In low level vision, an observation 
is a
rectangular array of pixel values. In some cases, the observation may be
sparse, that is, it is available at 
. Every pixel
takes a value 
in a set 
. In practice, 
is often the set
of integers encoded by a byte (8 bits), as the results of normalization
and quantization, so that 
.
An observation d can be considered as a transformed and degraded
version of an MRF realization f. The transformation may include
geometric transformation and blurring and the degradation is due to
random factors such as noise. These determine the conditional
distribution 
, or the likelihood of f. A general observation model can be expressed as
where B is a blurring effect, 
is a transformation which can
be linear or nonlinear, deterministic or probabilistic, e is the
sensor noise and 
is an operator of addition or multiplication.
In practice, a simple observation model of no blurring, linear
transformation and independent additive Gaussian noise is often assumed.
Each observed pixel value is assumed to be the sum of the true grey
value and independent Gaussian noise
where 
is a linear function and 
. The probability distribution of d conditional on f,
or the likelihood of f, is
where
is the likelihood energy. Obviously, the above is a special form of
Gibbs distribution whose energy is due purely to single-site cliques in
the zero-th order neighborhood system (where the radius r=0), with
clique potentials being 
. If the
noise is also homogeneous, then the deviations 
= are
the same for all 
.
The function 
maps a label 
to a real grey value
where 
may be numerical or symbolic, continuous or discrete. When
is symbolic, for example, representing a texture type, the
ordering of the MRF labels is usually not defined unless artificially.
Without loss of generality, we can consider that there is a unique
numerical value for a label 
and denote 
simply as 
. Then the likelihood energy becomes
for i.i.d. Gaussian noise.
A special observation model for discrete label set is random
replacement. An 
is transformed into 
according to the
following likelihood probability
where 
contain un-ordered labels for 
and 
, 
, 
and 
. In this model, a
label value remains unchanged with probability p and changes to any
other value with equal probability 
. This describes the
transition from one state to another. The simplest case of this is the
random flip-over of binary values.
A useful and convenient model for both the underlying MRF and the observation is the hierarchical GRF model [Derin and Cole 1986 ; Derin and Elliott 1987]. There two-levels of Gibbs distributions are used to represent noisy or textured regions. The higher level Gibbs distribution, which is usually an MLL, characterizes the blob-like region formation process while the lower level Gibbs distribution describes the filling-in, such as noise or texture, in each region. This will be described in detail in Section 2.4.1.
