Efficient recovery of smooth functions which are s-sparse with respect to the basis of so-called prolate spheroidal wave functions from a small number of random sampling points is considered. The main ingredient in the design of both the algorithms we propose here consists in establishing a uniform L? bound on the measurement ensembles which constitute the columns of the sensingmatrix. Such a bound provides us with the restricted isometry property for this rectangular random matrix, which leads to either the exact recovery property or the "best s-term approximation" of the original signal by means of the 1 minimization program. The first algorithm considers only a restricted number of columns for which the L? holds as a consequence of the fact that eigenvalues of the Bergman's restriction operator are close to 1 whereas the second one allows for a wider system of PSWF by taking advantage of a preconditioning technique. Numerical examples are spread throughout the text to illustrate the results.
Compressed sensing with preconditioning for sparse recovery with subsampled matrices of Slepian prolate functions
Laurent Gosse
2013
Abstract
Efficient recovery of smooth functions which are s-sparse with respect to the basis of so-called prolate spheroidal wave functions from a small number of random sampling points is considered. The main ingredient in the design of both the algorithms we propose here consists in establishing a uniform L? bound on the measurement ensembles which constitute the columns of the sensingmatrix. Such a bound provides us with the restricted isometry property for this rectangular random matrix, which leads to either the exact recovery property or the "best s-term approximation" of the original signal by means of the 1 minimization program. The first algorithm considers only a restricted number of columns for which the L? holds as a consequence of the fact that eigenvalues of the Bergman's restriction operator are close to 1 whereas the second one allows for a wider system of PSWF by taking advantage of a preconditioning technique. Numerical examples are spread throughout the text to illustrate the results.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.