Multiscale Optimal Filtering on the Sphere

TL;DR

This paper introduces a wavelet-based optimal filtering framework for spherical signals contaminated with anisotropic noise, improving denoising performance over existing methods like hard thresholding and weighted-SPHARM.

Contribution

It presents a novel optimal filter in the wavelet domain for spherical signals, including a simplified version for azimuthally symmetric wavelets, enhancing denoising accuracy.

Findings

The filter minimizes mean square error in wavelet domain.

It outperforms hard thresholding in denoising Earth topography maps.

The simplified filter is effective with azimuthally symmetric wavelets.

Abstract

We present a framework for the optimal filtering of spherical signals contaminated by realizations of an additive, zero-mean, uncorrelated and anisotropic noise process on the sphere. Filtering is performed in the wavelet domain given by the scale-discretized wavelet transform on the sphere. The proposed filter is optimal in the sense that it minimizes the mean square error between the filtered wavelet representation and wavelet representation of the noise-free signal. We also present a simplified formulation of the filter for the case when azimuthally symmetric wavelet functions are used. We demonstrate the use of the proposed optimal filter for denoising of an Earth topography map in the presence of additive, zero-mean, uncorrelated and white Gaussian noise, and show that the proposed filter performs better than the hard thresholding method and weighted spherical harmonic~(weighted-SPHARM) signal estimation framework.