Signal Reconstruction From the Magnitude of Subspace Components

We consider signal reconstruction from the norms of subspace components generalizing standard phase retrieval problems. In the deterministic setting, a closed reconstruction formula is derived when the subspaces satisfy certain cubature conditions, that require at least a quadratic number of subspac...

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Bibliographic Details
Published in:IEEE transactions on information theory 2015-07, Vol.61 (7), p.4015-4027
Main Authors: Bachoc, Christine, Ehler, Martin
Format: Article
Language:English
Subjects:
Electronics
Feasibility
fusion frame
Grassmannian cubature
Image reconstruction
Norms
Optical diffraction
Optical imaging
Optical variables measurement
phase retrieval
Polynomials
Probability
Problem solving
Signal reconstruction
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Summary:We consider signal reconstruction from the norms of subspace components generalizing standard phase retrieval problems. In the deterministic setting, a closed reconstruction formula is derived when the subspaces satisfy certain cubature conditions, that require at least a quadratic number of subspaces. Moreover, we address reconstruction under the erasure of a subset of the norms; using the concepts of p -fusion frames and list decoding, we propose an algorithm that outputs a finite list of candidate signals, one of which is the correct one. In the random setting, we show that a set of subspaces chosen at random and of cardinality scaling linearly in the ambient dimension allows for exact reconstruction with high probability by solving the feasibility problem of a semidefinite program.
ISSN:0018-9448
1557-9654
DOI:10.1109/TIT.2015.2429634