Towards theoretically-founded learning-based denoising

摘要

Denoising a stationary process (Xi)i∈ℤ corrupted by additive white Gaussian noise (Zi)i∈ℤ, i.e., recovering Xⁿ from Yⁿ = Xⁿ + Zⁿ, is a classic and fundamental problem in information theory and statistical signal processing. Theoretically-founded and computationally-efficient denoising algorithms which are applicable to general sources are yet to be found. In a Bayesian setup, given the distribution of Xⁿ, a minimum mean square error (MMSE) denoiser computes E[Xⁿ|Yⁿ]. However, for general sources, computing E[Xⁿ|Yⁿ] is computationally very challenging, if not infeasible. In this paper, starting from a Bayesian setup, a novel denoiser, namely, quantized maximum a posteriori (Q-MAP) denoiser, is proposed and its asymptotic performance is analyzed. Both for memoryless sources, and for structured first-order Markov sources, it is shown that, asymptotically, as σ² (noise variance) converges to zero, $rac{1}{{{sigma ^2}}}{ext{E}}left[ {{{left( {{X_i} - hat X_i^{{ext{Q - MAP}}}} ight)}^2}} ight]$ converges to the information dimension of the source. For the studied memoryless sources, this limit is known to be optimal. A key advantage of the Q-MAP denoiser is that, unlike a MMSE denoiser, it highlights the key properties of the source distribution that are to be used in its denoising. This naturally leads to a learning-based denoising algorithm. Using ImageNet database for training, initial simulation results exploring the performance of such a learning-based denoiser in image denoising are presented.