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On Reconstructing Functions from Binary Measurements

  • Robert Calderbank
  • Anders Hansen
  • Bogdan RomanEmail author
  • Laura Thesing
Chapter
Part of the Applied and Numerical Harmonic Analysis book series (ANHA)

Abstract

We consider the problem of reconstructing a function from linear binary measurements. That is, the samples of the function are given by inner products with functions taking only the values 0 and 1. We consider three particular methods for this problem, the parameterized-background data-weak (PBDW) method, generalized sampling and infinite-dimensional compressed sensing. The first two methods are dependent on knowing the stable sampling rate when considering samples by Walsh function and wavelet reconstruction. We establish linearity of the stable sampling rate, which is sharp, allowing for optimal use of these methods. In addition, we provide recovery guaranties for infinite-dimensional compressed sensing with Walsh functions and wavelets.

Notes

Acknowledgements

RC acknowledges support from the Air Force Office of Scientific Research through grant FA 9550-17-1-0291. AH acknowledges support from Royal Society UK, and EPSRC UK through grant EP/L003457/1. BR acknowledges support from EPSRC UK through grants EP/N014588/1 and EP/R008272/1. LT acknowledges support from EPSRC UK through grant EP/L016516/1.

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Robert Calderbank
    • 1
  • Anders Hansen
    • 2
  • Bogdan Roman
    • 3
    Email author
  • Laura Thesing
    • 2
  1. 1.Department of Electrical and Computer EngineeringDuke UniversityDurhamUSA
  2. 2.Department of Applied Mathematics and Theoretical PhysicsUniversity of CambridgeCambridgeUK
  3. 3.Department of Pure Mathematics and Mathematical StatisticsUniversity of CambridgeCambridgeUK

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