Self Consistency: A General Recipe for Wavelet Estimation With Irregularly-spaced and/or Incomplete Data

TL;DR

This paper introduces a versatile wavelet estimation method for irregularly-spaced or incomplete data, leveraging the self-consistency principle and EM-inspired algorithms, applicable to various noise types and data scenarios.

Contribution

It presents a novel, general framework for wavelet estimation with incomplete data, compatible with multiple shrinkage methods and noise models, supported by new algorithms.

Findings

Algorithms outperform common methods in numerical experiments

Effective with non-Gaussian and correlated noise

Applicable to Poisson data smoothing and image denoising

Abstract

Inspired by the key principle behind the EM algorithm, we propose a general methodology for conducting wavelet estimation with irregularly-spaced data by viewing the data as the observed portion of an augmented regularly-spaced data set. We then invoke the self-consistency principle to define our wavelet estimators in the presence of incomplete data. Major advantages of this approach include: (i) it can be coupled with almost any wavelet shrinkage methods, (ii) it can deal with non--Gaussian or correlated noise, and (iii) it can automatically handle other kinds of missing or incomplete observations. We also develop a multiple-imputation algorithm and fast EM-type algorithms for computing or approximating such estimates. Results from numerical experiments suggest that our algorithms produce favorite results when comparing to several common methods, and therefore we hope these empirical findings would motivate subsequent theoretical investigations. To illustrate the flexibility of our approach, examples with Poisson data smoothing and image denoising are also provided.