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## Tuesday, November 01, 2011

### Spin Spherical Harmonic Transforms (SSHT v 1.0)

Jason McEwen and Yves Wiaux just made available the first version of the Spin Spherical Harmonic toolbox, From the website:

## Introduction

The SSHT code provides functionality to perform fast and exact spin spherical harmonic transforms based on the sampling theorem on the sphere derived in our paper: A novel sampling theorem on the sphere (ArXiv | DOI). In some applications, adjoint forward and inverse spherical harmonic transforms are also required (for example, when solving convex optimisation problems). We provide functionality to perform fast and exact adjoint transforms, based on the fast algorithms derived in our paper: Efficient and compressive sampling on the sphere.
This documentation outlines the various harmonic transforms supported in SSHT, before describing installation details and documenting the C, Fortran and Matlab source code. Reference, version, and license information then follows.

The attendant paper is:A novel sampling theorem on the sphere by Jason McEwen and Yves Wiaux . The abstract reads:
We develop a novel sampling theorem on the sphere and corresponding fast algorithms by associating the sphere with the torus through a periodic extension. The fundamental property of any sampling theorem is the number of samples required to represent a band-limited signal. To represent exactly a signal on the sphere band-limited at L, all sampling theorems on the sphere require O(L^2) samples. However, our sampling theorem requires less than half the number of samples of other equiangular sampling theorems on the sphere and an asymptotically identical, but smaller, number of samples than the Gauss-Legendre sampling theorem. The complexity of our algorithms scale as O(L^3), however, the continual use of fast Fourier transforms reduces the constant prefactor associated with the asymptotic scaling considerably, resulting in algorithms that are fast. Furthermore, we do not require any precomputation and our algorithms apply to both scalar and spin functions on the sphere without any change in computational complexity or computation time. We make our implementation of these algorithms available publicly and perform numerical experiments demonstrating their speed and accuracy up to very high band-limits. Finally, we highlight the advantages of our sampling theorem in the context of potential applications, notably in the field of compressive sampling.

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