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

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 $α\geq0$ at the rim of their supporting ball $ρ\leq1$. 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 $α=0$, when the smoothness parameter $α$ 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.