Second-Generation Curvelets on the Sphere

Abstract

Curvelets are efficient to represent highly anisotropic signal content, such as a local linear and curvilinear structure. First-generation curvelets on the sphere, however, suffered from blocking artefacts. We present a new second-generation curvelet transform, where scale-discretised curvelets are constructed directly on the sphere. Scale-discretised curvelets exhibit a parabolic scaling relation, are well-localised in both spatial and harmonic domains, support the exact analysis and synthesis of both scalar and spin signals, and are free of blocking artefacts. We present fast algorithms to compute the exact curvelet transform, reducing computational complexity from $\mathcal{O}(L^5)$ to $\mathcal{O}(L^3{\log }_{2}L)$ for signals band-limited at $L$. The implementation of these algorithms is made publicly available. Finally, we present an illustrative application demonstrating the effectiveness of curvelets for representing directional curve-like features in natural spherical images.

10 pages, 7 figures, Code available at http://astro-informatics.github.io/s2let/

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• SILC: a new Planck Internal Linear Combination CMB temperature map using directional wavelets
• Sparse image reconstruction on the sphere: analysis and synthesis
• Wavelet-Based Techniques for the Gamma-Ray Sky
• Covariant polarized radiative transfer on cosmological scales for investigating large-scale magnetic field structures
• A new perspective on turbulent Galactic magnetic fields through comparison of linear polarisation decomposition techniques
• A Novel CMB Component Separation Method: Hierarchical Generalized Morphological Component Analysis
• Wavelet-Based Segmentation on the Sphere
• Sparse image reconstruction on the sphere: a general approach with uncertainty quantification
• Learning sparse representations on the sphere
• Directional polynomial wavelets on spheres
• Edge detection with polynomial frames on the sphere
• A Local Spectral Exterior Calculus for the Sphere and Application to the Shallow Water Equations