8,236 research outputs found
Radon transforms of twisted D-modules on partial flag varieties
In this paper we study intertwining functors (Radon transforms) for twisted
D-modules on partial flag varieties and their relation to the representations
of semisimple Lie algebras. We show that certain intertwining functors give
equivalences of derived categories of twisted D-modules. This is a
generalization of a result by Marastoni. We also show that these intertwining
functors from dominant to antidominant direction are compatible with taking
global sections
Aspects of Multilinear Harmonic Analysis Related to Transversality
The purpose of this article is to survey certain aspects of multilinear
harmonic analysis related to notions of transversality. Particular emphasis
will be placed on the multilinear restriction theory for the euclidean Fourier
transform, multilinear oscillatory integrals, multilinear geometric
inequalities, multilinear Radon-like transforms, and the interplay between
them.Comment: 28 pages. Article based on a short course given at the 9th
International Conference on Harmonic Analysis and Partial Differential
Equations, El Escorial, 201
Applications of microlocal analysis to some hyperbolic inverse problems
This thesis compiles my work on three inverse problems: ultrasound recovery in thermoacoustic tomography, cancellation of singularities in synthetic aperture radar, and the injectivity and stability of some generalized Radon transforms. Each problem is approached using microlocal methods. In the context of thermoacoustic tomography under the damped wave equation, I show uniqueness and stability of the problem with complete data, provide a reconstruction algorithm for small attenuation with complete data, and obtain stability estimates for visible singularities with partial data. The chapter on synthetic aperture radar constructs microlocally several infinite-dimensional families of ground reflectivity functions which appear microlocally regular when imaged using synthetic aperture radar. Finally, based on a joint work with Hanming Zhou, we show the analytic microlocal regularity of a class of analytic generalized Radon transforms, using this to show injectivity and stability for a generic class of generalized Radon transforms defined on analytic Riemannian manifolds
Approximation and Reconstruction from Attenuated Radon Projections
Attenuated Radon projections with respect to the weight function are shown to be closely related to the orthogonal
expansion in two variables with respect to . This leads to an algorithm
for reconstructing two dimensional functions (images) from attenuated Radon
projections. Similar results are established for reconstructing functions on
the sphere from projections described by integrals over circles on the sphere,
and for reconstructing functions on the three-dimensional ball and cylinder
domains.Comment: 25 pages, 3 figure
A range description for the planar circular Radon transform
The transform considered in the paper integrates a function supported in the
unit disk on the plane over all circles centered at the boundary of this disk.
Such circular Radon transform arises in several contemporary imaging
techniques, as well as in other applications. As it is common for transforms of
Radon type, its range has infinite co-dimension in standard function spaces.
Range descriptions for such transforms are known to be very important for
computed tomography, for instance when dealing with incomplete data, error
correction, and other issues. A complete range description for the circular
Radon transform is obtained. Range conditions include the recently found set of
moment type conditions, which happens to be incomplete, as well as the rest of
conditions that have less standard form. In order to explain the procedure
better, a similar (non-standard) treatment of the range conditions is described
first for the usual Radon transform on the plane.Comment: submitted for publicatio
On Radon transforms on tori
We show injectivity of the X-ray transform and the -plane Radon transform
for distributions on the -torus, lowering the regularity assumption in the
recent work by Abouelaz and Rouvi\`ere. We also show solenoidal injectivity of
the X-ray transform on the -torus for tensor fields of any order, allowing
the tensors to have distribution valued coefficients. These imply new
injectivity results for the periodic broken ray transform on cubes of any
dimension.Comment: 13 page
Fast hyperbolic Radon transform represented as convolutions in log-polar coordinates
The hyperbolic Radon transform is a commonly used tool in seismic processing,
for instance in seismic velocity analysis, data interpolation and for multiple
removal. A direct implementation by summation of traces with different moveouts
is computationally expensive for large data sets. In this paper we present a
new method for fast computation of the hyperbolic Radon transforms. It is based
on using a log-polar sampling with which the main computational parts reduce to
computing convolutions. This allows for fast implementations by means of FFT.
In addition to the FFT operations, interpolation procedures are required for
switching between coordinates in the time-offset; Radon; and log-polar domains.
Graphical Processor Units (GPUs) are suitable to use as a computational
platform for this purpose, due to the hardware supported interpolation routines
as well as optimized routines for FFT. Performance tests show large speed-ups
of the proposed algorithm. Hence, it is suitable to use in iterative methods,
and we provide examples for data interpolation and multiple removal using this
approach.Comment: 21 pages, 10 figures, 2 table
Fast algorithms and efficient GPU implementations for the Radon transform and the back-projection operator represented as convolution operators
The Radon transform and its adjoint, the back-projection operator, can both
be expressed as convolutions in log-polar coordinates. Hence, fast algorithms
for the application of the operators can be constructed by using FFT, if data
is resampled at log-polar coordinates. Radon data is typically measured on an
equally spaced grid in polar coordinates, and reconstructions are represented
(as images) in Cartesian coordinates. Therefore, in addition to FFT, several
steps of interpolation have to be conducted in order to apply the Radon
transform and the back-projection operator by means of convolutions.
Both the interpolation and the FFT operations can be efficiently implemented
on Graphical Processor Units (GPUs). For the interpolation, it is possible to
make use of the fact that linear interpolation is hard-wired on GPUs, meaning
that it has the same computational cost as direct memory access. Cubic order
interpolation schemes can be constructed by combining linear interpolation
steps which provides important computation speedup.
We provide details about how the Radon transform and the back-projection can
be implemented efficiently as convolution operators on GPUs. For large data
sizes, speedups of about 10 times are obtained in relation to the computational
times of other software packages based on GPU implementations of the Radon
transform and the back-projection operator. Moreover, speedups of more than a
1000 times are obtained against the CPU-implementations provided in the MATLAB
image processing toolbox
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