Generalized 3D Zernike functions for analytic construction of band-limited line-detecting wavelets

TL;DR

This paper develops generalized 3D Zernike functions that vanish at the boundary, enabling analytic construction of band-limited line-detecting wavelets for 3D data analysis, with improved decay and oscillation properties.

Contribution

It introduces a new class of 3D Zernike functions with prescribed boundary vanishing, facilitating analytic wavelet construction for 3D medical image analysis.

Findings

Fourier transform of generalized Zernike functions shows faster decay.

Constructed prewavelets enable analytic Fourier and Funk transforms.

Applications include line detection in 3D medical imaging.

Abstract

We consider 3D versions of the Zernike polynomials that are commonly used in 2D in optics and lithography. We generalize the 3D Zernike polynomials to functions that vanish to a prescribed degree $\alpha\geq0$ at the rim of their supporting ball $\rho\leq1$. The analytic theory of the 3D generalized Zernike functions is developed, with attention for computational results for their Fourier transform, Funk and Radon transform, and scaling operations. The Fourier transform of generalized 3D Zernike functions shows less oscillatory behaviour and more rapid decay at infinity, compared to the standard case $\alpha=0$, when the smoothness parameter $\alpha$ is increased beyond 0. The 3D generalized Zernike functions can be used to expand smooth functions, supported by the unit ball and vanishing at the rim and the origin of the unit ball, whose radial and angular dependence is separated. Particular instances of the latter functions (prewavelets) yield, via the Funk transform and the Fourier transform, an anisotropic function that can be used for a band-limited line-detecting wavelet transform, appropriate for analysis of 3D medical data containing elongated structures. We present instances of prewavelets, with relevant radial functions, that allow analytic computation of Funk and Fourier transform. A key step here is to identify the special form that is assumed by the expansion coefficients of a separable function on the unit ball with respect to generalized 3D Zernike functions. A further issue is how to scale a function on the unit ball while maintaining its supporting set, and this issue is solved in a particular form.