Computation and analysis of the N-Layer scalable rate-distortion function

Methods for determining and computing the rate-distortion (RD) bound for N-layer scalable source coding of a finite memoryless source are considered. Optimality conditions were previously derived for two layers in terms of the reproduction distributions q_y1 and q_y2|y₁. However, the ignored and seemingly insignificant boundary cases, where q_y1=0 and q_y2|y₁ is undefined, have major implications on the solution and its practical application. We demonstrate that, once the gap is filled and the result is extended to N-layers, it is, in general, impractical to validate a tentative solution, as one has to verify the conditions for all conceivable q_yi+1,...,y_N|y₁,...,y_i at each (y₁,...,y_i) such that q_y1,...,y_i=0. As an alternative computational approach, we propose an iterative algorithm that converges to the optimal joint reproduction distribution q_y1,...,y_N, if initialized with q_y1,...,y_N>0 everywhere. For nonscalable coding (N=1), the algorithm specializes to the Blahut-Arimoto (1972) algorithm. The algorithm may be used to directly compute the RD bound, or as an optimality testing procedure by applying it to a perturbed tentative solution q. We address two additional difficulties due to the higher dimensionality of the RD surface in the scalable (N>1) case, namely, identifying the sufficient set of Lagrangian parameters to span the entire RD bound; and the problem of efficient navigation on the RD surface to compute a particular RD point.