189 research outputs found
Linear-time nearest point algorithms for Coxeter lattices
The Coxeter lattices, which we denote , are a family of lattices
containing many of the important lattices in low dimensions. This includes
, , and their duals , and . We consider
the problem of finding a nearest point in a Coxeter lattice. We describe two
new algorithms, one with worst case arithmetic complexity and the
other with worst case complexity O(n) where is the dimension of the
lattice. We show that for the particular lattices and the
algorithms reduce to simple nearest point algorithms that already exist in the
literature.Comment: submitted to IEEE Transactions on Information Theor
New Shortest Lattice Vector Problems of Polynomial Complexity
The Shortest Lattice Vector (SLV) problem is in general hard to solve, except
for special cases (such as root lattices and lattices for which an obtuse
superbase is known). In this paper, we present a new class of SLV problems that
can be solved efficiently. Specifically, if for an -dimensional lattice, a
Gram matrix is known that can be written as the difference of a diagonal matrix
and a positive semidefinite matrix of rank (for some constant ), we show
that the SLV problem can be reduced to a -dimensional optimization problem
with countably many candidate points. Moreover, we show that the number of
candidate points is bounded by a polynomial function of the ratio of the
smallest diagonal element and the smallest eigenvalue of the Gram matrix.
Hence, as long as this ratio is upper bounded by a polynomial function of ,
the corresponding SLV problem can be solved in polynomial complexity. Our
investigations are motivated by the emergence of such lattices in the field of
Network Information Theory. Further applications may exist in other areas.Comment: 13 page
Compute-and-Forward: Finding the Best Equation
Compute-and-Forward is an emerging technique to deal with interference. It
allows the receiver to decode a suitably chosen integer linear combination of
the transmitted messages. The integer coefficients should be adapted to the
channel fading state. Optimizing these coefficients is a Shortest Lattice
Vector (SLV) problem. In general, the SLV problem is known to be prohibitively
complex. In this paper, we show that the particular SLV instance resulting from
the Compute-and-Forward problem can be solved in low polynomial complexity and
give an explicit deterministic algorithm that is guaranteed to find the optimal
solution.Comment: Paper presented at 52nd Allerton Conference, October 201
Image Sampling with Quasicrystals
We investigate the use of quasicrystals in image sampling. Quasicrystals
produce space-filling, non-periodic point sets that are uniformly discrete and
relatively dense, thereby ensuring the sample sites are evenly spread out
throughout the sampled image. Their self-similar structure can be attractive
for creating sampling patterns endowed with a decorative symmetry. We present a
brief general overview of the algebraic theory of cut-and-project quasicrystals
based on the geometry of the golden ratio. To assess the practical utility of
quasicrystal sampling, we evaluate the visual effects of a variety of
non-adaptive image sampling strategies on photorealistic image reconstruction
and non-photorealistic image rendering used in multiresolution image
representations. For computer visualization of point sets used in image
sampling, we introduce a mosaic rendering technique.Comment: For a full resolution version of this paper, along with supplementary
materials, please visit at
http://www.Eyemaginary.com/Portfolio/Publications.htm
Finding a closest point in a lattice of Voronoi's first kind
We show that for those lattices of Voronoi's first kind with known obtuse
superbasis, a closest lattice point can be computed in operations
where is the dimension of the lattice. To achieve this a series of relevant
lattice vectors that converges to a closest lattice point is found. We show
that the series converges after at most terms. Each vector in the series
can be efficiently computed in operations using an algorithm to
compute a minimum cut in an undirected flow network
Symmetric box-splines on root lattices
AbstractRoot lattices are efficient sampling lattices for reconstructing isotropic signals in arbitrary dimensions, due to their highly symmetric structure. One root lattice, the Cartesian grid, is almost exclusively used since it matches the coordinate grid; but it is less efficient than other root lattices. Box-splines, on the other hand, generalize tensor-product B-splines by allowing non-Cartesian directions. They provide, in any number of dimensions, higher-order reconstructions of fields, often of higher efficiency than tensored B-splines. But on non-Cartesian lattices, such as the BCC (Body-Centered Cubic) or the FCC (Face-Centered Cubic) lattice, only some box-splines and then only up to dimension three have been investigated.This paper derives and completely characterizes efficient symmetric box-spline reconstruction filters on all irreducible root lattices that exist in any number of dimensions n≥2 (n≥3 for Dn and Dn∗ lattices). In all cases, box-splines are constructed by convolution using the lattice directions, generalizing the known constructions in two and three variables. For each box-spline, we document the basic properties for computational use: the polynomial degree, the continuity, the linear independence of shifts on the lattice and optimal quasi-interpolants for fast approximation of fields
06271 Abstracts Collection -- Challenges in Symbolic Computation Software
From 02.07.06 to 07.07.06, the Dagstuhl Seminar 06271 ``Challenges in Symbolic Computation Software\u27\u27 was held in the International Conference and Research Center (IBFI), Schloss Dagstuhl.
During the seminar, several participants presented their current
research, and ongoing work and open problems were discussed. Abstracts of
the presentations given during the seminar as well as abstracts of
seminar results and ideas are put together in this paper. The first section
describes the seminar topics and goals in general.
Links to extended abstracts or full papers are provided, if available
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