Denoising of structured random processes
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Denoising stationary process (X_i)_i ∈ Z corrupted by additive white Gaussian noise is a classic and fundamental problem in information theory and statistical signal processing. Despite considerable progress in designing efficient denoising algorithms, for general analog sources, theoretically-founded computationally-efficient methods are yet to be found. For instance in denoising X^n corrupted by noise Z^n as Y^n=X^n+Z^n, given the full distribution of X^n, a minimum mean square error (MMSE) denoiser needs to compute E[X^n|Y^n]. However, for general sources, computing E[X^n|Y^n] is computationally very challenging, if not infeasible. In this paper, starting by a Bayesian setup, where the source distribution is fully known, a novel denoising method, namely, quantized maximum a posteriori (Q-MAP) denoiser, is proposed and its asymptotic performance in the high signal to noise ratio regime is analyzed. Both for memoryless sources, and for structured first-order Markov sources, it is shown that, asymptotically, as σ converges to zero, 1σ^2E[(X_i-X̂^ Q-MAP_i)^2] achieved by Q-MAP denoiser converges to the information dimension of the source, which, at least for the studied memoryless sources, is known to be the optimal. One key advantage of the Q-MAP denoiser is that, while it is designed for a Bayesian setup, it naturally leads to a learning-based denoising method, which learns the source structure from training data.
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