Optimal Lagrange Multipliers for Dependent Rate Allocation in Video Coding

In a typical video rate allocation problem, the objective is to optimally distribute a source rate budget among a set of (in)dependently coded data units to minimize the total distortion of all units. Conventional Lagrangian approaches convert the lone rate constraint to a linear rate penalty scaled by a multiplier in the objective, resulting in a simpler unconstrained formulation. However, the search for the “optimal” multiplier—one that results in a distortion-minimizing solution among all Lagrangian solutions that satisfy the original rate constraint—remains an elusive open problem in the general setting. To address this problem, we propose a computation-efficient search strategy to identify this optimal multiplier numerically. Specifically, we first formulate a general rate allocation problem where each data unit can be dependently coded at different quantization parameters (QP) using a previous unit as predictor, or left uncoded at the encoder and subsequently interpolated at the decoder using neighboring coded units. After converting the original rateconstrained problem to the unconstrained Lagrangian counterpart, we design an efficient dynamic programming (DP) algorithm that finds the optimal Lagrangian solution for a fixed multiplier. Finally, within the DP framework, we iteratively compute neighboring singular multiplier values, each resulting in multiple simultaneously optimal Lagrangian solutions, to drive the rates of the computed Lagrangian solutions towards the bit budget. We terminate when a singular multiplier value results in two Lagrangian solutions with rates below and above the bit budget. In extensive monoview and multiview video coding experiments, we show that, for fixed target rate constraints, our DP algorithm and selection of optimal multipliers on average outperform comparable rate control solutions used in video compression standards such as HEVC that do not skip frames in Y-PSNR.