9,916 research outputs found
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,π). In practice, however, we pre-filter discrete projections take
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
Automorphisms of graph products of groups from a geometric perspective
This article studies automorphism groups of graph products of arbitrary
groups. We completely characterise automorphisms that preserve the set of
conjugacy classes of vertex groups as those automorphisms that can be
decomposed as a product of certain elementary automorphisms (inner
automorphisms, partial conjugations, automorphisms associated to symmetries of
the underlying graph). This allows us to completely compute the automorphism
group of certain graph products, for instance in the case where the underlying
graph is finite, connected, leafless and of girth at least . If in addition
the underlying graph does not contain separating stars, we can understand the
geometry of the automorphism groups of such graph products of groups further:
we show that such automorphism groups do not satisfy Kazhdan's property (T) and
are acylindrically hyperbolic. Applications to automorphism groups of graph
products of finite groups are also included. The approach in this article is
geometric and relies on the action of graph products of groups on certain
complexes with a particularly rich combinatorial geometry. The first such
complex is a particular Cayley graph of the graph product that has a
quasi-median geometry, a combinatorial geometry reminiscent of (but more
general than) CAT(0) cube complexes. The second (strongly related) complex used
is the Davis complex of the graph product, a CAT(0) cube complex that also has
a structure of right-angled building.Comment: 36 pages. The article subsumes and vastly generalises our preprint
arXiv:1803.07536. To appear in Proc. Lond. Math. So
Noncommutative geometry, topology and the standard model vacuum
As a ramification of a motivational discussion for previous joint work, in
which equations of motion for the finite spectral action of the Standard Model
were derived, we provide a new analysis of the results of the calculations
herein, switching from the perspective of Spectral triple to that of Fredholm
module and thus from the analogy with Riemannian geometry to the pre-metrical
structure of the Noncommutative geometry. Using a suggested Noncommutative
version of Morse theory together with algebraic -theory to analyse the
vacuum solutions, the first two summands of the algebra for the finite triple
of the Standard Model arise up to Morita equivalence. We also demonstrate a new
vacuum solution whose features are compatible with the physical mass matrix.Comment: 24 page
A Galois Connection for Weighted (Relational) Clones of Infinite Size
A Galois connection between clones and relational clones on a fixed finite
domain is one of the cornerstones of the so-called algebraic approach to the
computational complexity of non-uniform Constraint Satisfaction Problems
(CSPs). Cohen et al. established a Galois connection between finitely-generated
weighted clones and finitely-generated weighted relational clones [SICOMP'13],
and asked whether this connection holds in general. We answer this question in
the affirmative for weighted (relational) clones with real weights and show
that the complexity of the corresponding valued CSPs is preserved
Representation by Integrating Reproducing Kernels
Based on direct integrals, a framework allowing to integrate a parametrised
family of reproducing kernels with respect to some measure on the parameter
space is developed. By pointwise integration, one obtains again a reproducing
kernel whose corresponding Hilbert space is given as the image of the direct
integral of the individual Hilbert spaces under the summation operator. This
generalises the well-known results for finite sums of reproducing kernels;
however, many more special cases are subsumed under this approach: so-called
Mercer kernels obtained through series expansions; kernels generated by
integral transforms; mixtures of positive definite functions; and in particular
scale-mixtures of radial basis functions. This opens new vistas into known
results, e.g. generalising the Kramer sampling theorem; it also offers
interesting connections between measurements and integral transforms, e.g.
allowing to apply the representer theorem in certain inverse problems, or
bounding the pointwise error in the image domain when observing the pre-image
under an integral transform
Mixture of Kernels and Iterated Semidirect Product of Diffeomorphisms Groups
In the framework of large deformation diffeomorphic metric mapping (LDDMM),
we develop a multi-scale theory for the diffeomorphism group based on previous
works. The purpose of the paper is (1) to develop in details a variational
approach for multi-scale analysis of diffeomorphisms, (2) to generalise to
several scales the semidirect product representation and (3) to illustrate the
resulting diffeomorphic decomposition on synthetic and real images. We also
show that the approaches presented in other papers and the mixture of kernels
are equivalent.Comment: 21 pages, revised version without section on evaluatio
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