A Geometric Approach to Quadratic Optimization in Computerized Tomography

The problem of image reconstruction from projections in computerized tomography, when cast as a system of linear equations, leads to an inconsistent system. The problem is studied under very adverse conditions, consisting of lowcontrast images and a strongly underdetermined system, where the number of equations is only about 25% of the number of variables (fewer equations enable the use of less radiation). Various algorithms are examined with respect to their performance in such cases: ART (Algebraic Reconstruction Technique), quadratic optimization (QUAD – a method that minimizes the L2-norm of the residual) and the relatively new component-averaging (CAV) and BICAV algorithms. A variant of QUAD, called NQUAD, is obtained by normalizing the equations before applying QUAD. This has a geometric significance since the resulting system is independent of any particular algebraic representation of the equations – a property shared by ART, CAV and BICAV. Experiments with phantom reconstructions show that NQUAD is always preferable to QUAD. Under the stated adverse conditions, NQUAD is much better than the other studied algorithms, in terms of image quality, runtime efficiency, and the achieved error measures.