6,681 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
Image reconstruction from scattered Radon data by weighted positive definite kernel functions
We propose a novel kernel-based method for image reconstruction from scattered Radon data. To this end, we employ generalized Hermite–Birkhoff interpolation by positive definite kernel functions. For radial kernels, however, a straightforward application of the generalized Hermite–Birkhoff interpolation method fails to work, as we prove in this paper. To obtain a well-posed reconstruction scheme for scattered Radon data, we introduce a new class of weighted positive definite kernels, which are symmetric but not radially symmetric. By our construction, the resulting weighted kernels are combinations of radial positive definite kernels and positive weight functions. This yields very flexible image reconstruction methods, which work for arbitrary distributions of Radon lines. We develop suitable representations for the weighted basis functions and the symmetric positive definite kernel matrices that are resulting from the proposed reconstruction scheme. For the relevant special case, where Gaussian radial kernels are combined with Gaussian weights, explicit formulae for the weighted Gaussian basis functions and the kernel matrices are given. Supporting numerical examples are finally presented
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
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