10,424 research outputs found
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
Reconstruction from non-uniform samples: A direct, variational approach in shift-invariant spaces
International audienceWe propose a new approach for signal reconstruction from non-uniform samples, without any constraint on their locations. We look for a function that minimizes a classical regularized least-squares criterion, but with the additional constraint that the solution lies in a chosen linear shift-invariant space--typically, a spline space. In comparison with a pure variational treatment involving radial basis functions, our approach is resolution de- pendent; an important feature for many applications. Moreover, the solution can be computed exactly by a fast non-iterative algorithm, that exploits at best the particular structure of the problem
Sampling and Reconstruction of Spatial Fields using Mobile Sensors
Spatial sampling is traditionally studied in a static setting where static
sensors scattered around space take measurements of the spatial field at their
locations. In this paper we study the emerging paradigm of sampling and
reconstructing spatial fields using sensors that move through space. We show
that mobile sensing offers some unique advantages over static sensing in
sensing time-invariant bandlimited spatial fields. Since a moving sensor
encounters such a spatial field along its path as a time-domain signal, a
time-domain anti-aliasing filter can be employed prior to sampling the signal
received at the sensor. Such a filtering procedure, when used by a
configuration of sensors moving at constant speeds along equispaced parallel
lines, leads to a complete suppression of spatial aliasing in the direction of
motion of the sensors. We analytically quantify the advantage of using such a
sampling scheme over a static sampling scheme by computing the reduction in
sampling noise due to the filter. We also analyze the effects of non-uniform
sensor speeds on the reconstruction accuracy. Using simulation examples we
demonstrate the advantages of mobile sampling over static sampling in practical
problems.
We extend our analysis to sampling and reconstruction schemes for monitoring
time-varying bandlimited fields using mobile sensors. We demonstrate that in
some situations we require a lower density of sensors when using a mobile
sensing scheme instead of the conventional static sensing scheme. The exact
advantage is quantified for a problem of sampling and reconstructing an audio
field.Comment: Submitted to IEEE Transactions on Signal Processing May 2012; revised
Oct 201
Sub-Nyquist Sampling: Bridging Theory and Practice
Sampling theory encompasses all aspects related to the conversion of
continuous-time signals to discrete streams of numbers. The famous
Shannon-Nyquist theorem has become a landmark in the development of digital
signal processing. In modern applications, an increasingly number of functions
is being pushed forward to sophisticated software algorithms, leaving only
those delicate finely-tuned tasks for the circuit level.
In this paper, we review sampling strategies which target reduction of the
ADC rate below Nyquist. Our survey covers classic works from the early 50's of
the previous century through recent publications from the past several years.
The prime focus is bridging theory and practice, that is to pinpoint the
potential of sub-Nyquist strategies to emerge from the math to the hardware. In
that spirit, we integrate contemporary theoretical viewpoints, which study
signal modeling in a union of subspaces, together with a taste of practical
aspects, namely how the avant-garde modalities boil down to concrete signal
processing systems. Our hope is that this presentation style will attract the
interest of both researchers and engineers in the hope of promoting the
sub-Nyquist premise into practical applications, and encouraging further
research into this exciting new frontier.Comment: 48 pages, 18 figures, to appear in IEEE Signal Processing Magazin
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
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