First-order convex feasibility algorithms for x-ray CT

• Published in: Medical physics (Lancaster) Vol. 40; no. 3; pp. 031115 - n/a
• Main Authors: Sidky, Emil Y., Jørgensen, Jakob S., Pan, Xiaochuan
• Format: Journal Article
• Language: English
• Published: United States American Association of Physicists in Medicine 01.03.2013
• Subjects: ALGORITHMS; APPROXIMATIONS; Computed tomography; Computerised tomographs; computerised tomography; COMPUTERIZED TOMOGRAPHY; convex optimization; Convex sets; DESIGN; diagnostic radiography; Digital computing or data processing equipment or methods, specially adapted for specific applications; Feasibility Studies; Image data processing or generation, in general; IMAGE PROCESSING; Image Processing, Computer-Assisted - methods; image reconstruction; INSTRUMENTATION RELATED TO NUCLEAR SCIENCE AND TECHNOLOGY; iterative image reconstruction; ITERATIVE METHODS; LEAST SQUARE FIT; least squares approximations; Linear equations; Medical image noise; medical image processing; Medical image quality; Medical image reconstruction; Medical imaging; MINIMIZATION; Numerical approximation and analysis; Numerical optimization; Online Only; optimisation; Optimization; Quality Control; Radiography; RADIOLOGY AND NUCLEAR MEDICINE; Signal-To-Noise Ratio; SIMULATION; Singular values; Tomography, X-Ray Computed - methods; X RADIATION; computed tomography; convex optimization; iterative image reconstruction
• ISSN: 0094-2405; 2473-4209; 1522-8541; 2473-4209; 0094-2405
• DOI: 10.1118/1.4790698

Purpose: Iterative image reconstruction (IIR) algorithms in computed tomography (CT) are based on algorithms for solving a particular optimization problem. Design of the IIR algorithm, therefore, is aided by knowledge of the solution to the optimization problem on which it is based. Often times, however, it is impractical to achieve accurate solution to the optimization of interest, which complicates design of IIR algorithms. This issue is particularly acute for CT with a limited angular-range scan, which leads to poorly conditioned system matrices and difficult to solve optimization problems. In this paper, we develop IIR algorithms which solve a certain type of optimization called convex feasibility. The convex feasibility approach can provide alternatives to unconstrained optimization approaches and at the same time allow for rapidly convergent algorithms for their solution—thereby facilitating the IIR algorithm design process. Methods: An accelerated version of the Chambolle−Pock (CP) algorithm is adapted to various convex feasibility problems of potential interest to IIR in CT. One of the proposed problems is seen to be equivalent to least-squares minimization, and two other problems provide alternatives to penalized, least-squares minimization. Results: The accelerated CP algorithms are demonstrated on a simulation of circular fan-beam CT with a limited scanning arc of 144°. The CP algorithms are seen in the empirical results to converge to the solution of their respective convex feasibility problems. Conclusions: Formulation of convex feasibility problems can provide a useful alternative to unconstrained optimization when designing IIR algorithms for CT. The approach is amenable to recent methods for accelerating first-order algorithms which may be particularly useful for CT with limited angular-range scanning. The present paper demonstrates the methodology, and future work will illustrate its utility in actual CT application.