Generalized Sampling Expansion for Functions on the Sphere

Hagai, Ilan Ben; Fazi, Filippo Maria; Rafaely, Boaz

doi:10.1109/tsp.2012.2210549

Cited by 15 publications

(8 citation statements)

References 31 publications

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“…Spherical harmonics are also used in medical imaging [26], optical tomography [3], several applications in physics such as solving potential problem in electrostatics [16] and the central potential Schrödinger equation in quantum mechanics [9]. Additional applications of spherical harmonics are sampling on the sphere [17,5] and more recently, compressed sensing [1] and sparse recovery [22,18]. In some sense, our work relates to these latter fields.…”

Section: Introductionmentioning

confidence: 99%

Exact Recovery of Dirac Ensembles from the Projection Onto Spaces of Spherical Harmonics

2014

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In this work we consider the problem of recovering an ensemble of Diracs on the sphere from its projection onto spaces of spherical harmonics. We show that under an appropriate separation condition on the unknown locations of the Diracs, the ensemble can be recovered through Total Variation norm minimization. The proof of the uniqueness of the solution uses the method of 'dual' interpolating polynomials and is based on [8], where the theory was developed for trigonometric polynomials. We also show that in the special case of non-negative ensembles, a sparsity condition is sufficient for exact recovery.

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Section: Introductionmentioning

confidence: 99%

Exact Recovery of Dirac Ensembles from the Projection Onto Spaces of Spherical Harmonics

2014

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“…5 where is given in (III.5). Now, The Bounded Real Lemma can be directly applied in our case, since the polynomial is causal and has the same magnitude as .…”

Section: ) Definementioning

confidence: 99%

“…Spherical harmonics are also used in medical imaging [20], [51], [54], optical tomography [1], wireless channel modeling [39] and several applications in physics such as solving potential problem in electrostatics [33], and the central potential Schrodinger equation in quantum mechanics [16]. Based on spherical harmonics analysis, new sampling theorems on the sphere for band-limited signals [5], [35] and for signals with finite rate of innovation [19] were suggested, and advanced analysis methods on the sphere were applied [28], [32]. Let denote the space of homogeneous spherical harmonics of degree , which is the restriction to the unit sphere of the homogeneous harmonic polynomials of degree in [2].…”

Section: Introductionmentioning

confidence: 99%

Super-Resolution on the Sphere Using Convex Optimization

Bendory

Dekel²,

Feuer

2015

IEEE Trans. Signal Process.

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This paper considers the problem of recovering an ensemble of Diracs on a sphere from its low resolution measurements. The Diracs can be located at any location on the sphere, not necessarily on a grid. We show that under a separation condition, one can recover the ensemble with high precision by a three-stage algorithm, which consists of solving a semi-definite program, root finding and least-square fitting. The algorithm's computation time depends solely on the number of measurements, and not on the required solution accuracy. We also show that in the special case of non-negative ensembles, a sparsity condition is sufficient for recovery. Furthermore, in the discrete setting, we estimate the recovery error in the presence of noise as a function of the noise level and the super-resolution factor.

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“…All above-mentioned sampling theorems associated with the LCT are considered as uniform sampling schemes [21][22][23][24][25][26]. However, in some practical engineering applications, more will be encountered in other ways such as non-uniform sampling [32][33][34][35][39][40][41] or multichannel sampling [34][35][36][37][38][42][43][44][45][46][47][48]. In many practical applications, periodic non-uniform sampling and multichannel sampling schemes for band-limited signal in the LCT domain are frequently introduced [32][33][34][35][36][37][38], such as flexible interleaving/ multiplexing analogue-to-digital (A/D) converter for band-limited signal in the LCT domain [35], the orthogonal frequency division multiplexing system based on the LCT for time-frequencyselective channels or multi-input-multi-output (MIMO) systems [8,9], the application in the context of the image super-resolution [37,38] or digital flight control [36].…”

Section: Introductionmentioning

confidence: 99%

Filterbank reconstruction of band‐limited signals from multichannel samples associated with the LCT

Wei

2017

IET signal process.

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The linear canonical transform (LCT) has been shown to be a powerful tool for optics and signal processing. This study addresses the problem of filterbank implementation for multichannel sampling in the LCT domain. First, the interpolation and sampling identities in the LCT domain are derived by the properties of LCT. The interpolation identity is the key result of the current study, which establishes the equivalence of two signal processing operations. One of these uses continuous-timedomain filters, whereas the other uses discrete processing. Then, applying the interpolation identity, the particularly efficient filterbank implementation for derivative sampling, periodic non-uniform sampling and multichannel sampling are obtained in the LCT domain. Moreover, the authors present filterbank implementation using polyphase structures in LCT domain. Furthermore, the simulations are carried out to verify the effectiveness of the results and the potential applications of the multichannel sampling are also presented. Finally, combining the sampling and interpolation identity, they explore the relationship between the multichannel sampling and the filterbank in the LCT domain.

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Generalized Sampling Expansion for Functions on the Sphere

Cited by 15 publications

References 31 publications

Exact Recovery of Dirac Ensembles from the Projection Onto Spaces of Spherical Harmonics

Exact Recovery of Dirac Ensembles from the Projection Onto Spaces of Spherical Harmonics

Super-Resolution on the Sphere Using Convex Optimization

Filterbank reconstruction of band‐limited signals from multichannel samples associated with the LCT

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