Fast Computation of Orthogonal Polar Harmonic Transforms

International audienceThis paper presents a method for the computation of polar harmonic transforms that is fast and efficient. The method is based on the inherent recurrence relations among harmonic functions that are used in the definitions of the radial and angular kernels of the transforms. The employment of these relations leads to recursive strategies for fast computation of harmonic function-based kernels. Polar harmonic transforms were recently proposed and have shown nice properties for image representation and pattern recognition. The proposed method is 10-time faster than direct computation and five-time faster than fast computation of Zernike moments

Fast generic polar harmonic transforms

International audienceGeneric polar harmonic transforms have recently been proposed to extract rotation-invariant features from images and their usefulness has been demonstrated in a number of pattern recognition problems. However, direct computation of these transforms from their definition is inefficient and is usually slower than some efficient computation strategies that have been proposed for other methods. This paper presents a number of novel computation strategies to compute these transforms rapidly. The proposed methods are based on the inherent recurrence relations among complex exponential and trigonometric functions used in the definition of the radial and angular kernels of these transforms. The employment of these relations leads to recursive and addition chain-based strategies for fast computation of harmonic function-based kernels. Experimental results show that the proposed method is about 10× faster than direct computation and 5× faster than fast computation of Zernike moments using the q-recursive strategy. Thus, among all existing rotation-invariant feature extraction methods, polar harmonic transforms are the fastest

Efficient Spherical Harmonic Transforms aimed at pseudo-spectral numerical simulations

In this paper, we report on very efficient algorithms for the spherical harmonic transform (SHT). Explicitly vectorized variations of the algorithm based on the Gauss-Legendre quadrature are discussed and implemented in the SHTns library which includes scalar and vector transforms. The main breakthrough is to achieve very efficient on-the-fly computations of the Legendre associated functions, even for very high resolutions, by taking advantage of the specific properties of the SHT and the advanced capabilities of current and future computers. This allows us to simultaneously and significantly reduce memory usage and computation time of the SHT. We measure the performance and accuracy of our algorithms. Even though the complexity of the algorithms implemented in SHTns are in $O(N^3)$ (where N is the maximum harmonic degree of the transform), they perform much better than any third party implementation, including lower complexity algorithms, even for truncations as high as N=1023. SHTns is available at https://bitbucket.org/nschaeff/shtns as open source software.Comment: 8 page

Fast directional continuous spherical wavelet transform algorithms

We describe the construction of a spherical wavelet analysis through the inverse stereographic projection of the Euclidean planar wavelet framework, introduced originally by Antoine and Vandergheynst and developed further by Wiaux et al. Fast algorithms for performing the directional continuous wavelet analysis on the unit sphere are presented. The fast directional algorithm, based on the fast spherical convolution algorithm developed by Wandelt and Gorski, provides a saving of O(sqrt(Npix)) over a direct quadrature implementation for Npix pixels on the sphere, and allows one to perform a directional spherical wavelet analysis of a 10^6 pixel map on a personal computer.Comment: 10 pages, 3 figures, replaced to match version accepted by IEEE Trans. Sig. Pro

A Fast and Accurate Algorithm for Spherical Harmonic Analysis on HEALPix Grids with Applications to the Cosmic Microwave Background Radiation

The Hierarchical Equal Area isoLatitude Pixelation (HEALPix) scheme is used extensively in astrophysics for data collection and analysis on the sphere. The scheme was originally designed for studying the Cosmic Microwave Background (CMB) radiation, which represents the first light to travel during the early stages of the universe's development and gives the strongest evidence for the Big Bang theory to date. Refined analysis of the CMB angular power spectrum can lead to revolutionary developments in understanding the nature of dark matter and dark energy. In this paper, we present a new method for performing spherical harmonic analysis for HEALPix data, which is a central component to computing and analyzing the angular power spectrum of the massive CMB data sets. The method uses a novel combination of a non-uniform fast Fourier transform, the double Fourier sphere method, and Slevinsky's fast spherical harmonic transform (Slevinsky, 2019). For a HEALPix grid with $N$ pixels (points), the computational complexity of the method is $\mathcal{O}(N\log^2 N)$, with an initial set-up cost of $\mathcal{O}(N^{3/2}\log N)$. This compares favorably with $\mathcal{O}(N^{3/2})$ runtime complexity of the current methods available in the HEALPix software when multiple maps need to be analyzed at the same time. Using numerical experiments, we demonstrate that the new method also appears to provide better accuracy over the entire angular power spectrum of synthetic data when compared to the current methods, with a convergence rate at least two times higher

Complex data processing: fast wavelet analysis on the sphere

In the general context of complex data processing, this paper reviews a recent practical approach to the continuous wavelet formalism on the sphere. This formalism notably yields a correspondence principle which relates wavelets on the plane and on the sphere. Two fast algorithms are also presented for the analysis of signals on the sphere with steerable wavelets.Comment: 20 pages, 5 figures, JFAA style, paper invited to J. Fourier Anal. and Appli

A novel sampling theorem on the rotation group

We develop a novel sampling theorem for functions defined on the three-dimensional rotation group SO(3) by connecting the rotation group to the three-torus through a periodic extension. Our sampling theorem requires $4L^3$ samples to capture all of the information content of a signal band-limited at $L$, reducing the number of required samples by a factor of two compared to other equiangular sampling theorems. We present fast algorithms to compute the associated Fourier transform on the rotation group, the so-called Wigner transform, which scale as $O(L^4)$, compared to the naive scaling of $O(L^6)$. For the common case of a low directional band-limit $N$, complexity is reduced to $O(N L^3)$. Our fast algorithms will be of direct use in speeding up the computation of directional wavelet transforms on the sphere. We make our SO3 code implementing these algorithms publicly available.Comment: 5 pages, 2 figures, minor changes to match version accepted for publication. Code available at http://www.sothree.or