52 research outputs found
Robust Recovery of Positive Stream of Pulses
The problem of estimating the delays and amplitudes of a positive stream of
pulses appears in many applications, such as single-molecule microscopy. This
paper suggests estimating the delays and amplitudes using a convex program,
which is robust in the presence of noise (or model mismatch). Particularly, the
recovery error is proportional to the noise level. We further show that the
error grows exponentially with the density of the delays and also depends on
the localization properties of the pulse
Recovery of Sparse Positive Signals on the Sphere from Low Resolution Measurements
This letter considers the problem of recovering a positive stream of Diracs
on a sphere from its projection onto the space of low-degree spherical
harmonics, namely, from its low-resolution version. We suggest recovering the
Diracs via a tractable convex optimization problem. The resulting recovery
error is proportional to the noise level and depends on the density of the
Diracs. We validate the theory by numerical experiments
A Least Squares Approach for Stable Phase Retrieval from Short-Time Fourier Transform Magnitude
We address the problem of recovering a signal (up to global phase) from its
short-time Fourier transform (STFT) magnitude measurements. This problem arises
in several applications, including optical imaging and speech processing. In
this paper we suggest three interrelated algorithms. The first algorithm
estimates the signal efficiently from noisy measurements by solving a simple
least-squares (LS) problem. In contrast to previously proposed algorithms, the
LS approach has stability guarantees and does not require any prior knowledge
on the sought signal. However, the recovery is guaranteed under relatively
strong restrictions on the STFT window. The second approach is guaranteed to
recover a non-vanishing signal efficiently from noise-free measurements, under
very moderate conditions on the STFT window. Finally, the third method
estimates the signal robustly from noisy measurements by solving a
semi-definite program (SDP). The proposed SDP algorithm contains an inherent
trade-off between its robustness and the restrictions on the STFT windows that
can be used
Exact recovery of non-uniform splines from the projection onto spaces of algebraic polynomials
In this work we consider the problem of recovering non-uniform splines from
their projection onto spaces of algebraic polynomials. We show that under a
certain Chebyshev-type separation condition on its knots, a spline whose
inner-products with a polynomial basis and boundary conditions are known, can
be recovered using Total Variation norm minimization. The proof of the
uniqueness of the solution uses the method of `dual' interpolating polynomials
and is based on \cite{SR}, where the theory was developed for trigonometric
polynomials. We also show results for the multivariate case
Stable Super-Resolution of Images: A Theoretical Study
We study the ubiquitous super-resolution problem, in which one aims at
localizing positive point sources in an image, blurred by the point spread
function of the imaging device. To recover the point sources, we propose to
solve a convex feasibility program, which simply finds a nonnegative Borel
measure that agrees with the observations collected by the imaging device.
In the absence of imaging noise, we show that solving this convex program
uniquely retrieves the point sources, provided that the imaging device collects
enough observations. This result holds true if the point spread function of the
imaging device can be decomposed into horizontal and vertical components, and
if the translations of these components form a Chebyshev system, i.e., a system
of continuous functions that loosely behave like algebraic polynomials.
Building upon recent results for one-dimensional signals [1], we prove that
this super-resolution algorithm is stable, in the generalized Wasserstein
metric, to model mismatch (i.e., when the image is not sparse) and to additive
imaging noise. In particular, the recovery error depends on the noise level and
how well the image can be approximated with well-separated point sources. As an
example, we verify these claims for the important case of a Gaussian point
spread function. The proofs rely on the construction of novel interpolating
polynomials---which are the main technical contribution of this paper---and
partially resolve the question raised in [2] about the extension of the
standard machinery to higher dimensions
Robust Recovery of Stream of Pulses using Convex Optimization
This paper considers the problem of recovering the delays and amplitudes of a
weighted superposition of pulses. This problem is motivated by a variety of
applications such as ultrasound and radar. We show that for univariate and
bivariate stream of pulses, one can recover the delays and weights to any
desired accuracy by solving a tractable convex optimization problem, provided
that a pulse-dependent separation condition is satisfied. The main result of
this paper states that the recovery is robust to additive noise or model
mismatch.Comment: Small modification
Edge Preserving Multi-Modal Registration Based On Gradient Intensity Self-Similarity
Image registration is a challenging task in the world of medical imaging.
Particularly, accurate edge registration plays a central role in a variety of
clinical conditions. The Modality Independent Neighbourhood Descriptor (MIND)
demonstrates state of the art alignment, based on the image self-similarity.
However, this method appears to be less accurate regarding edge registration.
In this work, we propose a new registration method, incorporating gradient
intensity and MIND self-similarity metric. Experimental results show the
superiority of this method in edge registration tasks, while preserving the
original MIND performance for other image features and textures
On Signal Reconstruction from FROG Measurements
Phase retrieval refers to recovering a signal from its Fourier magnitude.
This problem arises naturally in many scientific applications, such as
ultra-short laser pulse characterization and diffraction imaging.
Unfortunately, phase retrieval is ill-posed for almost all one-dimensional
signals. In order to characterize a laser pulse and overcome the ill-posedness,
it is common to use a technique called Frequency-Resolved Optical Gating
(FROG). In FROG, the measured data, referred to as FROG trace, is the Fourier
magnitude of the product of the underlying signal with several translated
versions of itself. The FROG trace results in a system of phaseless quartic
Fourier measurements. In this paper, we prove that it suffices to consider only
three translations of the signal to determine almost all bandlimited signals,
up to trivial ambiguities. In practice, one usually also has access to the
signal's Fourier magnitude. We show that in this case only two translations
suffice. Our results significantly improve upon earlier work
Unified Convex Optimization Approach to Super-Resolution Based on Localized Kernels
The problem of resolving the fine details of a signal from its coarse scale
measurements or, as it is commonly referred to in the literature, the
super-resolution problem arises naturally in engineering and physics in a
variety of settings. We suggest a unified convex optimization approach for
super-resolution. The key is the construction of an interpolating polynomial
based on localized kernels. We also show that the localized kernels act as the
connecting thread to another wide-spread problem of stream of pulses
Fourier Phase Retrieval: Uniqueness and Algorithms
The problem of recovering a signal from its phaseless Fourier transform
measurements, called Fourier phase retrieval, arises in many applications in
engineering and science. Fourier phase retrieval poses fundamental theoretical
and algorithmic challenges. In general, there is no unique mapping between a
one-dimensional signal and its Fourier magnitude and therefore the problem is
ill-posed. Additionally, while almost all multidimensional signals are uniquely
mapped to their Fourier magnitude, the performance of existing algorithms is
generally not well-understood. In this chapter we survey methods to guarantee
uniqueness in Fourier phase retrieval. We then present different algorithmic
approaches to retrieve the signal in practice. We conclude by outlining some of
the main open questions in this field
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