[Paper Review] Perfect simulation for Bayesian wavelet thresholding with correlated coefficients

Graeme K. Ambler, Bernard W. Silverman|ArXiv.org|Mar 15, 2009

Image and Signal Denoising Methods17 references3 citations

TL;DR

This paper proposes a Bayesian wavelet thresholding method that models correlated wavelet coefficients using an area-interaction point process prior, enabling exact posterior sampling via Coupling From The Past (CFTP). The approach outperforms standard methods like SureShrink and BayesThresh in mean-square error at moderate to high signal-to-noise ratios, especially for signals with clustered coefficients.

ABSTRACT

We introduce a new method of Bayesian wavelet shrinkage for reconstructing a signal when we observe a noisy version. Rather than making the common assumption that the wavelet coefficients of the signal are independent, we allow for the possibility that they are locally correlated in both location (time) and scale (frequency). This leads us to a prior structure which is analytically intractable, but it is possible to draw independent samples from a close approximation to the posterior distribution by an approach based on Coupling From The Past.

Motivation & Objective

• To address the limitation of independent coefficient assumptions in Bayesian wavelet thresholding, which fails to capture local clustering in true signals.
• To model wavelet coefficient dependencies using a spatial point process prior that reflects local correlation in time and scale.
• To develop a simulation-based inference method that guarantees exact samples from the posterior distribution, avoiding MCMC convergence issues.
• To evaluate the performance of the proposed method against established wavelet thresholding techniques on standard test signals.

Proposed method

• Uses an area-interaction point process as a prior for wavelet coefficients to induce local clustering, where nonzero coefficients increase the probability of neighboring coefficients being nonzero.
• Models the wavelet coefficients as a mixture of a point mass at zero and a normal distribution, with variance dependent on scale, and extends this with a spatial dependence structure.
• Applies Coupling From The Past (CFTP) to generate exact samples from the posterior distribution, overcoming the convergence uncertainty inherent in standard MCMC methods.
• Implements the CFTP algorithm with a block Gibbs sampler and monotonic coupling to ensure coalescence to a unique stationary state.
• Employs a likelihood model where observed noisy coefficients are conditionally normal given the true coefficients, with known variance.
• Uses a Metropolis-Hastings step within the CFTP framework to update coefficient states while preserving detailed balance and ensuring exact sampling.

Experimental results

Research questions

• RQ1 Can modeling local correlation between wavelet coefficients improve signal reconstruction accuracy in noisy settings?
• RQ2 Does using a spatial point process prior that encourages coefficient clustering lead to better performance than independent coefficient assumptions?
• RQ3 Can Coupling From The Past (CFTP) be effectively adapted to generate exact posterior samples in a high-dimensional, non-iid wavelet thresholding model?
• RQ4 How does the proposed method compare in mean-square error to established methods like SureShrink, cross-validation, BayesThresh, and FDR under varying signal-to-noise ratios?
• RQ5 To what extent does the performance of the method depend on the choice of hyperparameters such as γ, λ, τ, and σ?

Key findings

• The proposed area-interaction BayesThresh (AIBT) method achieved the lowest mean-square error (AMSE) among all methods at signal-to-noise ratios of 25 and 10, particularly excelling on the Bumps and Heavisine signals.
• At a signal-to-noise ratio of 25, AIBT achieved an AMSE of 84 (1) for the Bumps signal, compared to 131 (6) for SureShrink and 1651 (17) for ordinary BayesThresh.
• For the Heavisine signal at SNR 25, AIBT achieved 32 (1) AMSE, outperforming SureShrink (66 (2)) and FDR (64 (3)).
• At lower signal-to-noise ratios (e.g., SNR 3), AIBT remained competitive, with an AMSE of 153 (6) on Heavisine, compared to 148 (3) for FDR and 140 (4) for BayesThresh.
• The method showed particular strength in preserving signal structure in clustered coefficient regions, such as the blocky features in the Blocks signal and the sharp transitions in Heavisine.
• The results demonstrate that incorporating coefficient correlation via the area-interaction prior significantly improves reconstruction accuracy, especially when the true signal has sparse, clustered structure.