Mathematical Aspects of Computerized Tomography pp 160-178 | Cite as

# Continuous and Semicontinuous Analogues of Iterative Methods of Cimmino and Kaczmarz with Applications to the Inverse Radon Transform

## Abstract

*Kaczmarz’s* method and variants thereof have been used effectively in the numerical resolution of linear algebraic equations arising from tomography and other areas of reconstruction from projections. The method is applied for example, to the system of equations arising from *sampling* and *full discretization* of the *Radon transform*. In contrast, *Cimmino’s method* which has universal convergence properties similar (in theory) to Kaczmarz’s method, does not seem to have as extensively studied or advocated in practice. In this paper, we develop *continuous* and/or *semicontinuous* analogues of the iterative methods of Cimmino and Kaczmarz for linear operator equations on *infinite dimensional function spaces*. The formulation of these algorithms hinges upon expressing the operator equation in terms of a family of hyperplanes in an appropriate function space. We identify and analyze two wide classes of linear operators for which this is possible. The semicontinuous analogue which involves a *finite* number of these hyperplanes is studied in particular in the framework of moment discretization (rather than full discretization) which is germane to problems of integral and operator equations with discrete data or sampling. Convergence properties of variants of Cimmino’s method are established; continuous and semicontinuous analogues are developed in a manner that permits convergence analysis in a similar manner. Several examples are given.

## Keywords

Iterative Method Operator Equation Linear Algebraic Equation Generalize Inverse Algebraic Reconstruction Technique## Preview

Unable to display preview. Download preview PDF.

## References

- [1]E. Artzy, T. Elfving, and G. T. Herman, Quadratic Optimization for image reconstruction, II, Computer Graphics and Image Processing 11 (1979), 242–261.CrossRefGoogle Scholar
- [2]A. Ben-Israel and T. Greville, Generalized Inverses: Theory and Applications, Wiley-Interscience, New York, 1974.MATHGoogle Scholar
- [3]L. Cesari, Sulla risoluzione del sistemi di equazioni lineari per approssimazioni successive, Rend. R. Accad. Naz. Lincei CI. Sci. Fis. Math. Nat., Ser. 6A, 25, Rome, 1937.Google Scholar
- [4]G. Cimmino, Calcolo approssimato per le soluzioni dei sistemi di equazioni lineari, “La Ricerca Scientifica”, Roma, Serie II, (1938), 326–333.Google Scholar
- [5]G. Cimmino, Su uno speciale tipo di metodi probabilistici in analisi numerica, Symposia Mathematica, Vol, X, Institute Nazionale di Alta Matematica, 247–254, Academic Press, London and New York, 1972.Google Scholar
- [6]P. P. B. Eggermont, G. T. Herman, and A. Lent, Iterative algorithms for large partitioned linear systems, with applications to image reconstruction, to appear.Google Scholar
- [7]N. Gastinel, Linear Numerical Analysis, Herman and Academic Press, Paris and New York, 1970.Google Scholar
- [8]R. Gordon and G. T. Herman, Three-dimensional reconstructions from projections, a review of algorithms, Int. Review of Cytology 38 (1974), 111–151.CrossRefGoogle Scholar
- [9]C. W. Groetsch, Generalized Inverses of Linear Operators: Representation and Approximation, Dekker, New York, 1977.MATHGoogle Scholar
- [10]G. T. Herman, ed., Image Reconstruction from Projections: Implementation and Applications, Springer Verlag, Berlin, 1979.Google Scholar
- [11]G. T. Herman and A. Naparstek, Past image reconstruction based on a Radon inversion formula appropriate for rapidly collected data, SIAM J. Applied Math. 33 (1977), 511–533.CrossRefMATHMathSciNetGoogle Scholar
- [12]F. Jossa, Risoluzione progressiva di un sistema di equazioni lineari. Analogia con un problema meccanio, Rend. Accad. Sci. Fis. Mat. Napoli (4) 10 (1940), 346–352.MATHMathSciNetGoogle Scholar
- [13]S. Kaczmarz, Angenäherte Auflosing von Systemen linearer Gleichungen, Bull. Acad. Polon. Sciences et Lettres, A, (1937), 355–357.Google Scholar
- [14]W. J. Kammerer and M. Z. Nashed, Steepest descent for singular linear operators with nonclosed range, Applicable Analysis 1 (1971), 143–159.CrossRefMATHMathSciNetGoogle Scholar
- [151.W. J, Kammerer and M. Z. Nashed, Convergence of the conjugate gradient method for singular linear operator equations, SIAM J, Numer. Anal. 9 (1971), 165–181.CrossRefMathSciNetGoogle Scholar
- [16]W. J. Kammerer and M. Z. Nashed, Iterative methods for best approximate solutions of linear integral equations of the first and second kinds, J. Math. Anal. Appl. 40 (1972), 547–573.CrossRefMATHMathSciNetGoogle Scholar
- [17]W. J. Kammerer and M. Z. Nashed, A generalization of a matrix iterative method of G. Cimmino to best approximate solution of linear integral equations of the first kind, Rend. Accad. dei Lincei 48 (1970), 184–194.Google Scholar
- [18]H. B. Keller, The solution of singular and semidefinite linear systems by iteration, SIAM J. Numer. Anal. 2 (1965), 281–290.Google Scholar
- [19]S. F. McCormick and G. H. Rodrigue, A uniform approach to gradient methods for linear operator equations, J. Math. Anal. Appl. 49 (1975), 275–285.CrossRefMATHMathSciNetGoogle Scholar
- [20]M. Z. Nashed, On moment discretization and least-squares solutions of linear integral equations of the first kind, J. Math. Anal. Appl. 53 (1976), 359–366.CrossRefMATHMathSciNetGoogle Scholar
- [21]M. Z. Nashed, ed., Generalized Inverses and Applications, Academic Press, New York, 1976,MATHGoogle Scholar
- [22]M. Z. Nashed, Steepest descent for singular linear operator equations, SIAM J. Numer. Anal, 7 (1970), 479–492.CrossRefMathSciNetGoogle Scholar
- [23]M. Z. Nashed, Generalized inverses, normal solvability, and iterations for singular operator equations, in Nonlinear Functional Analysis and Applications (L. B. Hall, ed.), pp. 311–359, Academic Press, New York, 1971.Google Scholar
- [24]M. Z. Nashed, Perturbations and approximations for generalized inverses and linear operator equations, in [21], pp. 325–396.Google Scholar
- [25]M. Z. Nashed and L. B. Rail, Annotated bibliography on generalized inverses and applications, in [21], pp. 771–1041.Google Scholar
- [26]M. Z. Nashed and G. F. Votruba, A unified operator theory of generalized inverses, in [21], pp. 1–109.Google Scholar
- [27]M. Z. Nashed and G. F. Votruba, Convergence of a class of iterative methods of Cimmino-type to weighted least squares solutions, Notices Amer. Math. Soc., 21 (1974), A-245é.Google Scholar
- [28]M. Z. Nashed and G, Wahba, Rates of convergence of approximate least squares solutions of linear integral and operator equations, Math. Comp. 28 (1974), 69–80.CrossRefMATHMathSciNetGoogle Scholar
- [29]M. Z. Nashed and G. Wahba, Generalized inverses in reproducing kernel spaces: An approach to regularization of linear operator equations, SIAM J. Math. Anal. 5 (1974), 974–987.CrossRefMathSciNetGoogle Scholar
- [30]M. Z. Nashed and G. Wahba, Regularization and approximation of linear operator equations in reproducing kernel spaces, Bull. Amer. Math. Soc. 80 (1974), 1213–1218.CrossRefMATHMathSciNetGoogle Scholar
- [31]F. Natterer, A. Sobolev space analysis of picture reconstruction,
*SIAM J. Appl. Math*., to appear,Google Scholar - [32]W. V. Petryshyn, On generalized inverses and uniform convergence of (I-3K)n with applications to iterative methods, J. Math. Anal. Appl, 18 (1967), 417–439.CrossRefMATHMathSciNetGoogle Scholar
- [33]D. Showalter and A. Ben-Israel, Representation and computation of the generalized inverse of a bounded linear operator between two Hilbert spaces, Rend. Accad. dei Lincei 48 (1970), 184–194.MATHMathSciNetGoogle Scholar
- [34]K. T. Smith, D. C. Solomon and S. L. Wagner, Practical and mathematical aspects of the problem of reconstructing objects from radiographs, Bull. Amer. Math. Soc. 83 (1977), 1227–1270.CrossRefMATHMathSciNetGoogle Scholar
- [35]K. Tanabe, Projection method for solving a singular system of linear equations and its applications, Numer. Math. 17 (1971), 203–214.CrossRefMATHMathSciNetGoogle Scholar
- [36]R. S. Varga, Matrix Iterative Analysis, Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1962.Google Scholar