21,318 research outputs found
Analytic inversion of a Radon transform on double circular arcs with applications in Compton Scattering Tomography
In this work we introduce a new Radon transform which arises from a new
modality of Compton Scattering Tomography (CST). This new system is made of a
single detector rotating around a fixed source. Unlike some previous CST, no
collimator is used at the detector. Such a system allows us to collect
scattered photons coming from two opposite sides of the source-detector
segment, hence the manifold of the associated Radon transform is a family of
double circular arcs. As first main theoretical result, an analytic inversion
formula is established for this new Radon transform. This is achieved through
the formulation of the transform in terms of circular harmonic expansion
satisfying the consistency conditions in Cormack's sense. Moreover, a fast and
efficient numerical implementation via an alternative formulation based on
Hilbert transform is carried out. Simulation results illustrate the theoretical
feasibility of the new system. From a practical point of view, an uncollimated
detector system considerably increases the amount of collected data, which is
particularly significant in a scatter imaging system.Comment: 14 pages, 5 figure
Radon-based Structure from Motion Without Correspondences
We present a novel approach for the estimation of 3Dmotion directly from two images using the Radon transform. We assume a similarity function defined on the crossproduct of two images which assigns a weight to all feature pairs. This similarity function is integrated over all feature pairs that satisfy the epipolar constraint. This integration is equivalent to filtering the similarity function with a Dirac function embedding the epipolar constraint. The result of this convolution is a function of the five unknownmotion parameters with maxima at the positions of compatible rigid motions.
The breakthrough is in the realization that the Radon transform is a filtering operator: If we assume that images are defined on spheres and the epipolar constraint is a group action of two rotations on two spheres, then the Radon transform is a convolution/correlation integral. We propose a new algorithm to compute this integral from the spherical harmonics of the similarity and Dirac functions. The resulting resolution in the motion space depends on the bandwidth we keep from the spherical transform. The strength of the algorithm is in avoiding a commitment to correspondences, thus being robust to erroneous feature detection, outliers, and multiple motions. The algorithm has been tested in sequences of real omnidirectional images and it outperforms correspondence-based structure from motion
Radon-based Image Reconstruction for MPI using a continuously rotating FFL
Magnetic particle imaging is a relatively new tracer-based medical imaging
technique exploiting the non-linear magnetization response of magnetic
nanoparticles to changing magnetic fields. If the data are generated by using a
field-free line, the sampling geometry resembles the one in computerized
tomography. Indeed, for an ideal field-free line rotating only in between
measurements it was shown that the signal equation can be written as a
convolution with the Radon transform of the particle concentration. In this
work, we regard a continuously rotating field-free line and extend the forward
operator accordingly. We obtain a similar result for the relation to the Radon
data but with two additive terms resulting from the additional
time-dependencies in the forward model. We jointly reconstruct particle
concentration and corresponding Radon data by means of total variation
regularization yielding promising results for synthetic data.Comment: YRM & CSE Workshop on Modeling, Simulation & Optimization of Fluid
Dynamic Applications 202
On Radon transforms on finite groups
If is a finite group, is a function determined by its
sums over all cosets of cyclic subgroups of ? In other words, is the Radon
transform on injective? This inverse problem is a discrete analogue of
asking whether a function on a compact Lie group is determined by its integrals
over all geodesics. We discuss what makes this new discrete inverse problem
analogous to well-studied inverse problems on manifolds and we also present
some alternative definitions. We use representation theory to prove that the
Radon transform fails to be injective precisely on Frobenius complements. We
also give easy-to-check sufficient conditions for injectivity and
noninjectivity for the Radon transform, including a complete answer for abelian
groups and several examples for nonabelian ones.Comment: 23 page
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