Universal Denoising of Discrete-time Continuous-Amplitude Signals

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.

💡 Research Summary

The paper tackles the fundamental problem of denoising a discrete‑time signal whose samples take continuous values and are corrupted by a known memoryless channel. Unlike traditional approaches that rely on a specific probabilistic model of the clean signal, the authors develop a universal denoising scheme that does not need any prior knowledge of the underlying distribution and yet achieves asymptotically optimal performance.

Problem formulation and loss model
Let (X^{n}=(X_{1},\dots ,X_{n})) be the unknown clean sequence and let the channel be described by a conditional density (p_{Y|X}(y|x)) that is memoryless and fully known. The observed noisy sequence is (Y^{n}). Performance is measured by a per‑symbol loss function (L(x,\hat{x})) that is non‑negative, symmetric, and satisfies a mild Lipschitz‑type regularity condition. The goal is to construct an estimator (\hat{X}^{n}) that minimizes the average expected loss \