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3.1. DCT AND HOMOGENEOUS COMPONENTS 45
Definition of a random field. We define a random field on a lattice. Let Z d denote the
d-dimensional integers, or the lattice points, in d-dimensional space, which is denoted by
R d . The finite set D isasubsetofZ d : D ⊂ Z d . For two lattice points x, y ∈ Z d , let |x − y|
denote the Euclidean distance between x and y. ThesetD is connected if and only if for any
x, y ∈ D, there exists a finite subset {x 1 ,x 2 ,... ,x n } of D, n ∈ N, such that (1) |x−x 1 |≤1,
(2) |x i − x i+1 |≤1,i =1, 2,... ,n− 1, and (3) |x n − y| ≤1. We call a connected set D a
domain. The dimension of the set D is, by definition, the number of integer points in the
set D. We denote the dimension of D by dim(D). On each lattice point in set D, a real
value is assigned. The set R D , which is equivalent to R dim(D) , is called a state space. Follow
some conventions, we denote the state space by Ω, so we have Ω = R dim(D) .LetF be the
σ-algebra that is generated from the Borel sets in Ω. Let P be the Lebesgue measure. The
triple (Ω, F, P) is called a random field (RF) on the domain D.
Note that we define a random field on a subset of all the lattice points.
Now we give the definition of a neighbor. Intuitively, under Euclidean distance, two
integer (lattice) points x and y are neighbors when |x − y| ≤1. This definition can be
extended. We define a non-negative, symmetric and translation-invariant bivariate function
N(x, y) on domain D, such that for x, y ∈ D, the function N satisfies
1. N(x, x) =0,
2. N(x, y) ≥ 0 (non-negativity),
3. N(x, y) =N(y, x) (symmetry),
4. N(x, y) =N(0,y− x) (homogeneity, or translation invariance).
Any two points are called neighbors if and only if N(x, y) > 0. For example, in Euclidean
space, if we let N(x, y) =1when|x − y| =1,andN(x, y) = 0 elsewhere, then we have the
ordinary definition of neighbor that is mentioned at the beginning of this paragraph.
Definition of a Markov random field. The definition of a Markov random field is based
upon conditional probability. The key idea of Markovity is that conditional probability
should depend only on neighbors. To be more precise, we need some terminology. Let ω
denote an element of Ω. We call ω a realization. Let p(ω) denote the probability density
function of ω. The p.d.f. p is associated with the Lebesgue measure P. Let ω(x) be the
value of the realization ω at the point x. For a subset A ⊂ D, suppose the values at points