sparse image representation via combined transforms - Convex ...

3.1. DCT AND HOMOGENEOUS COMPONENTS 47
Theorem 3.4 (Hammersley-Clifford Theorem) Every Markov random field on a domain
D is a Gibbs random field on D and vice versa.
The rigorous statement and the actual proof is too long to be presented here, readers
are referred to [12, 129] for technical details.
Definition of a Gaussian Markov random field. In (3.13), if the p.d.f. p(ω) isalsothe
p.d.f. of a multivariate Normal distribution, then there are two consequences: first, from
the Hammersley-Clifford Theorem, it is a Markov random field; second, the random field is
also Gaussian. We call such a random field a Gaussian Markov random field (GMRF). Let
⃗ω denote a vector corresponding to a realization ω. (The vector ⃗ω is simply a list of all the
values taken by ω.) We further suppose the vector ⃗ω is a column vector. Let Σ denote the
covariance matrix of the corresponding multivariate Normal distribution. We have
(
p(ω) =(2π) − 1 2 dim(D) 1
det 1/2 (Σ) exp − 1 )
2 ⃗ωT Σ −1 ⃗ω .
This is a p.d.f. of a GMRF.
A Gaussian Markov random field is a way to model homogeneous images. In the image
case, the dimensionality of the domain is 2 (d = 2). The gray scale at each integer point in
Z 2 is a continuous value. The fact that an image is homogeneous and only has the second
order correlation, is equivalent to the fact that the image is a GMRF.
For a GMRF, if there is a transform that diagonalizes the covariance matrix Σ, then this
transform is the KLT of the GMRF. Since we are discussing the connection between DCT
and the homogeneous signal, the interesting question to ask is “Which kind of covariance
matrices does the DCT diagonalize?”. We answer in the next subsubsection.
There have been some efforts to extend the GMRF model; for example, the work by
Zhu, Wu and Mumford [145].
Covariance Matrix Diagonalization
In this subsubsection, we answer the question raised in the previous subsubsection. We
answer it in a converse manner. Instead of giving the conditions of the covariance matrices
and then discussing the diagonalizability of matrices associated with this type of DCT, we
try to find out which group of matrices can be diagonalized by which type of DCT.
The following Theorem is a general result. A reason we list it here is that we have not
seen it documented explicitly in any other references.