On the Information Theoretic Performance Comparison of Causal Video Coding and Predictive Video Coding

Causal video coding is a coding paradigm where video source frames X 1 , X 2 ,..., X N are encoded in a frame-by-frame manner, the encoder for each frame can use all previous source frames and all previous encoded frames, and the corresponding decoder can use only all previous encoded frames. In the special case where the encoder for each frame X k is further restricted to enlist help only from all previous encoded frames, causal video coding is reduced to predictive video coding, which all MPEG-series and H-series video coding standards proposed so far are based upon. In this paper, we compare the rate distortion performance of causal video coding with that of predictive video coding from an information theoretic perspective by modeling each frame X k itself as a source X k ={X k (i)} i=1 ∞ . Let R c *(D 1, ...,D N ) (R p *(D1,...,DN), respectively) denote the minimum total rate required to achieve a given distortion level D 1 ,...,D N in causal video coding (predictive video coding, respectively). We first show that like R c *(D1,..., D N ), for jointly stationary and totally ergodic sources X 1 , X 2 ,..., XN, R p *(D 1 ,...,D N ) is equal to the infimum of the nth order total rate distortion function R p,n (D1,...,DN) over all n, where R p,n (D 1 ,...,D N ) itself is given by the minimum of an information quantity over a set of auxiliary random variables. We then prove that if the jointly stationary and totally ergodic sources X 1 ,..., X N form a (first-order) Markov chain, we have R p *(D 1 ,...,D N )=R c *(D 1 ,...,D N ). However, this is not true in general if X 1 ,..., X N do not form a (first-order) Markov chain. Specifically, we demonstrate that for independent and identically distributed vector source (X 1 ,..., X N ), if X 1 ,..., X N do not form a (first-order) Markov chain, then under some conditions on source frames and distortion, R c *(D 1 ,..., D N ) is strictly less than R p *(D 1 ,..., D N ) in general. Our techniques allow us to compare R p *(D 1 ,..., D N ) with R c *(D 1 ,..., D N ) even when the single-letter characterization of R p *(D 1 ,..., D N ), if any, is unknown.