3 research outputs found
Phaseless Reconstruction from Space-Time Samples
Phaseless reconstruction from space-time samples is a nonlinear problem of
recovering a function in a Hilbert space from the modulus of
linear measurements , ,
, where
is a set of functionals on
, and is a bounded operator on that acts as an
evolution operator. In this paper, we provide various sufficient or necessary
conditions for solving this problem, which has connections to -ray
crystallography, the scattering transform, and deep learning.Comment: 23 pages, 4 figure
Local-to-global frames and applications to dynamical sampling problem
In this paper we consider systems of vectors in a Hilbert space
of the form where and
are countable sets of indices. We find conditions under which the local
reconstruction properties of such a system extend to global stable recovery
properties on the whole space. As a particular case, we obtain new
local-to-global results for systems of type arising in the dynamical sampling problem
Phase Retrieval and System Identification in Dynamical Sampling via Prony's Method
Phase retrieval in dynamical sampling is a novel research direction, where an
unknown signal has to be recovered from the phaseless measurements with respect
to a dynamical frame, i.e. a sequence of sampling vectors constructed by the
repeated action of an operator. The loss of the phase here turns the well-posed
dynamical sampling into a severe ill-posed inverse problem. In the existing
literature, the involved operator is usually completely known. In this paper,
we combine phase retrieval in dynamical sampling with the identification of the
system. For instance, if the dynamical frame is based on a repeated
convolution, then we want to recover the unknown convolution kernel in advance.
Using Prony's method, we establish several recovery guarantees for signal and
system, whose proofs are constructive and yield analytic recovery methods. The
required assumptions are satisfied by almost all signals, operators, and
sampling vectors. Moreover, these guarantees not only hold for the
finite-dimensional setting but also carry over to infinite-dimensional spaces.
Studying the sensitivity of the analytic recovery procedures, we also establish
error bounds for the applied approximate Prony method with respect to complex
exponential sums