MEDICAL IMAGING AND DIAGNOSTIC RADIOLOGY Received 7 October 2013; revised 17 December 2013; accepted 5 January 2014; Date of publication 16 January 2014; date of current version 28 January 2014. Digital Object Identifier 10.1109/JTEHM.2014.2300862 Constrained TpV Minimization for Enhanced Exploitation of Gradient Sparsity: Application to CT Image Reconstruction EMIL Y. SIDKY 1 (Member, IEEE), RICK CHARTRAND 2 (Senior Member, IEEE), JOHN M. BOONE 3 , AND XIAOCHUAN PAN 1 1 Department of Radiology, University of Chicago, Chicago, IL 60637, USA 2 Theoretical Division T-5, Los Alamos National Laboratory, Los Alamos, NM 87545, USA 3 Department of Radiology, University of California Davis Medical Center, Sacramento, CA 95817, USA CORRESPONDING AUTHOR: E. Y. SIDKY (sidky@uchicago.edu) The work of R. Chartrand was supported by the UC Laboratory Fees Research Program, and the U.S. Department of Energy through the LANL/LDRD Program. This work was supported by NIH R01 under Grants CA158446 (EYS), CA120540 (XP), EB000225 (XP), and EB002138 (JMB). ABSTRACT Exploiting sparsity in the image gradient magnitude has proved to be an effective means for reducing the sampling rate in the projection view angle in computed tomography (CT). Most of the image reconstruction algorithms, developed for this purpose, solve a nonsmooth convex optimization problem involving the image total variation (TV). The TV seminorm is the ℓ 1 norm of the image gradient magnitude, and reducing the ℓ 1 norm is known to encourage sparsity in its argument. Recently, there has been interest in employing nonconvex ℓ p quasinorms with 0<p<1 for sparsity exploiting image reconstruc- tion, which is potentially more effective than ℓ 1 because nonconvex ℓ p is closer to ℓ 0 —a direct measure of sparsity. This paper develops algorithms for constrained minimization of the total p-variation (TpV), ℓ p of the image gradient. Use of the algorithms is illustrated in the context of breast CT—an imaging modality that is still in the research phase and for which constraints on X-ray dose are extremely tight. The TpV-based image reconstruction algorithms are demonstrated on computer simulated data for exploiting gradient magnitude sparsity to reduce the projection view angle sampling. The proposed algorithms are applied to projection data from a realistic breast CT simulation, where the total X-ray dose is equivalent to two-view digital mammography. Following the simulation survey, the algorithms are then demonstrated on a clinical breast CT data set. INDEX TERMS Computed tomography, X-ray tomography, image reconstruction, iterative algorithms, optimization. I. INTRODUCTION Much research for iterative image reconstruction (IIR) in computed tomography (CT) has focused on exploiting gra- dient magnitude image (GMI) sparsity. Several theoretical investigations have demonstrated accurate CT image recon- struction from reduced data sampling employing various convex optimization problems involving total variation (TV) minimization [1]–[6]. Many of these algorithms have been adapted to use on actual scanner data for sparse-view CT [7]– [12] or gated/dynamic CT [7], [13]–[17]. While the volume of work on this topic speaks to the success of the idea of exploiting GMI sparsity, TV minimization is not the most direct method for taking advantage of this prior. The most direct measure of sparsity is totaling the number of nonzero pixels in an image. Mathematically, the number of nonzero components of a vector can be expressed as the ℓ 0 norm, which is understood to be the limit as p goes to zero of the pth power of the ℓ p norm: ‖v‖ p p ≡  i |v i | p . (1) As of yet, no algorithms have been developed for CT IIR that minimize ℓ 0 of the GMI, and sparsity exploiting IIR has focused on minimizing ℓ 1 of the GMI – also known as TV. Logically, p < 1 should improve on exploitation of GMI sparsity for sampling reduction, but optimization problems VOLUME 2, 2014 2168-2372  2014 IEEE. Translations and content mining are permitted for academic research only. Personal use is also permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. 1800418