Convolution on the $n$-Sphere With Application to PDF Modeling

Petrinovic, Davor

doi:10.1109/tsp.2009.2033329

Cited by 17 publications

(22 citation statements)

References 46 publications

(53 reference statements)

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“…Let

k_{1} MathClass-punc, k_{2} MathClass-rel\in L^{2} ({double-struckS}^{2})

and k = k 1 ⋆ k 2 . Then the Fourier coefficients of k are given by,

\begin{matrix} leftalign \\ rightalign-odd k ̃ MathClass-open(l, m MathClass-close) & align-even = 2 π \sqrt{\frac{4 π}{2 l + 1}} {k ̃}_{1} MathClass-open(l, m MathClass-close) {k ̃}_{2} MathClass-open(l, 0 MathClass-close) & rightalign-label (5) \end{matrix}

For a formal statement and proof of this theorem on

{double-struckS}^{2}

, see Driscoll and Healy () or see Dokmanić and Pertinović () for a general proof on

{double-struckS}^{d}

. This is an adaptation of the convolution theorem on

double-struckR

to convolutions of functions defined on

{double-struckS}^{2}

.…”

Section: Review Of Spherical Harmonicsmentioning

confidence: 99%

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Constructing valid spatial processes on the sphere using kernel convolutions

et al. 2014

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Remotely sensed data products are now routinely used to study various aspects of the Earth's atmosphere. These remote sensing datasets are typically very high dimensional, have near global coverage and exhibit nonstationary spatial correlation structures. Proper statistical analysis of these datasets should be sufficiently flexible to account for all these aspects. To this end, we develop a kernel convolution construction of spatial processes on a sphere. As is the case with kernel convolution constructions on the plane, we establish a link between stationary kernels and a stationary covariance function on the sphere via the spherical harmonic decomposition of the kernel. We also introduce the Kent distribution as an appropriate kernel with interpretable parameters to be used in the kernel convolution construction. We demonstrate the discrete kernel convolution model using a dataset of remotely sensed CO 2 concentrations over the globe.

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“…Let

k_{1} MathClass-punc, k_{2} MathClass-rel\in L^{2} ({double-struckS}^{2})

and k = k 1 ⋆ k 2 . Then the Fourier coefficients of k are given by,

\begin{matrix} leftalign \\ rightalign-odd k ̃ MathClass-open(l, m MathClass-close) & align-even = 2 π \sqrt{\frac{4 π}{2 l + 1}} {k ̃}_{1} MathClass-open(l, m MathClass-close) {k ̃}_{2} MathClass-open(l, 0 MathClass-close) & rightalign-label (5) \end{matrix}

For a formal statement and proof of this theorem on

{double-struckS}^{2}

, see Driscoll and Healy () or see Dokmanić and Pertinović () for a general proof on

{double-struckS}^{d}

. This is an adaptation of the convolution theorem on

double-struckR

to convolutions of functions defined on

{double-struckS}^{2}

.…”

Section: Review Of Spherical Harmonicsmentioning

confidence: 99%

“…For a formal statement and proof of this theorem on S 2 , see Driscoll and Healy (1994) or see Dokmanić and Pertinović (2010) for a general proof on S d . This is an adaptation of the convolution theorem on R to convolutions of functions defined on S 2 .…”

Section: Review Of Spherical Harmonicsmentioning

confidence: 99%

Constructing valid spatial processes on the sphere using kernel convolutions

et al. 2014

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“…Using the usual hyperspherical angular coordinates (see for example [5], [10]) we may write the hyperspherical harmonics on S n as [10], [11],…”

Section: N-spherementioning

confidence: 99%

“…Various applications require scaling (dilating or contracting) of the function on a sphere and if the function is represented by its Fourier coefficients, it would be beneficial to perform the scaling directly in the Fourier domain. See for example spherical wavelets, [1], [2], spherical filter banks, [3], illumination in computer graphics [4] or spherical point density estimation, [5], [6]. Spectral computation is further facilitated by the development of fast spherical transform algorithms [7], [8] analogous to the Euclidean fast Fourier transform.…”

Section: Introductionmentioning

confidence: 99%

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Efficient Approximate Scaling of Spherical Functions in the Fourier Domain With Generalization to Hyperspheres

Petrinovic

2010

IEEE Trans. Signal Process.

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Abstract-We propose a simple model for approximate scaling of spherical functions in the Fourier domain. The proposed scaling model is analogous to the scaling property of the classical Euclidean Fourier transform. Spherical scaling is used for example in spherical wavelet transform and filter banks or illumination in computer graphics. Since the function that requires scaling is often represented in the Fourier domain, our method is of significant interest. Furthermore, we extend the result to higher-dimensional spheres. We show how this model follows naturally from consideration of a hypothetical continuous spectrum. Experiments confirm the applicability of the proposed method for several signal classes. The proposed algorithm is compared to an existing linear operator formulation.

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