A geometric approach to quadratic optimization: an improved method for solving strongly underdetermined systems in CT

Abstract

The problem of image reconstruction from projections in computerized tomography, when cast as a system of linear equations, leads to an inconsistent system. Hence, a common approach to this problem is to seek a solution that minimizes some optimization criterion, such as minimization of the residual (least-squares solution) by quadratic optimization. The QUAD algorithm is one well-studied method for quadratic optimization which applies the conjugate gradient to a certain so-called ''normal equations'' system derived from the original system. The concept of a geometric algorithm is introduced and defined as a method that is independent of any particular algebraic representation of the hyperplanes defined by the equations. An example of a geometric algorithm is the well known ART algorithm which, when implemented with a small relaxation parameter, achieves excellent results. Any non-geometric algorithm can be converted to a geometric one by normalizing the equations before applying the algorithm. This concept, when applied to QUAD, results in a geometric algorithm called NQUAD. ART, QUAD, and NQUAD were tested on various phantom reconstructions and evaluated with respect to image quality, convergence measures, and runtime efficiency. The results indicate that NQUAD is always preferable to QUAD, thus proving the usefulness of the geometric approach. Although ART is somewhat better than NQUAD in most standard cases, NQUAD is much better when the system is strongly underdetermined (a situation corresponding to low dose radiation), and this is even more pronounced with low-contrast images.

Efficient parallel implementation of iterative reconstruction algorithms for electron tomography

“…Quadratic optimization is one such method; it seeks to minimize the L 2 -norm of the residual; see [1,13,18]. Other optimization approaches are based on statistical considerations, such as maximum likelihood (ML) and expectation maximization (EM).…”

A Note on the Behavior of the Randomized Kaczmarz Algorithm of Strohmer and Vershynin

“…Clearly, Kaczmarz’s method is a geometric algorithm (this term was used by Gordon and Mansour [14]) in the sense that the sequence of iterates generated by it depends only on the hyperplanes defined by the equations and not on any particular algebraic representation of the hyperplanes. One can always use real nonzero numbers, say c i , i = 1, 2, …, m , and divide through each equation 〈 a i , x 〉 = b i , without affecting the hyperplanes defined by the equations and the solution set of the system.…”

Row scaling as a preconditioner for some nonsymmetric linear systems with discontinuous coefficients

“…Also, there is no study on the general usefulness of geometric scaling for nonsymmetric problems with discontinuous coefficients. The idea of geometric scaling was found to be useful for certain problems in image reconstruction from projections; see [7].…”