914 research outputs found
On stable reconstructions from nonuniform Fourier measurements
We consider the problem of recovering a compactly-supported function from a
finite collection of pointwise samples of its Fourier transform taking
nonuniformly. First, we show that under suitable conditions on the sampling
frequencies - specifically, their density and bandwidth - it is possible to
recover any such function in a stable and accurate manner in any given
finite-dimensional subspace; in particular, one which is well suited for
approximating . In practice, this is carried out using so-called nonuniform
generalized sampling (NUGS). Second, we consider approximation spaces in one
dimension consisting of compactly supported wavelets. We prove that a linear
scaling of the dimension of the space with the sampling bandwidth is both
necessary and sufficient for stable and accurate recovery. Thus wavelets are up
to constant factors optimal spaces for reconstruction
Recovering piecewise smooth functions from nonuniform Fourier measurements
In this paper, we consider the problem of reconstructing piecewise smooth
functions to high accuracy from nonuniform samples of their Fourier transform.
We use the framework of nonuniform generalized sampling (NUGS) to do this, and
to ensure high accuracy we employ reconstruction spaces consisting of splines
or (piecewise) polynomials. We analyze the relation between the dimension of
the reconstruction space and the bandwidth of the nonuniform samples, and show
that it is linear for splines and piecewise polynomials of fixed degree, and
quadratic for piecewise polynomials of varying degree
Computing weak distances between the 2-sphere and its nonsmooth approximations
A novel algorithm is proposed for quantitative comparisons between compact
surfaces embedded in the three-dimensional Euclidian space. The key idea is to
identify those objects with the associated surface measures and compute
distances between them using the Fourier transform on the ambient space. In
particular, the inhomogeneous Sobolev norms of negative order are approximated
from data in the frequency space, which amounts to comparing measures after
appropriate smoothing. Such Fourier-based distances allow several advantages
including high accuracy due to fast-converging numerical quadrature rules,
acceleration by the nonuniform fast Fourier transform, parallelization on
massively parallel architectures. In numerical experiments, the 2-sphere, which
is an example whose Fourier transform is explicitly known, is compared with its
icosahedral discretization, and it is observed that the piecewise linear
approximations converge to the smooth object at the quadratic rate up to small
truncations.Comment: 14 pages, 4 figure
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