ABSTRACT
The simplest exponential families of distributions-independent random fields (IRF) of signals at the lattice sites-are of limited descriptive power, but due easy learning is often used in image analysis. An IRF is specified by site-wise marginal probability distributions of discrete signals or densities of continuous signals (called marginals), which generally vary over the lattice: Pr = (Pr(q) : q ∈ Q); ∑q∈Q Pr(q) = 1; r ∈ R. The probability or density Pirf(g) of an image g sampled from the IRF is factored over the lattice sites:
≡ exp ( ∑
) (3.1)
An inhomogeneous IRF with different site-wise marginals has the simplest sufficient statistics, namely, just the signal g(r) at each site, or |Q| its binary indicators: Sirf(g) = [(δ(g(r) − q) : q ∈ Q) ; r ∈ R]T. The negated logmarginals, Virf = [− ln Pr(q) : q ∈ Q; r ∈ R]T, act as weights, or potentials in Equation 2.17: Pirf(g) = exp
(−VTirfSirf(g)). Piecewise-homogeneous, or piecewise-uniform IRFs, which are very popu-
lar in image modeling, split the lattice into several regions Rl ; l ∈ L = {1, . . . , L}; L ≥ 2, characterized each by own conditional marginal Pl = (P(q | l) : q ∈ Q); ∑q∈Q P(q | l) = 1. The regions may be discontinuous and cover the entire lattice with no overlaps:
⋂ Rk = 0/ if l = k;
l, k ∈ L. A conditional piecewise-homogeneous IRF presumes that all images g ∈ G
have the same region map m = [m(r) : r ∈ R; m(r) ∈ L], so that Rl = {r : r ∈ R; m(r) = l} ⊂ R. The conditional probability or density of an image g, given a map m:
Modeling for Medical
and the log-likelihood of the marginal P = {Pl : l ∈ L}, given an image g and a map m:
depend on the conditional signal histogram, or the nonnormalized empirical conditional probability distribution of signals h(g |m) = [hg:m(q | l) : q ∈ Q; l ∈ L]. Each component
of this histogram is the cardinality of the subset Rq:l = {r : m(r) = l; g(r) = q} ⊆ Rl of sites supporting the signal q in the region Rl :
Given a map m, the components hg:m(q | l) are scalar sufficient statistics of the image g with the potentials Vq:l =− log P(q | l) in Equation 2.17.
