Learning circulant sensing kernels

Yangyang Xu, Wotao Yin, Stanley Osher

Research output: Contribution to journalArticlepeer-review

12 Scopus citations


In signal acquisition, Toeplitz and circulant matrices are widely used as sensing operators. They correspond to discrete convolutions and are easily or even naturally realized in various applications. For compressive sensing, recent work has used random Toeplitz and circulant sensing matrices and proved their effciency in theory, by computer simulations, as well as through physical optical experiments. Motivated by recent work [8], we propose models to learn a circulant sensing matrix/operator for one and higher dimensional signals. Given the dictionary of the signal(s) to be sensed, the learned circulant sensing matrix/operator is more effective than a randomly generated circulant sensing matrix/operator, and even slightly so than a (non-circulant) Gaussian random sensing matrix. In addition, by exploiting the circulant structure, we improve the learning from the patch scale in [8] to the much large image scale. Furthermore, we test learning the circulant sensing matrix/operator and the nonparametric dictionary altogether and obtain even better performance. We demonstrate these results using both synthetic sparse signals and real images.

Original languageEnglish (US)
Pages (from-to)901-923
Number of pages23
JournalInverse Problems and Imaging
Issue number3
StatePublished - Aug 2014


  • Circulant matrix
  • Compressive sensing
  • Dictionary learning
  • Sensing kernel learn-ing
  • Sensing operator learning
  • Toeplitz matrix

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