A Theorem on Multi-Objective Optimization Approach for Bit Allocation of Scalable Coding

TL;DR

This paper establishes conditions under which all Pareto optimal solutions in multi-objective bit allocation for scalable codecs can be obtained through scalarization, linking properties of rate-distortion functions to solution completeness.

Contribution

It provides a sufficient condition involving convexity and monotonicity of rate-distortion functions ensuring scalarization captures all Pareto optimal points.

Findings

All Pareto points can be derived via scalarization under specified conditions.

Convexity and strict decreasing nature of rate-distortion functions are key.

The Pareto front forms a continuous curve under these conditions.

Abstract

In the current work, we have formulated the optimal bit-allocation problem for a scalable codec of images or videos as a constrained vector-valued optimization problem and demonstrated that there can be many optimal solutions, called Pareto optimal points. In practice, the Pareto points are derived via the weighted sum scalarization approach. An important question which arises is whether all the Pareto optimal points can be derived using the scalarization approach? The present paper provides a sufficient condition on the rate-distortion function of each resolution of a scalable codec to address the above question. The result indicated that if the rate-distortion function of each resolution is strictly decreasing and convex and the Pareto points form a continuous curve, then all the optimal Pareto points can be derived by using the scalarization method.