Convex Codecells Research Articles

Randomized Quantization and Source Coding With Constrained Output Distribution

This paper studies fixed-rate randomized vector quantization under the constraint that the quantizer's output has a given fixed probability distribution. A general representation of randomized quantizers that includes the common models in the literature is introduced via appropriate mixtures of joint probability measures on the product of the source and reproduction alphabets. Using this representation and results from optimal transport theory, the existence of an optimal (minimum distortion) randomized quantizer having a given output distribution is shown under various conditions. For sources with densities and the mean square distortion measure, it is shown that this optimum can be attained by randomizing quantizers having convex codecells. For stationary and memoryless source and output distributions a rate-distortion theorem is proved, providing a single-letter expression for the optimum distortion in the limit of large block-lengths.

On Optimal Zero-Delay Coding of Vector Markov Sources

Optimal zero-delay coding (quantization) of a vector-valued Markov source driven by a noise process is considered. Using a stochastic control problem formulation, the existence and structure of optimal quantization policies are studied. For a finite-horizon problem with bounded per-stage distortion measure, the existence of an optimal zero-delay quantization policy is shown provided that the quantizers allowed are ones with convex codecells. The bounded distortion assumption is relaxed to cover cases that include the linear quadratic Gaussian problem. For the infinite horizon problem and a stationary Markov source the optimality of deterministic Markov coding policies is shown. The existence of optimal stationary Markov quantization policies is also shown provided randomization that is shared by the encoder and the decoder is allowed.

Jointly Optimal LQG Quantization and Control Policies for Multi-Dimensional Systems

For controlled R <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> -valued linear systems driven by Gaussian noise under quadratic cost criteria, we investigate the existence and the structure of optimal quantization and control policies. For fully observed and partially observed systems, we establish the global optimality of a class of predictive encoders and show that an optimal quantization policy exists, provided that the quantizers allowed are ones which have convex codecells. Furthermore, optimal control policies are linear in the conditional estimate of the state, and a form of separation of estimation and control holds.

On Codecell Convexity of Optimal Multiresolution Scalar Quantizers for Continuous Sources

It has been shown by earlier results that for fixed rate multiresolution scalar quantizers and for mean squared error distortion measure, codecell convexity precludes optimality for certain discrete sources. However it was unknown whether the same phenomenon can occur for any continuous source. In this paper, examples of continuous sources (even with bounded continuous densities) are presented for which optimal fixed rate multiresolution scalar quantizers cannot have only convex codecells, proving that codecell convexity precludes optimality also for such regular sources.

Lagrangian Optimization of Two-Description Scalar Quantizers

In this paper, we study the problem of optimal design of balanced two-description fixed-rate scalar quantizer (2DSQ) under the constraint of convex codecells. Using a graph-based approach to model the problem, we show that the minimum expected distortion of the 2DSQ is a convex function of the number of codecells in the side quantizers. This property allows the problem to be solved by Lagrangian minimization for which the optimal Lagrangian multiplier exists. Given a trial multiplier, we exploit a monotonicity of the objective function, and develop a simple and fast dynamic programming technique to solve the parameterized problem. To further improve the algorithm efficiency, we propose an RD-guided search strategy to find the optimal Lagrangian multiplier. In our experiments on distributions of interest for signal compression applications the proposed algorithm improves the speed of the fastest algorithm so far, by a factor of O(K/logK), where K is the number of codecells in each side quantizer. We also assess the impact on the optimality of the convex codecell constraint. Using a published performance analysis of 2DSQ at high rates, we show that asymptotically this constraint does not preclude optimality for L 2 distortion measure, when channels have a higher than 0.12 loss rate.

Optimal Two-Description Scalar Quantizer Design

Multiple description quantization is a signal compression technique for robust networked multimedia communication. In this paper we consider the problem of optimally quantizing a random variable into two descriptions, with each description being produced by a side quantizer of convex codecells. The optimization objective is to minimize the expected distortion given the probabilities of receiving either and both descriptions. The problem is formulated as one of shortest path in a weighted directed acyclic graph with constraints on the number and types of edges. An $O(K_1K_2N^3)$ time algorithm for designing the optimal two-description quantizer is presented, where $N$ is the cardinality of the source alphabet, and $K_1$, $K_2$ are the number of codewords of the two quantizers, respectively. This complexity is reduced to $O(K_1K_2N^2)$ by exploiting the Monge property of the objective function. Furthermore, if $K_1 = K_2 = K$ and the two descriptions are transmitted through two channels of the same statistics, then the optimal two-description quantizer design problem can be solved in $O(KN^2)$ time.

Codecell convexity in optimal entropy-constrained vector quantization

Properties of optimal entropy-constrained vector quantizers (ECVQs) are studied for the squared-error distortion measure. It is known that restricting an ECVQ to have convex codecells may preclude its optimality for some sources with discrete distribution. We show that for sources with continuous distribution, any finite-level ECVQ can be replaced by another finite-level ECVQ with convex codecells that has equal or better performance. We generalize this result to infinite-level quantizers, and also consider the problem of existence of optimal ECVQs for continuous source distributions. In particular, we show that given any entropy constraint, there exists an ECVQ with (possibly infinitely many) convex codecells that has minimum distortion among all ECVQs satisfying the constraint. These results extend analogous statements in entropy-constrained scalar quantization. They also generalize results in entropy-constrained vector quantization that were obtained via the Lagrangian formulation and, therefore, are valid only for certain values of the entropy constraint.