Stable and robust sampling strategies for compressive imaging

Felix Krahmer, Rachel Ward

Research output: Contribution to journalArticlepeer-review

112 Scopus citations

Abstract

In many signal processing applications, one wishes to acquire images that are sparse in transform domains such as spatial finite differences or wavelets using frequency domain samples. For such applications, overwhelming empirical evidence suggests that superior image reconstruction can be obtained through variable density sampling strategies that concentrate on lower frequencies. The wavelet and Fourier transform domains are not incoherent because low-order wavelets and low-order frequencies are correlated, so compressive sensing theory does not immediately imply sampling strategies and reconstruction guarantees. In this paper, we turn to a more refined notion of coherence - the so-called local coherence - measuring for each sensing vector separately how correlated it is to the sparsity basis. For Fourier measurements and Haar wavelet sparsity, the local coherence can be controlled and bounded explicitly, so for matrices comprised of frequencies sampled from a suitable inverse square power-law density, we can prove the restricted isometry property with near-optimal embedding dimensions. Consequently, the variable-density sampling strategy we provide allows for image reconstructions that are stable to sparsity defects and robust to measurement noise. Our results cover both reconstruction by $\ell1-minimization and total variation minimization. The local coherence framework developed in this paper should be of independent interest, as it implies that for optimal sparse recovery results, it suffices to have bounded average coherence from sensing basis to sparsity basis - as opposed to bounded maximal coherence - as long as the sampling strategy is adapted accordingly.

Original languageEnglish
Article number6651836
Pages (from-to)612-622
Number of pages11
JournalIEEE Transactions on Image Processing
Volume23
Issue number2
DOIs
StatePublished - Feb 2014
Externally publishedYes

Keywords

  • Compressive imaging
  • frequency
  • incoherence
  • local coherence
  • variable density sampling

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