Wavelets for $L^2(B(0,1))$ using Zernike polynomials

TL;DR

This paper develops wavelets based on Zernike polynomials for analyzing two-dimensional signals on circular domains, enabling multiresolution analysis of functions in $L^2(B(0,1))$ with applications to optical and corneal data.

Contribution

It introduces a novel wavelet construction on the unit disk using Zernike polynomials, extending previous one-dimensional methods to two-dimensional circular domains.

Findings

Wavelet basis effectively analyzes 2D signals on circular domains.

Application demonstrated on corneal data with promising results.

Provides a multiresolution framework for $L^2(B(0,1))$ functions.

Abstract

A set of orthogonal polynomials on the unit disk $B(0,1)$ known as Zernike polynomials are commonly used in the analysis and evaluation of optical systems. Here Zernike polynomials are used to construct wavelets for polynomial subspaces of $L^2(B(0,1)).$ This naturally leads to a multiresolution analysis of $L^2(B(0,1)).$ Previously, other authors have dealt with the one dimensional case, and used orthogonal polynomials of a single variable to construct time localized bases for polynomial subspaces of an $L^2$-space with arbitrary weight. Due to the nature of Zernike polynomials, the wavelet construction given here is well-suited for the analysis of two-dimensional signals defined on circular domains. This is shown by some experimental results done on corneal data.