motion estimation and compensation for very low bitrate video coding

Appendix C
Markov Random Fields
The use of Markovian techniques in digital image processing is based
on the possibility of modeling the image texture, the noise, and even
some other criteria, like stochastic processes. Since their remarkable
use in image restoration by Geman and Geman [39], the theory of
Markovian processes [99] has been extended to Markov Random Fields
(MRF) [40], and their eld of application extended to images [137] and
motion elds [37, 109].
The present appendix aims at giving a quick overview of MRF and the
di erent concepts involved by their use. It will rst present Bayesian
modeling, which is often related to MRF. Then, image description and
neighborhood relationship will introduce the de nition of MRF. The
link with the Gibbs distribution is also tackled. Finally, the possible
algorithms to solve the overall minimization problem are referred to.
C.1 Bayesian Model
Bayesian modeling assumes a priori statistical distribution for the solution
of estimation problems. This a priori model (or prior model),
p(u), is a probabilistic description of the solution u or of its properties
before any sense data is collected.
A second component, the sensor model (or likelihood model), p(dju),
is a description of the noisy or stochastic processes that connects the
original (unknown) state u to the sampled input image or sensor values,
the data d. Using Bayes' rule, both can be combined in order to obtain
an a posteriori model (or posterior model), p(ujd), which describes