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

Dan Gordon, Rawia Mansour

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish
Pages (from-to)811-826
Number of pages16
JournalInverse Problems in Science and Engineering
Issue number8
StatePublished - Jan 2007


  • Computerized tomography
  • Conjugate gradient
  • Image reconstruction
  • Iterative algorithms
  • Least squares
  • Normal equations
  • Quadratic optimization

ASJC Scopus subject areas

  • General Engineering
  • Computer Science Applications
  • Applied Mathematics


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