Deriving the Algebraic Reconstruction Technique (ART) by the Method of Projections onto Convex Sets (POCS)

Abstract

To solve the reconstruction problem iteratively, the image is defined by the vector X and the projections, obtained in N views, by the vector Y. The weights of each pixel j in projections i are entered in matrix A. The image can thus be related to its projection by: Y =A X This equation can be more conveniently written as: y =xa i jij j ∑
for each i. In computerized tomography this system of linear equations is huge and usually underdetermined (128*128 unknowns for 128*64 equations per slice is typical), so it has many solutions X. A popular method to solve it is ART. The most important contribution of this paper is to have demonstrated that ART can be derived by POCS by considering each linear equation as a convex set. Since ART is POCS, additional convex constraints or a priori knowledge can be introduced in order to reduce the solution space. The constraints we have used are the positivity constraint and the support constraint (the image is zero outside a prescribed region). Since POCS converges to the point belonging to the intersection of all the convex sets which is the closest to the initial solution, the choice of the latter is also of importance.

The performance of this method will be illustrated with numerical and physical phantoms and on cardiac patients. It will also be compared to other methods (MART, SIRT, MLEM and OSEM).