Optimization-based algorithm for solving the discrete x-ray transform with nonlinear partial volume effect

Purpose: Inverting the discrete x-ray transform (DXT) with the nonlinear partial volume (NLPV) effect, which we refer to as the NLPV DXT, remains of theoretical and practical interest. We propose an optimization-based algorithm for accurately and directly inverting the NLPV DXT. Methods: Formulating the inversion of the NLPV DXT as a nonconvex optimization program, we propose an iterative algorithm, referred to as the nonconvex primal-dual (NCPD) algorithm, to solve the problem. We obtain the NCPD algorithm by modifying a first-order primal-dual algorithm to address the nonconvex optimization. Subsequently, we perform quantitative studies to verify and characterize the NCPD algorithm. Results: In addition to proposing the NCPD algorithm, we perform numerical studies to verify that the NCPD algorithm can reach the devised numerically necessary convergence conditions and, under the study conditions considered, invert the NLPV DXT by yielding numerically accurate image reconstruction. Conclusion: We have developed and verified with numerical studies the NCPD algorithm for accurate inversion of the NLPV DXT. The study and results may yield insights into the effective compensation for the NLPV artifacts in CT imaging and into the algorithm development for nonconvex optimization programs in CT and other tomographic imaging technologies.