Scale-discretised ridgelet transform on the sphere

TL;DR

This paper introduces a novel spherical ridgelet transform that is invertible for antipodal signals, supports spin signals, and is effective for analyzing white matter fibers in brain imaging.

Contribution

It presents a new spherical ridgelet transform with exact inversion, supporting spin signals and avoiding blocking artifacts, which is unique among existing methods.

Findings

Exact inversion for antipodal signals

Supports spin signals and global analysis along great circles

Effective in diffusion MRI of brain white matter

Abstract

We revisit the spherical Radon transform, also called the Funk-Radon transform, viewing it as an axisymmetric convolution on the sphere. Viewing the spherical Radon transform in this manner leads to a straightforward derivation of its spherical harmonic representation, from which we show the spherical Radon transform can be inverted exactly for signals exhibiting antipodal symmetry. We then construct a spherical ridgelet transform by composing the spherical Radon and scale-discretised wavelet transforms on the sphere. The resulting spherical ridgelet transform also admits exact inversion for antipodal signals. The restriction to antipodal signals is expected since the spherical Radon and ridgelet transforms themselves result in signals that exhibit antipodal symmetry. Our ridgelet transform is defined natively on the sphere, probes signal content globally along great circles, does not exhibit blocking artefacts, supports spin signals and exhibits an exact and explicit inverse transform. No alternative ridgelet construction on the sphere satisfies all of these properties. Our implementation of the spherical Radon and ridgelet transforms is made publicly available. Finally, we illustrate the effectiveness of spherical ridgelets for diffusion magnetic resonance imaging of white matter fibers in the brain.