sparse image representation via combined transforms - Convex ...

46 CHAPTER 3. IMAGE TRANSFORMS AND IMAGE FEATURES
in A are given by a deterministic function f(·), the conditional p.d.f. at point x ∈ D given
values in A is denoted by p{ω(x)|ω(y) =f(y),y ∈ A ⊂ D}. To simplify, we let
p{ω(x)|ω(·) =f(·) onA} = p{ω(x)|ω(y) =f(y),y ∈ A ⊂ D}.
If the p.d.f. satisfies the following conditions:
1. positivity: p(ω) > 0, ∀ω ∈ D,
2. Markovity: p{ω(x)|ω(z) =f(z),z ∈ D,z ≠ x} =
p{ω(x)|ω(z) =f(z),z ∈ D,x and z are neighbor},
3. translation invariance: if two functions f,f ′ satisfy f(x + z) =f ′ (y + z) for all z ∈ D,
then p{ω(x)|ω(·) =f(·), on D −{x}} = p{ω(y)|ω(·) =f ′ (·), on D −{y}},
then the random field (Ω, F, P) isaMarkov random field (MRF).
Definition of a Gibbs random field. The key idea of defining a Gibbs random field is
writing the p.d.f. as an exponential function. Let p(ω) represent the same p.d.f. as in the
previous paragraph. A random field is a Gibbs random field (GRF) if function p(ω) has the
form
⎛
⎞
p(ω) =Z −1 exp ⎝− 1 ∑ ∑
ω(x)ω(y)U(x, y) ⎠ , (3.13)
2
x∈D y∈D
where the constant Z is a normalizing constant, and the function U(x, y) has the following
properties: for x, y ∈ D, wehave
1. U(x, y) =U(y, x) (symmetry),
2. U(x, y) =U(0,y− x) (homogeneity),
3. U(x, y) =0whenpointsx and y are not neighbors (nearest neighbor property).
The function U(x, y) is sometimes called pair potential [129].
Albeit the two types of random field are ostensibly different, they are actually equivalent.
The following theorem, which is attributed to Hammersley and Clifford, creates the
equivalence. Note that the theorem is not expressed in a rigorous form. We are just trying
to demonstrate the idea.