Universal Denoising of Discrete-time Continuous-Amplitude Signals

TL;DR

This paper introduces universal denoising algorithms for discrete-time signals with continuous amplitudes corrupted by known channels, achieving optimal performance without prior knowledge of the signal distribution.

Contribution

It develops practically implementable, distribution-agnostic denoisers that are universally optimal for large sequences in both semi-stochastic and stochastic settings.

Findings

Achieves universal optimality in denoising performance.

Effective in Gaussian and non-Gaussian noise settings.

Demonstrated success in image denoising applications.

Abstract

We consider the problem of reconstructing a discrete-time signal (sequence) with continuous-valued components corrupted by a known memoryless channel. When performance is measured using a per-symbol loss function satisfying mild regularity conditions, we develop a sequence of denoisers that, although independent of the distribution of the underlying `clean' sequence, is universally optimal in the limit of large sequence length. This sequence of denoisers is universal in the sense of performing as well as any sliding window denoising scheme which may be optimized for the underlying clean signal. Our results are initially developed in a ``semi-stochastic'' setting, where the noiseless signal is an unknown individual sequence, and the only source of randomness is due to the channel noise. It is subsequently shown that in the fully stochastic setting, where the noiseless sequence is a stationary stochastic process, our schemes universally attain optimum performance. The proposed schemes draw from nonparametric density estimation techniques and are practically implementable. We demonstrate efficacy of the proposed schemes in denoising gray-scale images in the conventional additive white Gaussian noise setting, with additional promising results for less conventional noise distributions.