On azimuthally symmetric 2-sphere convolution

Kennedy, Rodney A.; Lamahewa, Tharaka A.; Wei, Li-Yi

doi:10.1016/j.dsp.2011.05.002

Cited by 30 publications

(34 citation statements)

References 13 publications

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“…This reveals the SLSHT as being a hybrid of isotropic convolution [16] and the SHT (1). Because of the symmetry of the window, convolution present in the reformulation (5) and direct spatial weighting present in the original (2a) are equivalent.…”

Section: Slsht Reformulation With Isotropic Convolution Kernelmentioning

confidence: 98%

Efficient kernel-based formulations of spatio-spectral and related transformations on the 2-sphere

Kennedy

Khalid

Sadeghi

2014

2014 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)

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In this paper we show that the spatially localized spherical harmonic transform (SLSHT), which represents a signal on the 2-sphere in the spatio-spectral domain, can be efficiently computed using new kernel-based formulations. In addition to the standard spatiospectral domain, we show there are three other related transforms that provide alternative representations in the spatio-spatial, spectrospatial and spectro-spectral domains. We provide inversion results that extend available results for the SLSHT. We show that for signals on the 2-sphere band-limited to degree L, the computational complexity using our class of kernel-based SLSHT transforms is O(L 4 ) and outperforms the previous best known fast methods, which have complexity O(L 5 ).Index Terms-2-sphere; unit sphere; spherical harmonic transform; spatio-spectral domain; spatially localized spherical harmonic transform; fast transforms.

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Section: Slsht Reformulation With Isotropic Convolution Kernelmentioning

confidence: 98%

Efficient kernel-based formulations of spatio-spectral and related transformations on the 2-sphere

Kennedy

Khalid

Sadeghi

2014

2014 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)

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“…As we have shown in [7] that the spherical harmonic multiplication operator K v is developed as a convenient mathematical tool to weight the functions of interest in the spectral domain by a real, non-negative sequence {v m n } with n = 0, 1, · · · , m = −n, · · · , n. In this paper, for simplicity, let…”

Section: Spherical Harmonic Multiplication Operationmentioning

confidence: 99%

“…Our former work [7] has shown that when the weighting sequence v m n is chosen, v m n = v n for all n and m, the spherical harmonic multiplication operation is equivalent to the isotropic convolution, and the values of sequence can be constructed from an axisymmetric function,…”

Section: Special Case -Equivalence To Isotropic Convolutionmentioning

confidence: 99%

“…In this paper, we simply require {v m n } ∈ l 2 , the canonical Hilbert space, i.e., v m n is to be square summable. The harmonic multiplication operation applied to a function of interest f ∈ L 2 (S 2 ) is defined by [7] (…”

Section: Spherical Harmonic Multiplication Operationmentioning

confidence: 99%

“…In this paper, using the spherical harmonic multiplication operation which has been used to implement the frequency weighting to achieve some optimal functions [7,8], we first propose a zero-forcing optimal filter design, and then considering the additive white noise, we develop another optimal filter on the 2-sphere which achieves minimum mean square error.…”

Section: Problem Formulationmentioning

confidence: 99%

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Zero-forcing and MMSE filters design on the 2-sphere

Wei

Kennedy

2011

2011 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)

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We design a zero-forcing filter and a minimum mean square error (MMSE) filter on the 2-sphere by a spherical harmonic multiplication operator. Compared with the traditional optimal filters, our filters, determined by the signal and noise statistics, have such advantages: 1) can be asymmetric, and 2) simple calculation of our filters is involved.Index Terms-Isotropic convolution, spherical harmonic multiplication, minimum mean square error (MMSE)

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Towards unconstrained compartment modeling in white matter using diffusion‐relaxation MRI with tensor‐valued diffusion encoding

Lampinen

Szczepankiewicz

Mårtensson

et al. 2020

Magnetic Resonance in Med

111

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This is an open access article under the terms of the Creative Commons Attribution-NonCommercial-NoDerivs License, which permits use and distribution in any medium, provided the original work is properly cited, the use is non-commercial and no modifications or adaptations are made. Purpose:To optimize diffusion-relaxation MRI with tensor-valued diffusion encoding for precise estimation of compartment-specific fractions, diffusivities, and T 2 values within a two-compartment model of white matter, and to explore the approach in vivo. Methods: Sampling protocols featuring different b-values (b), b-tensor shapes (b Δ ), and echo times (TE) were optimized using Cramér-Rao lower bounds (CRLB).Whole-brain data were acquired in children, adults, and elderly with white matter lesions. Compartment fractions, diffusivities, and T 2 values were estimated in a model featuring two microstructural compartments represented by a "stick" and a "zeppelin." Results: Precise parameter estimates were enabled by sampling protocols featuring seven or more "shells" with unique b/b Δ /TE-combinations. Acquisition times were approximately 15 minutes. In white matter of adults, the "stick" compartment had a fraction of approximately 0.5 and, compared with the "zeppelin" compartment, featured lower isotropic diffusivities (0.6 vs. 1.3 μm 2 /ms) but higher T 2 values (85 vs. 65 ms). Children featured lower "stick" fractions (0.4). White matter lesions exhibited high "zeppelin" isotropic diffusivities (1.7 μm 2 /ms) and T 2 values (150 ms). Conclusions: Diffusion-relaxation MRI with tensor-valued diffusion encoding expands the set of microstructure parameters that can be precisely estimated and therefore increases their specificity to biological quantities. K E Y W O R D Sbrain microstructure, diffusion-relaxation MRI, Fisher information, tensor-valued diffusion encoding 1606 | LAMPINEN Et AL. S (u) = (K ⊛ P) (u) = ∫ |n|=1 K(u ⋅ n) P(n) dn,(2) K(u ⋅ n) = S 0 J

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On azimuthally symmetric 2-sphere convolution

Cited by 30 publications

References 13 publications

Efficient kernel-based formulations of spatio-spectral and related transformations on the 2-sphere

Efficient kernel-based formulations of spatio-spectral and related transformations on the 2-sphere

Zero-forcing and MMSE filters design on the 2-sphere

Towards unconstrained compartment modeling in white matter using diffusion‐relaxation MRI with tensor‐valued diffusion encoding

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