Sampling and reconstructing signals from a union of linear subspaces

Blumensath, Thomas (2011) Sampling and reconstructing signals from a union of linear subspaces. IEEE Transactions on Information Theory, 57, (7), 4660-4671.


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In this note we study the problem of sampling and reconstructing signals which are assumed to lie on or close to one of
several subspaces of a Hilbert space. Importantly, we here consider a very general setting in which we allow infinitely many
subspaces in infinite dimensional Hilbert spaces. This general approach allows us to unify many results derived recently in areas
such as compressed sensing, affine rank minimisation and analog compressed sensing.
Our main contribution is to show that a conceptually simple iterative projection algorithms is able to recover signals from
a union of subspaces whenever the sampling operator satisfies a bi-Lipschitz embedding condition. Importantly, this result holds
for all Hilbert spaces and unions of subspaces, as long as the sampling procedure satisfies the condition for the set of subspaces
considered. In addition to recent results for finite unions of finite dimensional subspaces and infinite unions of subspaces in finite
dimensional spaces, we also show that this bi-Lipschitz property can hold in an analog compressed sensing setting in which we
have an infinite union of infinite dimensional subspaces living in infinite dimensional space

Item Type: Article
ISSNs: 0018-9448 (print)
1557-9654 (electronic)
Related URLs:
Keywords: computer science, information theory
Subjects: Q Science > Q Science (General)
Divisions : University Structure - Pre August 2011 > School of Mathematics > Applied Mathematics
Faculty of Engineering and the Environment > Institute of Sound and Vibration Research > Signal Processing & Control Research Group
ePrint ID: 142497
Accepted Date and Publication Date:
July 2011Published
Date Deposited: 31 Mar 2010 15:58
Last Modified: 31 Mar 2016 13:18

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