SCOPE Shrinkage: A Unified Framework for Wavelet Denoising

2026-06-17

AI Summary

This study addresses the challenge of balancing signal structure preservation and effective noise suppression in wavelet denoising by proposing the SCOPE shrinkage framework. SCOPE introduces a sign-preserving shrinkage rule constructed from the central cumulative distribution function of symmetric unimodal distributions, governed by two interpretable parameters that independently control the threshold location and transition sharpness. This formulation yields a unified family of shrinkage operators that simultaneously offers theoretical interpretability and structural flexibility, with dual interpretations grounded in both Bayesian inference and penalized likelihood theory. Data-driven parameter selection is achieved via Stein’s unbiased risk estimate. Oracle experiments on the Donoho–Johnstone test functions demonstrate that SCOPE achieves denoising performance comparable to state-of-the-art methods while substantially enhancing interpretability and design adaptability.

Abstract

We introduce Symmetric CDF Oriented Probability Enhanced (SCOPE) shrinkage, a unified family of sign-preserving shrinkage rules constructed from centered cumulative distribution functions of symmetric unimodal distributions. The proposed framework generates a broad class of attenuation profiles that interpolate between strong local shrinkage near zero and asymptotically unbiased behavior in the tails. A general formulation is developed that separates scale and shape effects through two interpretable parameters, allowing effective threshold location and transition sharpness to be controlled independently. Under explicit regularity assumptions, structural properties of SCOPE shrinkage are established, including oddness, monotonicity, continuity, contractivity, and a mixture representation that connects the rules to softened thresholding operators. A Bayesian and penalized likelihood interpretation is also developed: SCOPE rules admit even penalty representations that are nondecreasing in coefficient magnitude, and suitable subclasses arise as exact maximum a posteriori estimators under proper symmetric unimodal priors. Representative examples based on logistic, uniform, and Cauchy distributions illustrate how probabilistic shape governs shrinkage behavior. Data driven parameter selection for smooth subclasses is discussed via Stein-type unbiased risk estimation. Oracle calibrated simulation studies on standard Donoho-Johnstone test functions show that SCOPE shrinkage performs competitively with several established wavelet denoising methods, while retaining a high degree of interpretability and structural flexibility. The results highlight centered distribution functions as a natural and versatile design principle for shrinkage in wavelet denoising and related estimation problems.

Problem

• wavelet denoising
• shrinkage
• thresholding
• symmetric unimodal distributions
• bias-variance tradeoff

Innovation

• wavelet denoising
• shrinkage rule
• symmetric unimodal distribution
• cumulative distribution function
• penalized likelihood