45,609 research outputs found
Fourier Transforms of Lorentz Invariant Functions
Fourier transforms of Lorentz invariant functions in Minkowski space, with
support on both the timelike and the spacelike domains are performed by means
of direct integration. The cases of 1+1 and 1+2 dimensions are worked out in
detail, and the results for 1+n dimensions are given.Comment: 15 pages, 1 figur
Weighted frames of exponentials and stable recovery of multidimensional functions from nonuniform Fourier samples
In this paper, we consider the problem of recovering a compactly supported
multivariate function from a collection of pointwise samples of its Fourier
transform taken nonuniformly. We do this by using the concept of weighted
Fourier frames. A seminal result of Beurling shows that sample points give rise
to a classical Fourier frame provided they are relatively separated and of
sufficient density. However, this result does not allow for arbitrary
clustering of sample points, as is often the case in practice. Whilst keeping
the density condition sharp and dimension independent, our first result removes
the separation condition and shows that density alone suffices. However, this
result does not lead to estimates for the frame bounds. A known result of
Groechenig provides explicit estimates, but only subject to a density condition
that deteriorates linearly with dimension. In our second result we improve
these bounds by reducing the dimension dependence. In particular, we provide
explicit frame bounds which are dimensionless for functions having compact
support contained in a sphere. Next, we demonstrate how our two main results
give new insight into a reconstruction algorithm---based on the existing
generalized sampling framework---that allows for stable and quasi-optimal
reconstruction in any particular basis from a finite collection of samples.
Finally, we construct sufficiently dense sampling schemes that are often used
in practice---jittered, radial and spiral sampling schemes---and provide
several examples illustrating the effectiveness of our approach when tested on
these schemes
Incremental Distance Transforms (IDT)
A new generic scheme for incremental implementations of distance transforms (DT) is presented: Incremental Distance Transforms (IDT). This scheme is applied on the cityblock, Chamfer, and three recent exact Euclidean DT (E2DT). A benchmark shows that for all five DT, the incremental implementation results in a significant speedup: 3.4×−10×. However, significant differences (i.e., up to 12.5×) among the DT remain present. The FEED transform, one of the recent E2DT, even showed to be faster than both city-block and Chamfer DT. So, through a very efficient incremental processing scheme for DT, a relief is found for E2DT’s computational burden
Maximum Scatter TSP in Doubling Metrics
We study the problem of finding a tour of points in which every edge is
long. More precisely, we wish to find a tour that visits every point exactly
once, maximizing the length of the shortest edge in the tour. The problem is
known as Maximum Scatter TSP, and was introduced by Arkin et al. (SODA 1997),
motivated by applications in manufacturing and medical imaging. Arkin et al.
gave a -approximation for the metric version of the problem and showed
that this is the best possible ratio achievable in polynomial time (assuming ). Arkin et al. raised the question of whether a better approximation
ratio can be obtained in the Euclidean plane.
We answer this question in the affirmative in a more general setting, by
giving a -approximation algorithm for -dimensional doubling
metrics, with running time , where . As a corollary we obtain (i) an
efficient polynomial-time approximation scheme (EPTAS) for all constant
dimensions , (ii) a polynomial-time approximation scheme (PTAS) for
dimension , for a sufficiently large constant , and (iii)
a PTAS for constant and . Furthermore, we
show the dependence on in our approximation scheme to be essentially
optimal, unless Satisfiability can be solved in subexponential time
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