motion estimation and compensation for very low bitrate video coding

184 Chapter C. Markov Random Fields
the estimate of the solution u according to the collected data d:
where p(d)= P u p(dju):p(u).
p(ujd)= p(dju):p(u)
p(d)
(C.1)
Bayesian modeling is often used to determine the Maximum A Posteriori
(MAP) estimate, i.e. the value of the solution u that maximizes
the conditional probability p(ujd). The Maximum Likelihood (ML)
estimate does not statistically describe u: it considers it as a vector of
parameters. The ML is a special case of the MAP, in which the a priori
model p(u) is constant (uniform distribution). The aim then becomes
to maximize the conditional probability p(dju).
Working with the logarithm of the posterior density, one obtains:
log(p(ujd)) = log(p(dju)) + log(p(u)) , log(p(d)); (C.2)
where the last term does not depend on u and can be neglected in
maximization processes with respect to u. The MAP estimate is thus:
[ @log(p(dju))
@u
+ @log(p(u))
]ju=^uMAP =0; (C.3)
@u
where u =^uMAP is the solution at the maximum. Similarly, the ML
estimate is, with u =^uML the solution at the maximum:
C.2 Markov Random Fields
[ @log(p(dju))
]ju=^uML =0: (C.4)
@u
Let consider an image I, in which pixels de ne a lattice S of sites s:
S = fs =(i; j)g (C.5)
To every site s is associated a random variable [99] As, whose values
as belong to . For instance, = f0; :::; 255g represents the possible
luminance values of a black and white picture and = f0; :::; 255g q the
values that can be associated with any pixel of a mutispectral image
with q channels.