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 X₁, X₂,..., 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 D₁,...,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 X₁, X₂,..., XN, R_p*(D₁,...,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(D₁,...,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 X₁,..., X_N form a (first-order) Markov chain, we have R_p*(D₁,...,D_N)=R_c*(D₁,...,D_N). However, this is not true in general if X₁,..., X_N do not form a (first-order) Markov chain. Specifica- ly, we demonstrate that for independent and identically distributed vector source (X₁,..., X_N), if X₁,..., X_N do not form a (first-order) Markov chain, then under some conditions on source frames and distortion, R_c*(D₁,..., D_N) is strictly less than R_p*(D₁,..., D_N) in general. Our techniques allow us to compare R_p*(D₁,..., D_N) with R_c*(D₁,..., D_N) even when the single-letter characterization of R_p*(D₁,..., D_N), if any, is unknown.