by an MRF model, and that the observation model 
where 
is additive identical independent zero-mean Gaussian noise
and 
is the grey level for type 
regions. Then
given f and d, the ML estimate of the noise variance for type I
MRF regions can be expressed explicitly as a function of f and d
where 
is the number of pixels in type I regions. When
given as 
,
(6.48) is reduced to
However, the minimization over the f-
space is a still difficult one. Therefore, it is divided into the two
sub-problems
Thus, the procedure iteratively alternates between the two equations.
The estimate 
thus found is suboptimal with respect to
(6.50).
As in the case for simultaneous restoration and estimation, there are
several options for choosing methods for solving
(6.51) and (6.52).
For example, one may choose a simulated annealing procedure if he can
afford to find a good 
for (6.51) or use
heuristic ICM to find a local solution. Pseudo-likelihood, the coding
method or the mean field method may be used for approximating 
in (6.52).
Now we turn to the case of textured images. Suppose that the
hierarchical Gibbs model of [Derin and Cole 1986,Derin and Elliot 1987] is used to represent
textured images. The higher level MRF corresponds to a region process
with distribution 
. When modeled as a homogeneous
and isotropic MLL model, 
in which 
have an effect of controlling the relative
percentage of type-I pixels and 
controls the interaction
between neighboring regions. At the lower level, there are 
types of MRFs for the filling-in textures with p.d.f. 
with 
. When a
filling-in is modeled as an auto-normal MRF
[Besag 1974], its parameters consists of the mean and the interaction
matrix 
. After all, note that the
region process is a much ``slower varying" MRF than the texture MRFs.
Because it is difficult to compute the true joint conditional p.d.f.
, we use the pseudo-likelihood, 
, to approximate it (The lower case letters pl is used to denote the PL for the continuous random variables d.). The
conditional p.d.f. for 
takes the form (1.44).
When the MRF is also homogeneous, the conditional
p.d.f. can be written as
where 
. The above is a meaningful formula only when all the
sites in the block 
have the same texture label. To
approximate, we may pretend that the all the sites in 
have the
same label as i even when this is not true; in this case, the formula
is only an precise one.
The conditional p.d.f. 
has a very
limited power for texture description because it is about the datum on a
single site i given the neighborhood. Therefore, a scheme based on a
block of data may be used to enhance the capability [Won and Derin 1992].
Define a block centered at i,
for each 
. Regard the data in 
as a vector
of 
dependent random variables, denoted 
. Assume
that that the all the sites in 
have the same label, 
, as
i. Then the joint p.d.f. of 
given f and 
is multivariate normal
where 
and 
are the mean vector and covariance
matrix of 
, respectively. Note that 
is an
vector, as 
, and 
is an 
matrix.
The parameters, 
and 
, represent texture
features for type-I texture. They can be estimated using sample mean
and covariance. For only a single block centered at i, they are
estimated as 
and
. For
all blocks in 
, they are taken as the averages
and
Obviously, the estimates have larger errors at or near boundaries.
Obviously, 
is symmetric. When 
, which is
generally true, 
is positive definite with probability 1.
Therefore, the validity of 
is
generally guaranteed. It is not difficult to establish relationships
between the elements in the inverse of the covariance matrix
and those in the interaction matrix 
[Won and Derin 1992].
Based on the conditional p.d.f.'s for the blocks, the pseudo-likelihood for type-I texture can be defined as
where 
is defined in (2.51), and
is there to compensate for the fact that
is the joint p.d.f. of n elements in
. The overall pseudo-likelihood is then
Given f, 
can be explicitly expressed as
(6.56 and (6.57) and hence 
is 
-free. Then, the problem reduces to
finding optimal f and 
.
Given initial values for the parameters 
and the segmentation,
such a procedure alternately solves the two sub-problems:
until convergence. In the above, 
may be
replaced by the corresponding pseudo-likelihood. Fig.6.6
show a segmentation result.
Figure 6.6: Texture segmentation with unknown parameters.
(Upper-left) True regions. (Upper-right) Observed noisy textured regions.
(Lower-left) Segmentation error. (Lower-right) Segmentation result.
From (Won and Derin 1992) with permission; © 1992 Academic Press.
