2,295 research outputs found
Sparse non-negative super-resolution -- simplified and stabilised
The convolution of a discrete measure, , with
a local window function, , is a common model for a measurement
device whose resolution is substantially lower than that of the objects being
observed. Super-resolution concerns localising the point sources
with an accuracy beyond the essential support of
, typically from samples , where indicates an inexactness in the sample
value. We consider the setting of being non-negative and seek to
characterise all non-negative measures approximately consistent with the
samples. We first show that is the unique non-negative measure consistent
with the samples provided the samples are exact, i.e. ,
samples are available, and generates a Chebyshev system. This is
independent of how close the sample locations are and {\em does not rely on any
regulariser beyond non-negativity}; as such, it extends and clarifies the work
by Schiebinger et al. and De Castro et al., who achieve the same results but
require a total variation regulariser, which we show is unnecessary.
Moreover, we characterise non-negative solutions consistent with
the samples within the bound . Any such
non-negative measure is within of the discrete
measure generating the samples in the generalised Wasserstein distance,
converging to one another as approaches zero. We also show how to make
these general results, for windows that form a Chebyshev system, precise for
the case of being a Gaussian window. The main innovation of these
results is that non-negativity alone is sufficient to localise point sources
beyond the essential sensor resolution.Comment: 59 pages, 7 figure
The dual approach to non-negative super-resolution: perturbation analysis
We study the problem of super-resolution, where we recover the locations and
weights of non-negative point sources from a few samples of their convolution
with a Gaussian kernel. It has been shown that exact recovery is possible by
minimising the total variation norm of the measure, and a practical way of
achieve this is by solving the dual problem. In this paper, we study the
stability of solutions with respect to the solutions dual problem, both in the
case of exact measurements and in the case of measurements with additive noise.
In particular, we establish a relationship between perturbations in the dual
variable and perturbations in the primal variable around the optimiser and a
similar relationship between perturbations in the dual variable around the
optimiser and the magnitude of the additive noise in the measurements. Our
analysis is based on a quantitative version of the implicit function theorem.Comment: 35 pages, 5 figure
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