Skip to main content
SpringerLink
Log in

The Method of Alternating Orthogonal Projections

  • Chapter
Approximation Theory, Spline Functions and Applications

Part of the book series: NATO ASI Series ((ASIC,volume 356))

Abstract

The method of alternating orthogonal projections is discussed, and some of its many and diverse applications are described.

Supported by NSF Grant DMS-9100228

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  • S. Agmon, [1954], The relaxation method for linear inequalities, Canadian J. Math. 6, 382–392.

    Article  MathSciNet  MATH  Google Scholar 

  • R. Aharoni and Y. Censor, [1988], Block-iterative projection methods for parallel computation of solutions to convex feasibility problems., preprint.

    Google Scholar 

  • N. Aronszajn, [1950], Theory of reproducing kernels, Trans. Amer. Math. Soc. 68, 337–404.

    Article  MathSciNet  MATH  Google Scholar 

  • B. Atlestam and F. Sullivan, [1976], Iteration with best approximation operators, Rev. Roumaine Math. Pures Appl. 21, 125–131.

    MathSciNet  MATH  Google Scholar 

  • G. Aumann, [1959], Über approximative Nomographie, II, Bayer. Akad. Wiss., Math.-Natur. Kl. Sitzundsber, 103–109.

    Google Scholar 

  • A. P. Bosznay, [1986], A remark on the alternating algorithm, Periodica Math. Hungarica 17, 241–244.

    Article  MathSciNet  MATH  Google Scholar 

  • D. Braess, [1981], The contraction number of a multigrid method for solving the Poisson equation, Numer. Math. 37, 387–404.

    Article  MathSciNet  MATH  Google Scholar 

  • L. M. Bregman, [1965], The method of successive projection for finding a common point of convex sets, Sov. Math. Dok. 6, 688–692.

    MATH  Google Scholar 

  • L. Breiman and J. H. Friedman, [1985], Estimating optimal transformations for multiple regression and correlation, J. Amer. Statist. bd80, 580–598.

    Article  MathSciNet  Google Scholar 

  • D. L. Burkholder, [1962], Successive conditional expectations of an integrable function, Ann. Math. Statist. bd33, 887–893.

    Article  MathSciNet  Google Scholar 

  • D. L. Burkholder and Y. S. Chow, [1961], Iterates of conditional expectation operators, Proc. Amer. Math. Soc. 12, 490–495.

    Article  MathSciNet  MATH  Google Scholar 

  • Y. Censor, [1984], Iterative methods for the convex feasibility problem, Annals of Discrete Math. 20, 83–91.

    MathSciNet  Google Scholar 

  • —, [1988], Parallel application of block-iterative methods in medical imaging and radiation Therapy, Math. Programming 42, 307–325.

    Article  MathSciNet  MATH  Google Scholar 

  • Y. Censor and G. T. Herman, [1987], On some optimization techniques in image reconstruction from projections, Appl. Numer. Math. 3, 365–391.

    Article  MathSciNet  MATH  Google Scholar 

  • W. Cheney and A. A. Goldstein, [1959], Proximity maps for convex sets, Proc. Amer. Math. Soc. 10, 448–450.

    Article  MathSciNet  MATH  Google Scholar 

  • J. E. Dennis, Jr., [1972], On some methods bases on Broyden’s secant approximation to the Hessian, Numerical Methods for Non-linear Optimization, (F. A. Lootsma, ed.), Academic Press, London.

    Google Scholar 

  • J. E. Dennis, Jr. and R. B. Schnabel, [1979], Least change secant updates for quasi-Newton methods, SIAM Review 21, 443–459.

    Article  MathSciNet  MATH  Google Scholar 

  • J. E. Dennis, Jr. and H. F. Walker, [1981], Convergence theorems for least-change secant update methods, SIAM J. Numer. Anal. 18, 949–987.

    Article  MathSciNet  MATH  Google Scholar 

  • F. Deutsch, [1979], The alternating method of von Neumann, “Multivariate Approximation Theory”, (W. Schempp and K. Zeller, eds.), Birkhauser-Verlag, Basel.

    Google Scholar 

  • —, [1983a], Applications of von Neumann’s alternating projections algorithm, “Mathematical Methods in Operations Research”, (P. Kenderov, ed.), Sofia, Bulgaria, pp. 44–51.

    Google Scholar 

  • —, [1983b], Von Neumann’s alternating method: The rate of convergence, “Approximation Theory IV”, (C. K. Chui, L. L. Schumaker, and J. D. Ward, eds.), Academic Press, New York, pp. 427–434.

    Google Scholar 

  • D. P. Diliberto and E. G. Straus, [1951], On the approximation of a function of several variables by the sum of functions of fewer variables., Pacific J. Math. 1, 195–210.

    Article  MathSciNet  MATH  Google Scholar 

  • J. Dyer, [1965], Acceleration of the Convergence of the Kaczmarz Method and Iterated Homogeneous Transformations, doctoral dissertation, Univ. Calif. at Los Angeles.

    Google Scholar 

  • R. L. Dykstra, [1983], An algorithm for restricted least squares regression, J. Amer. Statist. Assoc. 78, 837–842.

    Article  MathSciNet  MATH  Google Scholar 

  • N. Dyn, [1980], A straightforward generalization of Diliberto and Straus’ algorithm does not work, J. Approx. Theory 30, 247–250.

    Article  MathSciNet  MATH  Google Scholar 

  • P. P. B. Eggermont, G. T. Herman, and A. Lent, [1981], Iterative algorithms for large partitioned linear systems, with applications to image reconstruction, Linear Alg. and its Applic. 40, 37–67.

    Article  MathSciNet  MATH  Google Scholar 

  • L. Flatto, [1965], The approximation of certain functions of several variables by sums of functions of fewer variables, Amer. Math. Monthly 71, 131–132.

    Google Scholar 

  • C. Franchetti, [1973], On the alternating approximation method, Boll. Un. Mat. Ital. 7 no. (4), 169–175.

    MathSciNet  MATH  Google Scholar 

  • C. Franchetti and W. Light, [1984], The alternating algorithm in uniformly convex spaces, J. London Math. Soc. 29 no. (2), 545–555.

    Article  MathSciNet  MATH  Google Scholar 

  • W. Light —, [1986], On the von Neumann alternating algorithm in Hilbert space, J. Math. Anal. Appl. 114, 305–314.

    Article  MathSciNet  MATH  Google Scholar 

  • K. Friedricks, [1937], On certain inequalities and characteristic value problems for analytic functions and for functions of two variables, Trans. Amer. Math. Soc. 41, 321–364.

    Article  MathSciNet  Google Scholar 

  • T. B. Gatski, C. E. Grosch, and M. E. Rose, [1982], A numerical study of the two-dimensional Navier-Stokes equations in vorticity-velocity variables, J. Comput. Physics 48, 1–22.

    Article  MATH  Google Scholar 

  • C. E. Grosch, and M. E. Rose —, [1988], The numerical solution of the Navier-Stokes equations for S-dimensional, unsteady, incompressible flows by compact schemes, J. Comput. Physics 82, 298–329.

    Article  MathSciNet  Google Scholar 

  • W. B. Gearhart and M. Kosky, [1989], Acceleration schemes for the method of alternating projections, J. Comp. Appl. Math. 26, 235–249.

    Article  MATH  Google Scholar 

  • J. Gilbert and W. A. Light, [1986], Multigrid methods and the alternating algorithm.

    Google Scholar 

  • M. von Golitschek and E. W. Cheney, [1979], On the algorithm of Diliberto and Straus for approximating bivariate functions by univariate ones, Num. Funct. Anal. Approx. 1, 341–363.

    Article  MATH  Google Scholar 

  • E. W. Cheney —, [1983], Failure of the alternating algorithm for best approximation of multivariate Functions, J. Approx. Theory 38, 139–143.

    Article  MathSciNet  MATH  Google Scholar 

  • M. von Golitschek and W. A. Light, [1987], Some properties of the Diliberto-Straus algorithms in C(S × T), “Numerical Methods of Approximation Theory”, (L. Collatz, G. Meinardus, and G. Nürnberger, eds.), vol. 8, Birkhäuser-Verlag, Basel.

    Google Scholar 

  • M. Golomb, [1959], Approximation by functions of fewer variables, “On Numerical Approximation”, (R. E. Langer, ed.), Univ. Wisconsin Press, Madison, pp. 275–327.

    Google Scholar 

  • L. G. Gubin, B. T. Polyak, and E. V. Raik, [1967], The method of projections for finding the common point of convex sets, USSR Comput. Math. and Math. Phys. 7, 1–24.

    Article  Google Scholar 

  • R. Gordon, R. Bender, and G. T. Herman, [1970], Algebraic reconstruction techniques (ART) for three-dimensional electron microscopy and X-ray photography, J. Theoretical Biol. 29, 471–481.

    Article  Google Scholar 

  • I. Halperin, [1962], The product of projection operators, Acta Sci. Math. (Szeged) 23, 96–99.

    MathSciNet  MATH  Google Scholar 

  • C. Hamaker and D. C. Solmon, [1978], The angles between null spaces of X-rays, J. Math. Anal. Appl. 62, 1–23.

    Article  MathSciNet  MATH  Google Scholar 

  • G. T. Herman, A. Lent, and P. H. Lutz, [1978], Iterative relaxation methods for image reconstruction, Communications of the ACM 21, 152–158.

    Article  MathSciNet  MATH  Google Scholar 

  • G. N. Hounsfield, [1973], Computerized transverse axial scanning (tomography): Part I Description of system, British J. Radiol. 46, 1016–1022.

    Article  Google Scholar 

  • S. Kaczmarz, [1937], Angenäherte Auflösung von Systemen linearer Gleichungen, Bull. Internat. Acad. Pol. Sci. Lett. A 35, 355–357.

    Google Scholar 

  • S. Kayalar and H. L. Weinert, [1988], Error bounds for the method of alternating projections, Math. Control Signals Systems 1, 43–59.

    Article  MathSciNet  MATH  Google Scholar 

  • C. T. Kelley, [1981], A note on the approximation of functions of several variables by sums of functions of one variable, J. Approx. Theory 33, 179–189.

    Article  MathSciNet  MATH  Google Scholar 

  • W. A. Light, [1983], The Diliberto-Straus algorithm in L 1(X × Y), J. Approx. Theory 38, 1–8.

    Article  MathSciNet  MATH  Google Scholar 

  • W. A. Light and E. W. Cheney, [1980], On the approximation of a bivariate function by the sum of univariate functions, J. Approx. Theory 29, 305–322.

    Article  MathSciNet  MATH  Google Scholar 

  • W. A. Light and S. M. Holland, [1984], The L 1-version of the Diliberto-Straus algorithm in C(T × S), Proc. Edinburgh Math. Soc. 27, 31–45.

    Article  MathSciNet  Google Scholar 

  • T. S. Motzkin and I. J. Schoenberg, [1954], The relaxation method for linear inequalities, Canadian J. Math. 6, 393–404.

    Article  MathSciNet  MATH  Google Scholar 

  • H. Nakano, [1953], Spectral theory in the Hilbert space, Japan Soc. Promotion Sc., Tokyo.

    Google Scholar 

  • J. von Neumann, [1933], Functional Operators-Vol. II. The Geometry of Orthogonal Spaces (,This is a reprint of mineographed lecture notes first distributed in 1933), Annals of Math. Studies #22, Princeton University Press, 1950.

    Google Scholar 

  • P. L. Papini, [1988], Alternating methods in approximation, “Functional Analysis and Approximation”, (P. L. Papini, ed.), Bagni di Lucca, Italy, pp. 219–229.

    Google Scholar 

  • M. J. D. Powell, [1970], A new algorithm for unconstrained optimization. Nonlinear Programming, (J. B. Rosen, O. L. Mangasarian, K. Ritter, eds.), Academic Press, New York.

    Google Scholar 

  • I. P. Ramadanov and M. L. Skwarczynski, [1984a], Bull. Polish Acad. Sci. Math. 32, 653–659.

    MathSciNet  MATH  Google Scholar 

  • M. L. Skwarczynski —, [1984b], Constructive Theory of Functions’ 84, Sofia, 726–730.

    Google Scholar 

  • T. J. Rivlin and R. J. Sibner, [1965], The degree of approximation of certain functions of two variables by a sum of functions of one variable, Amer. Math. Monthly 72, 1101–1103.

    Article  MathSciNet  MATH  Google Scholar 

  • G.-C. Rota, [1962], An “alternierende Verfahren” for general positive operators, Bull. Amer. Math. Soc. 68, 95–102.

    Article  MathSciNet  MATH  Google Scholar 

  • H. Salehi, [1967], On the alternating projections theorem and bivariate stationary stochastic Processes, Trans. Amer. Math. Soc. 128, 121–134.

    Article  MathSciNet  MATH  Google Scholar 

  • H. A. Schwarz, [1870], Heber einen Grenzübergang durch alternirendes Verfahren, Vierteljahrsschrift der Naturforschenden Gesellschaft in Zürich 15, 272–286.

    Google Scholar 

  • I. Singer, [1974], The Theory of Best Approximation and Functional Analysis, CBMS #13, SIAM, Philadelphia.

    Book  MATH  Google Scholar 

  • M. Skwarczynski, [1985a], Alternating projections in complex analysis, Complex Analysis and Applications’ 83, Sofia.

    Google Scholar 

  • —, [1985b], A general description of the Bergman projection, Annales Polonici Math. 46.

    Google Scholar 

  • K. T. Smith, D. C. Solman and S. L. Wagner, [1977], Practical and mathematical aspects of the problem of reconstructing objects from radiographs, Bull. Amer. Math. Soc. 83, 1227–1270.

    Article  MathSciNet  MATH  Google Scholar 

  • J. E. Spingarn, [1985], A primal-dual method for solving systems of linear inequalities, Linear Alg. Applic. 65, 45–62.

    Article  MathSciNet  MATH  Google Scholar 

  • —, [1987], A projection method for least-squares solutions to overdetermined systems of linear inequalities, Linear Alg. Applic. 86, 211–236.

    Article  MathSciNet  MATH  Google Scholar 

  • W. J. Stiles, [1965a], Closest-point maps and their products, Nieuw Archief voor Wiskunde 13 no. (3), 19–29.

    MathSciNet  MATH  Google Scholar 

  • —, [1965b], A solution to Hirschfeld’s problem, Nieuw Archief voor Wiskunde 13 no. (3), 116–119.

    MathSciNet  MATH  Google Scholar 

  • K. Tanabe, [1971], Projection method for solving a singular system of linear equations and its Applications, Numer. Math. 17, 203–214.

    Article  MathSciNet  MATH  Google Scholar 

  • R. Wegmann, [1989], Conformai mapping by the method of alternating projections, Numer. Math. 56, 291–307.

    Article  MathSciNet  MATH  Google Scholar 

  • N. Wiener, [1955], On the factorization of matrices, Comment. Math. Helv. 29, 97–111.

    Article  MathSciNet  MATH  Google Scholar 

  • N. Wiener and P. Masani, [1957], The prediction theory of multivariate stochastic processes, II: The linear predictor, Acta Math. 93, 95–137.

    MathSciNet  Google Scholar 

  • D. C. Youla, [1978], Generalized image restoration by the method of alternating orthogonal Projections, IEEE Trans. Circuits Syst. CAS-25, 694–702.

    Article  MathSciNet  Google Scholar 

  • D. C. Youla and H. Webb, [1982], Image restoration by the method of convex projections: Part 1-Theory, IEEE Trans. Medical Imaging M1-1, 81–94.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, The Pennsylvania State University, University Park, PA, 16802, USA

    Frank Deutsch

Authors
  1. Frank Deutsch
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Memorial University of Newfoundland, St. John’s, NF, Canada

    S. P. Singh

Rights and permissions

Reprints and permissions

Copyright information

© 1992 Springer Science+Business Media Dordrecht

About this chapter

Cite this chapter

Deutsch, F. (1992). The Method of Alternating Orthogonal Projections. In: Singh, S.P. (eds) Approximation Theory, Spline Functions and Applications. NATO ASI Series, vol 356. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-2634-2_5

Download citation

  • DOI: https://doi.org/10.1007/978-94-011-2634-2_5

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-010-5164-4

  • Online ISBN: 978-94-011-2634-2

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation