Sparse deconvolution via off-grid T.V minimization

• Published in: Signal processing Vol. 170; p. 107406
• Main Authors: Mahata, Kaushik, Hyder, Md Mashud
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01-05-2020
• Subjects: Prolate spheroidal wave functions; Sparse deconvolution; Super resolution; Sparse deconvolution; Prolate spheroidal wave functions; Super resolution
• ISSN: 0165-1684; 1872-7557
• DOI: 10.1016/j.sigpro.2019.107406

•A grid-less solution is proposed for sparse deconvolution problem.•The algorithm works in time domain, thereby overcome the leakage error that generally affects frequency domain algorithms.•We demonstrate its utility via numerical simulation studies as well as real world signal reconstruction examples.
We present a grid-less solution to the total variation minimization approach to the sparse deconvolution problem. Unlike the existing methods [1, 2], our algorithm works directly with time domain data, and does not suffer from the consequences of leakage errors incurred while transforming the time domain measurements to frequency domain. The total variation minimization problem requires us to optimize the total variation of a function. This is an infinite dimensional problem. To deal with the underlying infinite dimensionality, we consider its finite dimensional semi-infinite duals with point-wise constraints. We provide a finite parameterization to handle the point-wise constraints in a computationally tractable manner. This parameterization exploits the underlying bandlimitedness of the convolution kernel by employing Prolate Spheroidal Wave Functions. Consequently, we are able to reduce the total variation minimization problem into a semidefinite program, which can be solved in polynomial time. We demonstrate its utility via numerical simulation studies as well as real world signal reconstruction examples.