Abstract
We consider in this paper the problem of reconstructing 3D Computed Tomography images from limited data. The problem is modeled as a nonnegatively constrained minimization problem of very large size. In order to obtain an acceptable image in short time, we propose a scaled gradient projection method, accelerated by exploiting a suitable scaling matrix and efficient rules for the choice of the step-length. In particular, we select the step-length either by alternating Barzilai-Borwein rules or by exploiting a limited number of back gradients for approximating second-order information. Numerical results on a 3D Shepp-Logan phantom are presented and discussed.
Highlights
The numerical model3. The Scaled Gradient Projection (SGP) algorithm for Computed Tomography (CT) image reconstruction Due to the possible huge size of the matrix A and the very simple constraints of problem (2), firstorder algorithms exploiting only the gradient of f (x) and the projection onto the feasible region are very appealing approaches
3D Computed Tomography (CT) is a well known technique used in different areas, such as medicine, industry or art, to visually represent the interior of a 3D object
The red and blue lines refer to the scaled methods with the step-length based on the BB rules (SGP BB) and the Ritz-like values (SGP R), respectively; the black and green lines denote the non-scaled methods (GP BB and GP R, respectively), that is, the Scaled Gradient Projection (SGP) versions with Dk equal to the identity matrix at each iteration
Summary
3. The SGP algorithm for CT image reconstruction Due to the possible huge size of the matrix A and the very simple constraints of problem (2), firstorder algorithms exploiting only the gradient of f (x) and the projection onto the feasible region are very appealing approaches. The SGP algorithm for CT image reconstruction Due to the possible huge size of the matrix A and the very simple constraints of problem (2), firstorder algorithms exploiting only the gradient of f (x) and the projection onto the feasible region are very appealing approaches Among these algorithms, the gradient projection methods are the most popular and, thanks to recent ideas for accelerating their convergence rate, they have given rise to very effective image reconstruction algorithms in many different areas.
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