Numerical Solution of Underdetermined Systems from Partial Linear Circulant Measurements
11th International Conference on Sampling Theory and Applications, Washington, DC, May 25-29, 2015
Sparse matrices, Algorithm design and analysis, Optimization, Compressed sensing, Matching pursuit algorithms, Noise, Eigenvalues and egofunctions
We consider the traditional compressed sensing problem of recovering a sparse solution from undersampled data. We are in particular interested in the case where the measurements arise from a partial circulant matrix. This is motivated by practical physical setups that are usually implemented through convolutions. We derive a new optimization problem that stems from the traditional ℓ 1 minimization under constraints, with the added information that the matrix is taken by selecting rows from a circulant matrix. With this added knowledge it is possible to simulate the full matrix and full measurement vector on which the optimization acts. Moreover, as circulant matrices are well-studied it is known that using Fourier transform allows for fast computations. This paper describes the motivations, formulations, and preliminary results of this novel algorithm, which shows promising results.
Bouchot, Jean-Luc and Cao, Lei, "Numerical Solution of Underdetermined Systems from Partial Linear Circulant Measurements" (2015). Mathematics Faculty Proceedings, Presentations, Speeches, Lectures. 396.