On the Construction of Invertible Filter Banks on the 2-Sphere

Abstract

The theories of signal sampling, filter banks, wavelets, and "overcomplete wavelets" are well established for the Euclidean spaces and are widely used in the processing and analysis of images. While recent advances have extended some filtering methods to spherical images, many key challenges remain. In this paper, we develop theoretical conditions for the invertibility of filter banks under continuous spherical convolution. Furthermore, we present an analogue of the Papoulis generalized sampling theorem on the 2-Sphere. We use the theoretical results to establish a general framework for the design of invertible filter banks on the sphere and demonstrate the approach with examples of self-invertible spherical wavelets and steerable pyramids. We conclude by examining the use of a self-invertible spherical steerable pyramid in a denoising experiment and discussing the computational complexity of the filtering framework.

Shape Analysis with Overcomplete Spherical Wavelets

“…While several related formulations of wavelet transforms on the sphere exist, we follow the continuous spherical filter bank and sampling framework of [16].…”

“…For a general non-axisymmetric analysis-synthesis filter bank, the relationship between the input and reconstructed image is as follows [16]:…”

“…It turns out that for axisymmetric analysis-synthesis filter banks of a finite spherical harmonic degree, latitude-longitude (lat-lon) sampling -at a sufficient rate dependent on the maximum spherical harmonic degree -guarantees that the filter bank has the same frequency response under the continuous and discrete convolution [16]. Thus an invertible continuous filter bank remains invertible under sampling and discrete convolution.…”

“…In this paper, we explore the use of overcomplete spherical wavelets [1,16] for shape analysis of closed 2D surfaces. Wavelets offer a tradeoff between entirely local and global features more commonly used in shape analysis.…”

“…Unfortunately, the bi-orthogonal wavelet transform on the sphere suffers from the same aliasing problems observed in Euclidean images. Overcomplete spherical wavelets [1,16] overcome these problems by ensuring sufficient sampling at each scale and have been shown to be both more robust and more sensitive to group differences than bi-orthogonal spherical wavelets in shape analysis [18].…”