Evaluation of the rotation matrices in the basis of real spherical harmonics

Blanco, María del Carmen del Vando; Flórez, M.; Bermejo, M.

doi:10.1016/s0166-1280(97)00185-1

Cited by 172 publications

(160 citation statements)

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“…D (l) m m stands for the real Wigner rotation matrix expressed in terms of the Euler angles (α, β, γ) in the zyz convention [59] (refer to appendix F for more details).…”

Section: Optimization For Angular Featuresmentioning

confidence: 99%

Efficient and robust computation of PDF features from diffusion MR signal

Assemlal

Tschumperlé

Brun

2009

Medical Image Analysis

131

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We present a method for the estimation of various features of the tissue micro-architecture using the diffusion magnetic resonance imaging. The considered features are designed from the displacement probability density function (PDF). The estimation is based on two steps: first the approximation of the signal by a series expansion made of Gaussian-Laguerre and Spherical Harmonics functions; followed by a projection on a finite dimensional space. Besides, we propose to tackle the problem of the robustness to Rician noise corrupting in-vivo acquisitions. Our feature estimation is expressed as a variational minimization process leading to a variational framework which is robust to noise. This approach is very flexible regarding the number of samples and enables the computation of a large set of various features of the local tissues structure. We demonstrate the effectiveness of the method with results on both synthetic phantom and real MR datasets acquired in a clinical time-frame.

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“…D (l) m m stands for the real Wigner rotation matrix expressed in terms of the Euler angles (α, β, γ) in the zyz convention [59] (refer to appendix F for more details).…”

Section: Optimization For Angular Featuresmentioning

confidence: 99%

Efficient and robust computation of PDF features from diffusion MR signal

Assemlal

Tschumperlé

Brun

2009

Medical Image Analysis

131

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“…A very efficient recursive procedure can be found in [26]. The following formula will be enough to reproduce the results presented in this paper.…”

Section: Wigner D Functionsmentioning

confidence: 99%

“…The second term in the right-hand side is the topological signature in the covariance function. The expressions we present in the following are analogous to (26) and are also rather obvious. It is convenient to introduce a notation, so natural, that has been used in (26) without any previous definition.…”

Section: Decomposition Of ÿ In Cyclic Subgroupsmentioning

confidence: 99%

“…The expressions we present in the following are analogous to (26) and are also rather obvious. It is convenient to introduce a notation, so natural, that has been used in (26) without any previous definition. Let S be any subset of isometries of the covering spaceM, then a superscript S in the covariance function means…”

Section: Decomposition Of ÿ In Cyclic Subgroupsmentioning

confidence: 99%

“…The first one, a trivial decomposition given by (26), defines the topological signature of the covariance function. When written for the correlation matrix of the harmonic expansion coefficients, it yields the topological signature in the temperature anisotropy maps, as illustrated for the cylinder by (34).…”

Section: A the Formalismmentioning

confidence: 99%

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Topological signatures in CMB temperature anisotropy maps

Hipólito-Ricaldi

Gomero

2005

Phys. Rev. D

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We propose an alternative formalism to simulate cosmic microwave background (CMB) temperature maps in CDM universes with nontrivial spatial topologies. This formalism avoids the need to explicitly compute the eigenmodes of the Laplacian operator in the spatial sections. Instead, the covariance matrix of the coefficients of the spherical harmonic decomposition of the temperature anisotropies is expressed in terms of the elements of the covering group of the space. We obtain a decomposition of the correlation matrix that isolates the topological contribution to the CMB temperature anisotropies out of the simply connected contribution. A further decomposition of the topological signature of the correlation matrix for an arbitrary topology allows us to compute it in terms of correlation matrices corresponding to simpler topologies, for which closed quadrature formulas might be derived. We also use this decomposition to show that CMB temperature maps of (not too large) multiply connected universes must show ''patterns of alignment,'' and propose a method to look for these patterns, thus opening the door to the development of new methods for detecting the topology of our Universe even when the injectivity radius of space is slightly larger than the radius of the last scattering surface. We illustrate all these features with the simplest examples, those of flat homogeneous manifolds, i.e., tori, with special attention given to the cylinder, i.e., T 1 topology.

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Strength, Depth, and Geometry of Magnetic Sources in the Crust of the Moon From Localized Power Spectrum Analysis

Wieczorek

2018

JGR Planets

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Spacecraft observations show that weak magnetic fields of crustal origin are ubiquitous across the surface of the Moon. To investigate the origin of these magnetic anomalies, a model was developed for the magnetic power spectrum that consists of ensembles of randomly magnetized sills or prisms. Localized spectrum analyses constrained how the parameters of this model vary with position, including the size of the sources, a quantity proportional to their mean‐squared dipole moment, and the depth to the top and bottom of the magnetized region. The depth to the top of the magnetized region varies from the surface to about 25 km. The magnetic carriers in the deep crust likely formed at the same time as the crust itself, implying that a core‐generated dynamo field must have existed when the crust was cooling during the first 100 Myr of lunar evolution. The parameter related to the strength of magnetization shows the existence of a prominent region on the nearside hemisphere that is largely unmagnetized and that correlates with a region of extremely low surface field strengths. This region lies entirely within a geological province that is highly enriched in heat‐producing elements (the Procellarum KREEP Terrane), suggesting that this region escaped being magnetized because of prolonged high crustal temperatures. The nearside magnetic low may be representative of the size of that portion of the crust that is highly enriched in heat‐producing elements, which is almost one third the size of the Procellarum KREEP Terrane based on surface thorium abundances.

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Evaluation of the rotation matrices in the basis of real spherical harmonics

Cited by 172 publications

References 4 publications

Efficient and robust computation of PDF features from diffusion MR signal

Efficient and robust computation of PDF features from diffusion MR signal

Topological signatures in CMB temperature anisotropy maps

Strength, Depth, and Geometry of Magnetic Sources in the Crust of the Moon From Localized Power Spectrum Analysis

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