motion estimation and compensation for very low bitrate video coding

174 Chapter B. Rate Distortion theory
measures the average a priori uncertainty onX.
Considering two alphabets AM and AN , one can de ne a product space
AMN:
AMN : f(j; k)jj 2 AM;k2 AN g : (B.6)
The joint distribution p(j; k) and the joint ensemble (AMN;p(j; k))
help de ning the marginal distributions:
p(j)=
N,1 X
k=0
p(j; k); q(k) =
and the conditional distributions:
p(jjk)=
p(j; k)
q(k)
M,1 X
j=0
p(j; k); (B.7)
p(j; k)
; q(kjj)= : (B.8)
p(j)
A r.v. Z that assumes the event (j; k) with a probability p(j; k) maybe
de ned as: Z =(X; Y ) with X : p(j) and Y : q(k). The conditional
self-information is the information one receives when told that the
event X = j has occurred if one already knows the occurrence of the
event Y = k:
and the conditional entropy is:
I(jjk)=, log p(jjk); (B.9)
H(XjY )=, X
p(j; k) log p(jjk): (B.10)
The mutual information of both r.v.'s X and Y is thus:
j;k
I(j; k)=I(j) , I(jjk)=I(k; j); (B.11)
and the average mutual information or average amount of transmitted
information:
H(X; Y )=H(X) , H(XjY )= X
p(j; k)
p(j; k) log : (B.12)
p(j)q(k)
j;k