5,055 research outputs found
Fast Mojette Transform for Discrete Tomography
A new algorithm for reconstructing a two dimensional object from a set of one
dimensional projected views is presented that is both computationally exact and
experimentally practical. The algorithm has a computational complexity of O(n
log2 n) with n = N^2 for an NxN image, is robust in the presence of noise and
produces no artefacts in the reconstruction process, as is the case with
conventional tomographic methods. The reconstruction process is approximation
free because the object is assumed to be discrete and utilizes fully discrete
Radon transforms. Noise in the projection data can be suppressed further by
introducing redundancy in the reconstruction. The number of projections
required for exact reconstruction and the response to noise can be controlled
without comprising the digital nature of the algorithm. The digital projections
are those of the Mojette Transform, a form of discrete linogram. A simple
analytical mapping is developed that compacts these projections exactly into
symmetric periodic slices within the Discrete Fourier Transform. A new digital
angle set is constructed that allows the periodic slices to completely fill all
of the objects Discrete Fourier space. Techniques are proposed to acquire these
digital projections experimentally to enable fast and robust two dimensional
reconstructions.Comment: 22 pages, 13 figures, Submitted to Elsevier Signal Processin
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
The Discrete radon transform: A more efficient approach to image reconstruction
The Radon transform and its inversion are the mathematical keys that enable tomography. Radon transforms are defined for continuous objects with continuous projections at all angles in [0,Ď€). In practice, however, we pre-filter discrete projections take
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
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
Quantum Fourier transform, Heisenberg groups and quasiprobability distributions
This paper aims to explore the inherent connection among Heisenberg groups,
quantum Fourier transform and (quasiprobability) distribution functions.
Distribution functions for continuous and finite quantum systems are examined
first as a semiclassical approach to quantum probability distribution. This
leads to studying certain functionals of a pair of "conjugate" observables,
connected via the quantum Fourier transform. The Heisenberg groups emerge
naturally from this study and we take a rapid look at their representations.
The quantum Fourier transform appears as the intertwining operator of two
equivalent representation arising out of an automorphism of the group.
Distribution functions correspond to certain distinguished sets in the group
algebra. The marginal properties of a particular class of distribution
functions (Wigner distributions) arise from a class of automorphisms of the
group algebra of the Heisenberg group. We then study the reconstruction of
Wigner function from the marginal distributions via inverse Radon transform
giving explicit formulas. We consider applications of our approach to quantum
information processing and quantum process tomography.Comment: 39 page
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