Resolution-Adaptive Compact-Support Priors for Bayesian Wavelet Denoising

Abstract

We propose a resolution-adaptive Bayesian wavelet denoising method for noisy one-dimensional signals. Its central innovation is a spike-and-slab prior whose continuous slab mixes a compactly supported Wendland-type polynomial density with the more dispersed semicircle density. A low-dimensional empirical-Bayes trend produces data-adaptive mixture weights by resolution, while a data-adaptive support scale controls the common bounded interval. Thus, the method combines sparsity, explicit support control, and interpretable resolution-dependent shrinkage. Under squared-error loss, we derive the posterior-mean estimator and establish symmetry, boundedness, continuity, and limiting properties. We define fixed-hyperparameter bias, variance, and risk and develop an empirical-Bayes fitting procedure. Under a Laplace working likelihood, the Wendland contribution has finite-sum expressions, while the semicircle contribution is evaluated by stable one-dimensional integration. Simulations with the Bumps, Blocks, Doppler, and HeaviSine signals compare Gaussian- and Laplace-likelihood versions with universal thresholding, false-discovery-rate (FDR) thresholding, cross-validation (CV), Stein's unbiased risk estimate (SURE), the Bayesian adaptive multiresolution shrinker (BAMS), and a nonlocal-prior (NLP) method. In the primary Gaussian-error study, WS--Gaussian was the best non-NLP method in 24 of 36 cells, including 11 of 12 low-SNR cells, with a much more favorable computational profile than WS--Laplace. A real seismic acceleration record from the 2008 Chino Hills earthquake illustrates attenuation of rapid fluctuations and preservation of the dominant event. A semi-synthetic study using the processed trace as surrogate truth showed improvement over the noisy observation at lower and moderate SNRs, but not at the highest SNR.