Maximal multiplicative spatial-spectral concentration on the sphere: Optimal basis

Khalid, Zubair; Kennedy, Rodney A.

doi:10.1109/icassp.2015.7178754

Cited by 3 publications

(7 citation statements)

References 15 publications

(49 reference statements)

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“…For signals on the sphere, the Slepian concentration problem [37]- [40], to find the band-limited (or space-limited) functions with optimal energy concentration in the spatial (or spectral) domain, has been extensively investigated [17], [25], [27], [28], [30], [41]. In order to maximize the spatial concentration of a band-limited signal h ∈ H L within the spatial region R ⊂ S 2 , we seek to maximize the spatial concentration (energy) ratio λ given by [28]…”

Section: B Slepian Functions On the Spherementioning

confidence: 99%

Efficient Computation of Slepian Functions for Arbitrary Regions on the Sphere

Bates¹,

Khalid²,

Kennedy³

2017

IEEE Trans. Signal Process.

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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.

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Section: B Slepian Functions On the Spherementioning

confidence: 99%

Efficient Computation of Slepian Functions for Arbitrary Regions on the Sphere

Bates¹,

Khalid²,

Kennedy³

2017

IEEE Trans. Signal Process.

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“…The function g u is a unit energy function. The space-limited and band-limited eigenfunctions satisfy the following orthogonality relations [15]…”

Section: A Design Of Optimal Basis Functionsmentioning

confidence: 99%

“…From (14) it can be seen that the functions g u,1 and g v,2 for u, v ∈ [1, P L 2 ], u = v are orthonormal (unit energy constraint on g u,j ) for any values of α u,1 , α v,2 , β u,1 and β v,2 . Also from (15), it can be easily shown that g u,1 and g u,2 become orthonormal for each u ∈ [1, P L 2 ]. The joint subspace has dimension 2P L 2 (each of the subspaces H R and H P L is of dimension P L 2 ), therefore, the 2P L 2 number of orthonormal functions g u,j ∈ H P L + H R for u ∈ [1, P L 2 ], and j = 1, 2 completely span the joint subspace H P L + H R .…”

Section: A Design Of Optimal Basis Functionsmentioning

confidence: 99%

“…Using the method described in [15] and employing the orthonormality of Fourier-Laguerre basis functions, it can be shown that…”

Section: B Integral Operator Formulationmentioning

confidence: 99%

“…To support these applications, we consider the problem of maximizing the product of concentration of energy of signals defined on the ball. The problem of finding optimal basis with maximum energy concentration in spatial and harmonic domains, has been considered for Euclidean [14] and spherical (unit sphere) [15] domains.…”

Section: Introductionmentioning

confidence: 99%

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Signal analysis on the ball: Design of optimal basis functions with maximal multiplicative concentration in spatial and spectral domains

Nafees

Khalid

Kennedy

2017

2017 International Conference on Systems, Signals and Image Processing (IWSSIP)

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In this work, we design a set of complete orthonormal optimal basis functions for signals defined on the ball. We design the basis functions by maximizing the product of their energy concentration in some spatial region and that in some spectral region. The optimal basis functions are designed as a linear combination of space-limited functions with maximal concentration in the spectral region and band-limited functions with maximal concentration in the spatial region. The proposed optimal basis functions are shown to form a complete set for signal representation in a subspace formed by the vector sum of the subspaces spanned by space-limited and band-limited functions. We also formulate an integral operator which projects the signal to the joint subspace and maximizes the product of energy concentrations in harmonic as well as spatial domains. Furthermore, with the help of some properties of proposed optimal basis functions we show that these functions are the only eigenfunction of the integral operator.

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Achievable simultaneous time and frequency domain energy concentration for finite length sequences

Singh

Pradhan

2019

IET signal process.

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For numerous applications in the field of signal processing, it is desired to design a compact window that can simultaneously concentrate maximum energy in finite time interval and frequency band. Although zero-order discrete prolate spheroidal sequence (DPSS) meets this requirement, it is of infinite support. Limiting this sequence to finite support no longer guarantees the optimality property. This study aims at designing a discrete finite length window that can maximise the energy simultaneously in narrow time interval and frequency band. A multi-objective optimisation approach is adopted to obtain the upper bound of maximum achievable time and frequency domain energy concentrations for finite length sequences. The optimal sequence thus obtained is termed as the optimal window with finite support (OWFS), and its various associated properties are discussed. It is shown analytically that as the support of OWFS approaches infinity, it converges to zero-order DPSS. In order to illustrate its optimality, the compactness of the proposed OWFS is compared with those of various window functions. Extending the proposed OWFS, this study also discusses the formulation and associated properties of the zero-order periodic DPSS. Further, the closed form expression for the upper bound of achievable time and frequency domain energy concentrations is also derived.

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Maximal multiplicative spatial-spectral concentration on the sphere: Optimal basis

Cited by 3 publications

References 15 publications

Efficient Computation of Slepian Functions for Arbitrary Regions on the Sphere

Efficient Computation of Slepian Functions for Arbitrary Regions on the Sphere

Signal analysis on the ball: Design of optimal basis functions with maximal multiplicative concentration in spatial and spectral domains

Achievable simultaneous time and frequency domain energy concentration for finite length sequences

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