641 research outputs found
Sifting convolution on the sphere
A novel spherical convolution is defined through the sifting property of the Dirac delta on the sphere. The so-called sifting convolution is defined by the inner product of one function with a translated version of another, but with the adoption of an alternative translation operator on the sphere. This translation operator follows by analogy with the Euclidean translation when viewed in harmonic space. The sifting convolution satisfies a variety of desirable properties that are lacking in alternate definitions, namely: it supports directional kernels; it has an output which remains on the sphere; and is efficient to compute. An illustration of the sifting convolution on a topographic map of the Earth demonstrates that it supports directional kernels to perform anisotropic filtering, while its output remains on the sphere
Sifting Convolution on the Sphere
A novel spherical convolution is defined through the sifting property of the
Dirac delta on the sphere. The so-called sifting convolution is defined by the
inner product of one function with a translated version of another, but with
the adoption of an alternative translation operator on the sphere. This
translation operator follows by analogy with the Euclidean translation when
viewed in harmonic space. The sifting convolution satisfies a variety of
desirable properties that are lacking in alternate definitions, namely: it
supports directional kernels; it has an output which remains on the sphere; and
is efficient to compute. An illustration of the sifting convolution on a
topographic map of the Earth demonstrates that it supports directional kernels
to perform anisotropic filtering, while its output remains on the sphere.Comment: 5 pages, 3 figure
Scale-discretised ridgelet transform on the sphere
We revisit the spherical Radon transform, also called the Funk-Radon
transform, viewing it as an axisymmetric convolution on the sphere. Viewing the
spherical Radon transform in this manner leads to a straightforward derivation
of its spherical harmonic representation, from which we show the spherical
Radon transform can be inverted exactly for signals exhibiting antipodal
symmetry. We then construct a spherical ridgelet transform by composing the
spherical Radon and scale-discretised wavelet transforms on the sphere. The
resulting spherical ridgelet transform also admits exact inversion for
antipodal signals. The restriction to antipodal signals is expected since the
spherical Radon and ridgelet transforms themselves result in signals that
exhibit antipodal symmetry. Our ridgelet transform is defined natively on the
sphere, probes signal content globally along great circles, does not exhibit
blocking artefacts, supports spin signals and exhibits an exact and explicit
inverse transform. No alternative ridgelet construction on the sphere satisfies
all of these properties. Our implementation of the spherical Radon and ridgelet
transforms is made publicly available. Finally, we illustrate the effectiveness
of spherical ridgelets for diffusion magnetic resonance imaging of white matter
fibers in the brain.Comment: 5 pages, 4 figures, matches version accepted by EUSIPCO, code
available at http://www.s2let.or
Nonparametric estimation of composite functions
We study the problem of nonparametric estimation of a multivariate function
that can be represented as a composition of two
unknown smooth functions and . We suppose that and belong to known smoothness classes of
functions, with smoothness and , respectively. We obtain the
full description of minimax rates of estimation of in terms of and
, and propose rate-optimal estimators for the sup-norm loss. For the
construction of such estimators, we first prove an approximation result for
composite functions that may have an independent interest, and then a result on
adaptation to the local structure. Interestingly, the construction of
rate-optimal estimators for composite functions (with given, fixed smoothness)
needs adaptation, but not in the traditional sense: it is now adaptation to the
local structure. We prove that composition models generate only two types of
local structures: the local single-index model and the local model with
roughness isolated to a single dimension (i.e., a model containing elements of
both additive and single-index structure). We also find the zones of (,
) where no local structure is generated, as well as the zones where the
composition modeling leads to faster rates, as compared to the classical
nonparametric rates that depend only to the overall smoothness of .Comment: Published in at http://dx.doi.org/10.1214/08-AOS611 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Higher Order Statistsics of Stokes Parameters in a Random Birefringent Medium
We present a new model for the propagation of polarized light in a random
birefringent medium. This model is based on a decomposition of the higher order
statistics of the reduced Stokes parameters along the irreducible
representations of the rotation group. We show how this model allows a detailed
description of the propagation, giving analytical expressions for the
probability densities of the Mueller matrix and the Stokes vector throughout
the propagation. It also allows an exact description of the evolution of
averaged quantities, such as the degree of polarization. We will also discuss
how this model allows a generalization of the concepts of reduced Stokes
parameters and degree of polarization to higher order statistics. We give some
notes on how it can be extended to more general random media
Nonparametric estimation of the heterogeneity of a random medium using Compound Poisson Process modeling of wave multiple scattering
In this paper, we present a nonparametric method to estimate the
heterogeneity of a random medium from the angular distribution of intensity
transmitted through a slab of random material. Our approach is based on the
modeling of forward multiple scattering using Compound Poisson Processes on
compact Lie groups. The estimation technique is validated through numerical
simulations based on radiative transfer theory.Comment: 23 pages, 8 figures, 21 reference
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