An Iterative Approach to Two-Dimensional Laplacian Smoothing with Application to Image Restoration

Abstract

Smoothing methods are increasingly important in modern graphically oriented data analysis. There are several fast and reliable univariate cross-validated smoothing algorithms, but comparable methods are not available in higher dimensions. This article develops an iterative approach to two-dimensional Laplacian spline smoothing based on tensor-product cubic B splines. Estimating equations are developed and solved iteratively using two-line symmetric successive over relaxation with conjugate gradient acceleration. An asymptotic approximation for the generalized cross-validation score is described. The resulting algorithm is naturally suited to two-dimensional scatterplot smoothing but has potential value for image restoration as well. This is illustrated on a multicolor image example. Along with an image restoration an uncertainty assessment based on the entropy of the marginal posterior pixel value distribution is produced. This analysis particularly highlights the statistical uncertainty in resolving object boundaries in the image.

Key Words:

• A- and B-mode imaging
• Bicubic B splines
• Conjugate gradient acceleration
• Cross-validation
• Laplacian smoothing
• Leverage
• Posterior pixel entropy
• Symmetric successive overrelaxation