# Parallel application of block-iterative methods in medical imaging and radiation therapy

## Abstract

Some row-action algorithms which exploit special objective function and constraints structure have proven advantageous for solving huge and sparse feasibility or optimization problems. Recently developed block-iterative versions of such special-purpose methods enable parallel computation when the underlying problem is appropriately decomposed. This opens the door for parallel computation in image reconstruction problems of computerized tomography and in the inverse problem of radiation therapy treatment planning, all in their fully discretized modelling approach. Since there is more than one way of deriving block-iterative versions of any row-action method, the choice has to be made with reference to the underlying real-world problem.

## Key words

Full discretization computerized tomography image reconstruction radiotherapy treatment planning block-iterative algorithms parallel computations Cimmino's algorithm entropy maximization classification of algorithms## Preview

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## References

- [1]R. Aharoni, A. Berman and Y. Censor, “An interior points algorithm for the convex feasibility problem,”
*Advances in Applied Mathematics*4 (1983) 479–489.Google Scholar - [2]S. Agmon, “The relaxation method for linear inequalities,”
*Canadian Journal of Mathematics*6 (1954) 382–392.Google Scholar - [3]M.D. Altschuler, Y. Censor, P.P.B. Eggermont, G.T. Herman, Y.H. Kuo, R.M. Lewitt, M. McKay, H.K. Tuy, J.K. Udupa and M.M. Yau, “Demonstration of a software package for the reconstruction of the dynamically changing structure of the human heart from cone beam X-ray projections,”
*Journal of Medical Systems*4 (1980) 289–304.Google Scholar - [4]A. Auslender,
*Optimisation: Methods Numeriques*(Masson, Paris, 1976).Google Scholar - [5]L.M. Bregman, “The relaxation method of finding the common point of convex sets and its application to the solution of problems in convex programming,”
*USSR Computational Mathematics and Mathematical Physics*7 (1967) 200–217.Google Scholar - [6]Y. Censor, “Row-action methods for huge and sparse systems and their applications,”
*SIAM Review*23 (1981) 444–466.Google Scholar - [7]Y. Censor, “Finite series-expansion reconstruction methods,”
*Proceedings of the IEEE*71 (1983) 409–419.Google Scholar - [8]Y. Censor, “An automatic relaxation method for solving interval linear inequalities,”
*Journal of Mathematical Analysis and Applications*105 (1985) 19–25.Google Scholar - [9]Y. Censor and A. Lent, “An iterative row-action method for interval convex programming,”
*Journal of Optimization Theory and Applications*34 (1981) 321–353.Google Scholar - [10]Y. Censor and A. Lent, “Cyclic subgradient projections,”
*Mathematical Programming*24 (1982) 233–235.Google Scholar - [11]Y. Censor and T. Elfving, “New methods for linear inequalities,”
*Linear Algebra and Its Applications*42 (1982) 199–211.Google Scholar - [12]Y. Censor, P.P.B. Eggermont and D. Gordon, “Strong underrelaxation in Kaczmarz's method for inconsistent systems,”
*Numerische Mathematik*41 (1983) 83–92.Google Scholar - [13]Y. Censor and G.T. Herman, “On some optimization techniques in image reconstruction from projections,”
*Applied Numerical Mathematics*3 (1987) 365–391.Google Scholar - [14]Y. Censor, W. D. Powlis and M.D. Altschuler, “On the fully discretized model for the inverse problem of radiation therapy treatment planning,” in: K.R. Foster, ed.,
*Proceedings of the Thirteenth Annual Northeast Bioengineering Conference*, Vol. 1 (Institute of Electrical and Electronics Engineers, Inc., New York, 1987) pp. 211–214.Google Scholar - [15]Y. Censor and A. Lent, “Optimization of ‘log
*x*’ entropy over linear equality constrains,”*SIAM Journal on Control and Optimization*25 (1987) 921–933.Google Scholar - [16]Y. Censor, M.D. Altschuler and W.D. Powlis, “A computational solution of the inverse problem in radiation therapy treatment planning,”
*Applied Mathematics and Computation*25 (1988) 57–87.Google Scholar - [17]Y. Censor and J. Segman, “On block-iterative entropy maximization,”
*Journal of Information and Optimization Sciences*8 (1987) 275–291.Google Scholar - [18]Y. Censor, A.R. DePierro and A.N. Iusem, “On maximization of entropies and a generalization of Bregman's method for convex programming,” Technical Report MIPG 113, September 1986.Google Scholar
- [19]Y. Censor, M.D. Altschuler and W.D. Powlis, “On the use of Cimmino's simultaneous projections methods for computing a solution of the inverse problem in radiation therapy treatment planning,”
*Inverse Problems*(1988), in print.Google Scholar - [20]Y. Censor, A.R. DePierro, T. Elfving, G.T. Herman and A.N. Iusem, “On iterative methods for linearly constrained entropy maximization,” in: A. Wakulicz, ed.,
*Numerical Analysis and Mathematical Modelling*, Banach Center Publications, Vol. XXIV, Stefan Banach International Mathematical Center, Warsaw, 1988, in print.Google Scholar - [21]G. Cimmino, “Calcolo Approssimato per le Soluzioni dei Sistemi di Equazioni Lineari,”
*La Ricerca Scientifica*, Roma XVI, Ser. II, Anno IX, Vol. 1, pp. 326–333, 1938.Google Scholar - [22]J.N. Darroch and D. Ratcliff, “Generalized iterative scaling for log-linear models,”
*The Annals of Mathematical Statistics*43 (1972) 1470–1480.Google Scholar - [23]A.R. De Pierro and A.N. Iusem, “A simultaneous projection method for linear inequalities,”
*Linear Algebra and Its Applications*64 (1985) 243–253.Google Scholar - [24]P.P.B. Eggermont, G.T. Herman and A. Lent, “Iterative algorithms for large partitioned linear systems, with applications to image reconstruction,”
*Linear Algebra and Its applications*40 (1981) 37–67.Google Scholar - [25]T. Elfving, “Block-iterative methods for consistent and inconsistent linear equations,”
*Numerische Mathematik*35 (1980) 1–12.Google Scholar - [26]I.I. Eremin, “On some iterative methods in convex programming,”
*Ekonomika i Matematichesky Methody*2 (1966) 870–886 (In Russian).Google Scholar - [27]I.I. Eremin, “Fejer mappings and convex programming,”
*Siberian Mathematical Journal*10 (1969) 762–772.Google Scholar - [28]A. George, G.W. Stewart and R. Voigt (Editors), “Special Volume of Linear Algebra and Its Applications on Parallel Computing,”
*Linear Algebra and Its Applications*, Vol. 77, May 1986.Google Scholar - [29]P. Gilbert, “Iterative methods for the three-dimensional reconstruction of an object from projections,”
*Journal of Theoretical Biology*36 (1972) 105–117.Google Scholar - [30]R. Gordon, R. Bender and G.T. Herman, “Algebraic reconstruction techniques (ART) for three-dimensional electron microscopy and X-ray photography,”
*Journal of Theoretical Biology*29 (1970) 471–481.Google Scholar - [31]R. Gordon and G.T. Herman, “Three-dimensional reconstruction from projections: A review of algorithms,”
*International Review of Cytology*38 (1974) 111–151.Google Scholar - [32]L.G. Gubin, B.T. Polyak and E.V. Raik, “The method of projections for finding the common point of convex sets,”
*USSR Computational Mathematics and Mathematical Physics*7 (1967) 1–24.Google Scholar - [33]G.T. Herman, “A relaxation method for reconstructing objects from noisy X-rays,”
*Mathematical Programming*8 (1975) 1–19.Google Scholar - [34]G.T. Herman,
*Image Reconstruction from Projections: The Fundamentals of Computerized Tomography*(Academic Press, New York, 1980).Google Scholar - [35]G.T. Herman and A. Lent, “Iterative reconstruction algorithms,”
*Computers in Biology and Medicine*6 (1976) 273–294.Google Scholar - [36]G.T. Herman, A. Lent and P.H. Lutz, “Iterative relaxation methods for image reconstruction,”
*Communications of the ACM*21 (1978) 152–158.Google Scholar - [37]G.T. Herman and A. Lent, “A family of iterative quadratic optimization algorithms for pairs of inequalities with application in diagnostic radiology,”
*Mathematical Programming Study*9 (1978) 15–29.Google Scholar - [38]G.T. Herman, H. Levkowitz, H.K. Tuy and S. McCormick, “Multilevel image reconstruction,” in: A. Rosenfeld, ed.,
*Multiresolution Image Processing and Analysis*(Springer-Verlag, Berlin, Heidelberg, New York, Tokyo, 1984) pp. 121–135.Google Scholar - [39]G.T. Herman and H. Levkowitz, “Initial performance of block-iterative reconstruction algorithms,” in M.A. Viergever and A. Todd-Porkopek, eds.,
*Mathematics and Computer Science of Medical Imaging*(Springer-Verlag, Berlin, Heidelberg, New York, Tokyo, 1987) pp. 305–318.Google Scholar - [40]C. Hildreth, “A quadratic programming procedure,”
*Naval Research Logistics Quarterly*4 (1957) 79–85. Erratum,*Ibid*, p. 361.Google Scholar - [41]E.B. Hinkle, J.L.C. Sanz, A.K. Jain and D. Petkovic, “
*P*^{3}*E*: New Life for projection-based image processing,”*Jorunal of Parallel and Distributed Computing*4 (1987) 45–78.Google Scholar - [42]A.N. Iusem and A.R. De Pierro, “Convergence results for an accelerated nonlinear Cimmino algorithm,”
*Numerische Mathematik*49 (1986) 367–378.Google Scholar - [43]A.N. Iusem and A.R. De Pierro, “A simultaneous iterative method for computing projections on polyhedra,”
*SIAM Journal of Control and Optimization*25 (1987) 231–243.Google Scholar - [44]L.H. Jamieson and S.L. Tanimoto, “Special issue on parallel image processing and pattern recognition: Guest editors' introduction,”
*Journal of Parallel and Distributed Computing*, 4 (1987) 1–6.Google Scholar - [45]S. Kaczmarz, “Angenäherte Auflösung von Systemen Linearer Gleichungen,”
*Bull. Acad. Polon. Sci. Lett. A*. 35 (1937) 355–357.Google Scholar - [46]A.V. Lakshminarayanan and A. Lent, “Methods of least squares and SIRT in reconstruction,”
*Journal of Theoretical Biology*76 (1979) 267–295.Google Scholar - [47]A. Lent, “A convergent algorithm for maximum entropy image restoration with a medical X-ray application,” in: R. Shaw, ed.,
*Image Analysis and Evaluation*(Society of Photographic Scientists and Engineers, SPSE, Washington, DC, 1977), pp. 249–257.Google Scholar - [48]A. Lent, Private discussions, July 1987.Google Scholar
- [49]A. Lent and Y. Censor, “Extensions of Hildreth's row-action method for quadratic programming,”
*SIAM Journal on Control and Optimization*18 (1980) 444–454.Google Scholar - [50]R.M. Lewitt, “Reconstruction algorithms: Transform methods,”
*Proceedings of the IEEE*71 (1983) 390–408.Google Scholar - [51]F.A. Lootsma and K.M. Ragsdell, “State-of-the-art in parallel nonlinear optimization,”
*Parallel Computing*, 6 (1988) 133–155.Google Scholar - [52]J.M. Martinez and R.J.B. De Sampaio, “Parallel and sequential Kaczmarz methods for solving underdetermined nonlinear equations,”
*Journal of Computational and Applied Mathematics*15 (1986) 311–321.Google Scholar - [53]S.F. McCormick, “The method of Kaczmarz and row orthogonalization for solving linear equations and least squares problems in Hilbert space,”
*Indiana University Mathematics Journal*26 (1977) 1137–1150.Google Scholar - [54]Y.I. Merzlyakov, “On a relaxation method of solving systems of linear inequalities,”
*USSR Computational Mathematics and Mathematical Physics*3 (1963) 504–510.Google Scholar - [55]T.S. Motzkin and I.J. Schoenberg, “The relaxation method for linear inequalities,”
*Canadian Journal of Mathematics*6 (1954) 393–404.Google Scholar - [56]F. Natterer,
*The Mathematics of Computerized Tomography*(B.G. Teubner, Stuttgart, 1986).Google Scholar - [57]L.A. Shepp and Y. Vardi, “Maximum likelihood reconstruction in positron emission tomography,”
*IEEE Transactions on Medical Imaging MI-1*(1982) 113–122.Google Scholar - [58]K. Tanabe, “Projection method for solving a singular system of linear equations and its applications,”
*Numerische Mathematik*17 (1971) 203–214.Google Scholar - [59]P. Tseng and D.P. Bertsekas, “Relaxation methods for problems with strictly convex separable costs and linear constraints,”
*Mathematical Programming*38 (1987) 303–321.Google Scholar - [60]Y. Vardi, L.A. Shepp and L. Kaufman, “A statistical model for positron emission tomography,”
*Journal of the American Statistical Association*80 (1985) 8–37.Google Scholar - [61]S.A. Zenios, Private discussions, April 1987.Google Scholar
- [62]A.R. De Pierro and A.N. Iusem, “A relaxed version of Bregman's method for convex programming“,
*Journal of Optimization Theory and Applications*51 (1986) 421–440.Google Scholar - [63]R. Aharoni and Y. Censor, “Block-iterative projection methods for parallel computation of solutions to convex feasibility problems”, Technical Report, June 1988.Google Scholar