A novel sampling theorem on the sphere (1110.6298v1)

Abstract: 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.

• The paper introduces a novel sampling theorem and fast algorithms for analyzing band-limited signals on spherical domains by innovatively linking spherical and toroidal geometries.
• The theorem significantly reduces the number of samples required, roughly half compared to existing methods, and provides computationally efficient algorithms optimized with FFTs.
• Key applications include processing CMB data in cosmology, reducing acquisition time in diffusion MRI for neuroscience, and enhancing compressive sensing techniques on the sphere.

A Novel Sampling Theorem on the Sphere

The paper "A Novel Sampling Theorem on the Sphere" by Jason D. McEwen and Yves Wiaux introduces a new sampling theorem, accompanied by fast algorithms, for analyzing band-limited signals on spherical domains. This work is distinguished by its innovative association between spherical and toroidal geometries, allowing for periodic extensions to be utilized effectively in signal processing. The novelty of this approach lies in its reduced computational demands and applicability to both scalar and spin functions on the sphere.

Key Contributions

1. Sample Efficiency: The sampling theorem reduces the number of samples needed to exactly represent a band-limited signal on the sphere, requiring approximately half the number of samples compared to existing equiangular methods. Specifically, the theorem requires $O(2L^2)$ samples rather than $O(4L^2)$ from Driscoll-Healy type theorems, and effectively fewer samples than Gauss-Legendre-based approaches for asymptotic conditions.
2. Computational Complexity: The derived algorithms exhibit computational complexity scaling as $O(L^3)$, consistent with separable algorithms used for spherical harmonics transforms. However, the authors have optimized this complexity by leveraging Fast Fourier Transforms (FFTs), reducing the overhead associated with the asymptotic evaluation constants significantly.
3. Numerical Stability and Implementation: Numerical experiments confirm the stability and accuracy of these algorithms for high-resolution datasets, up to $L=4096$. The implementation does not require precomputation, thus avoiding infeasible storage demands typical in other methodologies.

Implications and Applications

The implications of this sampling theorem are notable across multiple domains:

• Cosmology: In the analysis of Cosmic Microwave Background (CMB) data, characterized by high-band limits, this sampling theorem provides exact harmonic transforms reliably, facilitating the processing of datasets comprising millions of samples with enhanced speed and decreased computational load.
• Neuroscience: For diffusion MRI, where spherical sampling is prevalent, the reduction in required samples can significantly decrease acquisition times, thus improving the feasibility of the process in clinical settings.
• Compressive Sensing (CS): The dimensionality reduction benefits inherent to this sampling theorem can enhance CS techniques on the sphere, which are utilized in data acquisition, noise reduction, and signal reconstruction tasks.

Speculations on Future Developments

This research paves the way for further innovations in harmonic analysis methodologies on spherical domains. Future advancements might focus on integrating compressive sensing more deeply, exploiting reduced sampling requirements for increasingly sparse datasets. There is also potential to explore applications in other scientific and engineering domains where spherical data analysis is requisite.

In conclusion, the paper represents a significant step in spherical harmonic analysis, particularly promising for applications requiring efficient, accurate processing of high-resolution spherical data sets. The potential for broader adoption and adaptation of these algorithms in diverse fields remains high, contingent on their scalability and integration with existing computational frameworks.