motion estimation and compensation for very low bitrate video coding

B.3 Rate Distortion Function 177
B.3 Rate Distortion Function
The Rate Distortion function R(D) (cf. gure B.2) speci es the minimum
rate that enables one to reproduce the source with an average
distortion that does not exceed D. To design the R(D) of a d.m.s. with
respect to a single-letter delity criterion F , one needs to know the
average distortion d(q) associated with the transition probabilities
q(kjj) of the channel:
d(q)= X
p(j)q(kjj) (j; k): (B.22)
j;k
The channel de ned by q(kjj) is called D-admissible if and only if
d(q) D. QD denotes the set of all D-admissible transition probability
assignments: QD = fq(kjj)jd(q) Dg. In parallel, every assignment
gives rise to an average mutual information:
H(q) = X
j;k
p(j)q(kjj) log q(kjj)
: (B.23)
q(k)
The rate distortion function R(D) offXt;pg with respect to F is then
de ned by:
R(D) = min H(q); (B.24)
q2QD
with D 2 [0; 1). Since the source is given and not the channel, such
an equation determines the minimum rate at which the information
about the source must be conveyed to the user in order to achieve a
prescribed delity (the reverse - given channel - generates a distortion
rate function). It means that R(D) C is a necessary condition for
the existence of a communication system that operates with
delity D.
As illustrated on gure B.2, R(D) is a monotonic, decreasing, convex U
function in the interval of interest from D =0toD = Dmax. R(D) =0
for D = Dmax. Dmax is the average distortion achieved when only the
statistics of the source are known but when nothing at all about the
particular realization of the source has been transmitted. In this case,
the decoder can only guess what the source could have been.