Efficient Computation of Slepian Functions for Arbitrary Regions on the Sphere

📝 Abstract

In this paper, we develop a new method for the fast and memory-efficient computation of Slepian functions on the sphere. Slepian functions, which arise as the solution of the Slepian concentration problem on the sphere, have desirable properties for applications where measurements are only available within a spatially limited region on the sphere and/or a function is required to be analyzed over the spatially limited region. Slepian functions are currently not easily computed for large band-limits for an arbitrary spatial region due to high computational and large memory storage requirements. For the special case of a polar cap, the symmetry of the region enables the decomposition of the Slepian concentration problem into smaller subproblems and consequently the efficient computation of Slepian functions for large band-limits. By exploiting the efficient computation of Slepian functions for the polar cap region on the sphere, we develop a formulation, supported by a fast algorithm, for the approximate computation of Slepian functions for an arbitrary spatial region to enable the analysis of modern datasets that support large band-limits. For the proposed algorithm, we carry out accuracy analysis of the approximation, computational complexity analysis, and review of memory storage requirements. We illustrate, through numerical experiments, that the proposed method enables faster computation, and has smaller storage requirements, while allowing for sufficiently accurate computation of the Slepian functions.

💡 Analysis

In this paper, we develop a new method for the fast and memory-efficient computation of Slepian functions on the sphere. Slepian functions, which arise as the solution of the Slepian concentration problem on the sphere, have desirable properties for applications where measurements are only available within a spatially limited region on the sphere and/or a function is required to be analyzed over the spatially limited region. Slepian functions are currently not easily computed for large band-limits for an arbitrary spatial region due to high computational and large memory storage requirements. For the special case of a polar cap, the symmetry of the region enables the decomposition of the Slepian concentration problem into smaller subproblems and consequently the efficient computation of Slepian functions for large band-limits. By exploiting the efficient computation of Slepian functions for the polar cap region on the sphere, we develop a formulation, supported by a fast algorithm, for the approximate computation of Slepian functions for an arbitrary spatial region to enable the analysis of modern datasets that support large band-limits. For the proposed algorithm, we carry out accuracy analysis of the approximation, computational complexity analysis, and review of memory storage requirements. We illustrate, through numerical experiments, that the proposed method enables faster computation, and has smaller storage requirements, while allowing for sufficiently accurate computation of the Slepian functions.

📄 Content

IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 65, NO. 16, AUGUST 15, 2017 4379 Efficient Computation of Slepian Functions for Arbitrary Regions on the Sphere Alice P. Bates, Member, IEEE, Zubair Khalid, Member, IEEE, and Rodney A. Kennedy, Fellow, IEEE Abstract—In this paper, we develop a new method for the fast and memory-efficient computation of Slepian functions on the sphere. Slepian functions, which arise as the solution of the Slepian con- centration problem on the sphere, have desirable properties for applications where measurements are only available within a spa- tially limited region on the sphere and/or a function is required to be analyzed over the spatially limited region. Slepian functions are currently not easily computed for large band-limits for an arbitrary spatial region due to high computational and large memory storage requirements. For the special case of a polar cap, the symmetry of the region enables the decomposition of the Slepian concentration problem into smaller subproblems and consequently the efficient computation of Slepian functions for large band-limits. By exploit- ing the efficient computation of Slepian functions for the polar cap region on the sphere, we develop a formulation, supported by a fast algorithm, for the approximate computation of Slepian functions for an arbitrary spatial region to enable the analysis of modern datasets that support large band-limits. For the proposed algorithm, we carry out accuracy analysis of the approximation, computational complexity analysis, and review of memory storage requirements. We illustrate, through numerical experiments, that the proposed method enables faster computation, and has smaller storage requirements, while allowing for sufficiently accurate com- putation of the Slepian functions. Index Terms—Spatial-spectral concentration problem, Slepian functions, 2-sphere (unit sphere), spherical harmonics. I. INTRODUCTION S IGNALS are naturally defined on a sphere in a large num- ber of real-world applications found in various and di- verse branches of science and engineering; including medical imaging [1]–[3], cosmology [4]–[6], acoustics [7], [8], geo- physics [9], [10], planetary sciences [11], [12], wireless commu- nication [13], [14] and computer graphics [15], [16], to name a few. In these applications, signals and/or data-sets on the sphere are often analyzed in the harmonic domain which is enabled by Manuscript received August 18, 2016; revised January 10, 2017, April 2, 2017, and April 28, 2017; accepted May 27, 2017. Date of publication June 5, 2017; date of current version June 23, 2017. The associate editor coordinating the review of this manuscript and approving it for publication was Dr. Ian Clarkson. The work of Alice P. Bates was supported by the Australian Research Council’s Discovery Projects Funding Scheme (Project no. DP150101011). The work of Rodney A. Kennedy was supported by the Australian Research Council’s Discovery Projects Funding Scheme (Project no. DP170101897). (Corresponding author: Alice P. Bates.) The authors are with the Research School of Engineering, The Australian National University, Canberra ACT 0200, Australia (e-mail: alice.bates@anu.edu.au; zubair.khalid@lums.edu.pk; rodney.kennedy@anu. edu.au). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org . Digital Object Identifier 10.1109/TSP.2017.2712122 the spherical harmonic transform which serves as a well-known counterpart of the Fourier transform [17]. Spherical harmonic functions, or spherical harmonics for short, form an orthonormal basis [17] for signals on the sphere. Signals on the sphere can be reconstructed from a finite number of measurements by expan- sion in the spherical harmonic basis, provided that the spherical harmonic transform can be accurately computed which requires the samples to be taken on a grid (on the whole sphere) defined by sampling schemes [18], [19]. However, it is common for signals to be measured, recon- structed and/or analyzed within a region of the sphere in many fields including medical imaging [20], signal processing [21], [22], geological studies [9], [10], acoustics [7] and cosmo- logical studies [23], [24], to name a few. For example, the samples are unavailable (or unreliable) at the North and South pole for satellite measurements of the Earth’s magnetic or gravi- tational field [25]. Since the data-sets/measurements are defined over the spatially limited region, the use of the globally de- fined spherical harmonic basis may not be suitable for signal analysis in these applications. Alternatively, Slepian functions, which arise as the solution of the Slepian concentration prob- lem on the sphere [26]–[28] to find the band-limited functions with optimal energy concentration within a spatial region on the sphere, serve as an orthonormal basis of the space formed by band-limited functions, and therefore are well suited for signal analysis [21], [24], [29], [30] a

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