On the global geometry of sphere-constrained sparse blind deconvolution

Yuqian Zhang, Yenson Lau, Han Wen Kuo, Sky Cheung, Abhay Pasupathy, John Wright

Research output: Chapter in Book/Report/Conference proceedingConference contribution

27 Scopus citations

Abstract

Blind deconvolution is the problem of recovering a convolutional kernel a0 and an activation signal x0 from their convolution y = a0 O x0. This problem is ill-posed without further constraints or priors. This paper studies the situation where the nonzero entries in the activation signal are sparsely and randomly populated. We normalize the convolution kernel to have unit Frobenius norm and cast the sparse blind deconvolution problem as a nonconvex optimization problem over the sphere. With this spherical constraint, every spurious local minimum turns out to be close to some signed shift truncation of the ground truth, under certain hypotheses. This benign property motivates an effective two stage algorithm that recovers the ground truth from the partial information offered by a suboptimal local minimum. This geometry-inspired algorithm recovers the ground truth for certain microscopy problems, also exhibits promising performance in the more challenging image deblurring problem. Our insights into the global geometry and the two stage algorithm extend to the convolutional dictionary learning problem, where a superposition of multiple convolution signals is observed.

Original languageEnglish (US)
Title of host publicationProceedings - 30th IEEE Conference on Computer Vision and Pattern Recognition, CVPR 2017
PublisherInstitute of Electrical and Electronics Engineers Inc.
Pages4381-4389
Number of pages9
ISBN (Electronic)9781538604571
DOIs
StatePublished - Nov 6 2017
Externally publishedYes
Event30th IEEE Conference on Computer Vision and Pattern Recognition, CVPR 2017 - Honolulu, United States
Duration: Jul 21 2017Jul 26 2017

Publication series

NameProceedings - 30th IEEE Conference on Computer Vision and Pattern Recognition, CVPR 2017
Volume2017-January

Other

Other30th IEEE Conference on Computer Vision and Pattern Recognition, CVPR 2017
Country/TerritoryUnited States
CityHonolulu
Period7/21/177/26/17

All Science Journal Classification (ASJC) codes

  • Software
  • Computer Vision and Pattern Recognition

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