  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.
