A General-Thresholding Solution for ℓp (0 < p

Abstract

It is well known that ℓ1 minimization can be used to recover sufficiently sparse unknown signals in the compressive sensing field. The ℓp regularization method, a generalized version between the well-known ℓ1 regularization and the ℓ0 regularization, has been proposed for a sparser solution. In this paper, we derive several quasi-analytic thresholding representations for the ℓp(0 < p < 1) regularization. The derived representations are exact matches for the well-known soft-threshold filtering for the ℓ1 regularization and the hard-threshold filtering for the ℓ0 regularization. The error bounds of the approximate general formulas are analyzed. The general-threshold representation formulas are incorporated into an iterative thresholding framework for a fast solution of an ℓp regularized computed tomography (CT) reconstruction. A series of simulated and realistic data experiments are conducted to evaluate the performance of the proposed general-threshold filtering algorithm for CT reconstruction, and it is also compared with the well-known re-weighted approach. Compared with the re-weighted algorithm, the proposed general-threshold filtering algorithm can substantially reduce the necessary view number for an accurate reconstruction of the Shepp-Logan phantom. In addition, the proposed general-threshold filtering algorithm performs well in terms of image quality, reconstruction accuracy, convergence speed, and sensitivity to parameters.