Reconstructing pictures from projections: On the convergence of the ART algorithm with relaxation

Trummer, Manfred R.

doi:10.1007/bf02243477

Cited by 31 publications

(13 citation statements)

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“…Kaczmarz's method is well understood even in the infeasible case; we refer the interested reader to Tanabe's [96] and Trummer's [97,99]. The iteration described in Example 6.21 is also known as "ART" (algebraic reconstruction technique).…”

Section: (*)mentioning

confidence: 99%

On Projection Algorithms for Solving Convex Feasibility Problems

1996

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Due to their extraordinary utility and broad applicability in many areas of classical mathematics and modem physical sciences (most notably, computerized tomography), algorithms for solving convex feasibility problems continue to receive great attention. To unify, generalize, and review some of these algorithms, a very broad and flexible framework is investigated. Several crucial new concepts which allow a systematic discussion of questions on behaviour in general Hilbert spaces and on the quality of convergence are brought out. Numerous examples are given.

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Section: (*)mentioning

confidence: 99%

On Projection Algorithms for Solving Convex Feasibility Problems

1996

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“…The convergence speed of the Kaczmarz method was investigated in many papers [20][21]. It mainly depends on the orthogonality of the system matrix rows.…”

Section: Resultsmentioning

confidence: 99%

Reconstruction of High Resolution Computed Tomography Image from Sinogram Space Using Adaptive Row Projection

Omer

2014

IJBSBT

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“…It is shown in [11] that the addition of a small relaxation parameter into ART produces excellent results. The convergence of ART with relaxation parameters, for consistent systems, has been shown by Herman et al [13] and by Trummer [23]. Tanabe [22] has shown that when the system is inconsistent, ART converges cyclically, i.e., for each hyperplane, the sequence of projections on that hyperplane converges to a limit.…”

Section: Introductionmentioning

confidence: 93%

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

Gordon

2007

Inverse Problems in Science and Engineering

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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.

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Reconstructing pictures from projections: On the convergence of the ART algorithm with relaxation

Cited by 31 publications

References 4 publications

On Projection Algorithms for Solving Convex Feasibility Problems

On Projection Algorithms for Solving Convex Feasibility Problems

Reconstruction of High Resolution Computed Tomography Image from Sinogram Space Using Adaptive Row Projection

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

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