Abstract
Given a collection of closed subspaces of a Hilbert space, the method of alternating projections produces a sequence which converges to the orthogonal projection onto the intersection of the subspaces. A large class of problems in medical and geophysical image reconstruction can be solved using this method. A sharp error bound will enable the userto estimate accurately the number of iterations necessary to achieve a desired relative error. We obtain the sharpest possible upper bound for the case of two subspaces, and the sharpest known upper bound for more than two subspaces.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
[AA] V. M. Adamyan and D. Z. Arov, A general solution of a problem in linear prediction of stationary processes,Theory Probab. Appl.,13 (1968), 394–407.
[A] N. Aronszain, Theory of reproducing kernels,Trans. Amer. Math. Soc.,68 (1950), 337–404.
[C] Y. Censor, Finite series-expansion reconstruction methods,Proc. IEEE,71 (1983), 409–419.
F. Deutsch, Applications of von Neumann's alternating projections algorithm, inMathematical Methods in Operations Research (P. Kendrov, ed.), pp. 44–51, Sofia, 1983.
[D2] F. Deutsch, Rate of convergence of the method of alternating projections, inParametric Optimization and Approximation (B. Brosowski and F. Deutsch, eds.), pp. 96–107, Birkhauser, Basel, 1985.
C. Franchetti and W. Light, On the von Neumann Alternating Algorithm in Hilbert Space, Report 28, Center for Approximation Theory, Texas A&M University, 1982.
[F] K. Friedrichs, On certain inequalities and characteristic value problems for analytic functions and for functions of two variables,Trans. Amer. Math. Soc.,41 (1937), 321–364.
[H] I. Halperin, The product of projection operators,Acta Sci. Math.,23 (1962), 96–99.
[HS] C. Hamaker and D. C. Solmon, The angles between the null spaces of X-rays,J. Math. Anal. Appl.,62 (1978), 1–23.
[I] S. Ivansson, Seismic borehole tomography—theory and computational methods,Proc. IEEE,74 (1986), 328–338.
[J] C. Jordan, Essay on the geometry ofn dimensions,Bull. Soc. Math. France,3 (1875), 103–174.
S. Kaczmarz, Angenaherte auflosung von systemen linearer gleichungen,Bull. Acad. Polon. Sci. Lett.,A (1937), 355–357.
[K2] T. Kato,A Short Introduction to Perturbation Theory for Linear Operators, Springer-Verlag, New York, 1982.
[N1] H. Nakano,Spectral Theory in Hilbert Space, Japanese Society for the Promotion of Science, Tokyo, 1953.
[N2] J. von Neumann, On rings of operators. Reduction theory,Ann. of Math.,50 (1949), 401–485.
[N3] J. von Neumann,Functional Operators, Vol. II, Princeton University Press, Princeton, NJ, 1950.
[P] M. Pavon, New results on the interpolation problem for continuous-time stationary-increments processes,SIAM J. Control Optim.,22 (1984), 133–142.
[S1] H. Salehi, On the alternating projections theorem and bivariate stationary stochastic processes,Trans. Amer. Math. Soc.,128 (1967), 121–134.
[S2] R. R. Stoll,Set Theory and Logic, Dover, New York, 1961.
[T] K. Tanabe, Projection method for solving a singular system of linear equations and its applications,Numer. Math.,17 (1971), 203–214.
[W] N. Wiener, On the factorization of matrices,Comment. Math. Helv.,29 (1955), 97–111.
[Y] D. C. Youla, Generalized image restoration by the method of alternating projections,IEEE Trans. Circuits and Systems,25 (1978), 694–702.
Author information
Authors and Affiliations
Additional information
This work was supported by the Office of Naval Research under Contract N00014-85-K-0255.
Rights and permissions
About this article
Cite this article
Kayalar, S., Weinert, H.L. Error bounds for the method of alternating projections. Math. Control Signal Systems 1, 43–59 (1988). https://doi.org/10.1007/BF02551235
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02551235