Algoritmos Numéricos para el Problema de Restauración de Imágenes usando el Método de las Proyecciones Alternantes

• Escalante, René ^[1]
  1. [1] Universidad Central de Venezuela

     Universidad Central de Venezuela

     Venezuela

     
• Localización: Revista de Matemática: Teoría y Aplicaciones, ISSN 2215-3373, ISSN-e 2215-3373, Vol. 11, Nº. 1, 2004, págs. 41-58
• Idioma: español
• DOI: 10.15517/rmta.v11i1.236
• Enlaces
  • Texto completo (pdf)
• Resumen
  • español

    Los algoritmos de proyección han evolucionado a partir del algoritmo de proyección alternante propuesto por J. von Neumann en 1933, donde el espacio solución es la intersección de un número finito de subespacios o conjuntos convexos. Investigaciones recientes se han centrado en técnicas para acelerar la convergencia del método y explotar el multiprocesamiento. En este trabajo consideramos el problema de restauración de imágenes. La mayoría de las técnicas desarrolladas para resolverlo han usado algoritmos iterativos; una de ellas consiste en usar proyecciones ortogonales alternantes. Llevamos a cabo una revisión cronológica de las diferentes técnicas en las que se ha aplicado el método de las proyecciones ortogonales alternantes al problema de restauración de imágenes, hasta llegar al enfoque reciente de Combettes (1997-1999), en donde la restauración se basa en el cálculo de proyecciones aproximadas, en lugar de proyecciones exactas.

  • English

    The projection algorithms have evolved from the alternating projection method proposed by J. von Neumann in 1933, who treated the problem of finding the projection of a given point in a Hilbert space onto the intersection of two closed subspaces. Recent researches have been centered in techniques for accelerate the convergence ofthe method and to exploit the multiprocessing. In this work we considered the image restoration problem. In most techniques developed to solved it have used iterative algorithms; one of them consists of using alternating orthogonal projections. We carried out one chronological looking back of different techniques in which has been applied the method of the alternating orthogonal projections to the problem of imagen restoration, until arriving at the recent approach of Combettes (1997-1999), on where the restoration process is based on the computation of approximate projections (i.e., subgradient projections), instead of exact projections

• Referencias bibliográficas
  • Andersen, A.H. (1989) “Algebraic reconstruction in CT from limited views”, IEEE Trans. Medical Imaging 8: 50–55.
  • Andrews, H.C.; Hunt, B.R. (1977) Digital Image Restoration. Prentice-Hall, Englewood Cliffs, NJ.
  • Arjona, J. (1998) Algoritmos Numéricos para el Problema de Restauración de Imágenes a partir de Métodos de Proyecciones Ortogonales Alternantes....
  • Aubin, J.P. (1993) An Introduction to Nonlinear Analysis. Springer-Verlag, New York.
  • Bauschke, H.H.; Borwein, J.M. (1996) “On projection algorithms for solving convex feasibility problems”, SIAM Rev. 38: 367–426.
  • Bazaraa, M.S.; Sherali, H.D.; Shetty, C.M. (1993) Nonlinear Programmig, Theory and Algorithms, 2nd. Edition. John Wiley, New York.
  • Censor, Y. (1988) “Parallel application of block-iterative methods in medical imagin and radiation therapy”, Math. Prog. 42(2): 307–325.
  • Censor, Y.; Zenios,S.A. (1997) Parallel Optimization. Theory, Algorithms, and Applications. Oxford University Press, New York.
  • Combettes, P.L. (1997) “Convex Set Theoretic Image Recovery by Extrapolated Iterations of Parallel Subgradient Projections”, IEEE Trans. Image...
  • Combettes, P.L. (1997) “Hilbertian Convex Feasibility Problem: Convergence of Projection Methods”, Appl. Math. Optim 35: 311–330.
  • Combettes, P.L. (1996) “The convex feasibility problem in image recovery”, Advances in Imaging and Electron Physics, Academic Press, Editor:...
  • Combettes, P.L. (1993) “The foundations of set theoric estimation”, Proc. IEEE81:182–208.
  • Combettes, P.L. (1994) “Inconsistent signal feasibility problems: Least-squares solutions in a product space”, IEEE Trans. Signal Processing...
  • P. L. Combettes, P.L.; Puh, H. (1994) “Iterations of Parallel convex projections in Hilbert spaces”, Numer. Funct. Anal. Optim. 15: 225-243.
  • Deutsch, F. (1992) “The method of alternating orthogonal projections”, in S.P. Singh (Ed.) Approximation Theory, Spline Functions and Applications,...
  • Escalante, R.; Raydan, M. (2000) Alternating Projection Methods: Theory and Applications, Book in preparation, CCCT.
  • R. Escalante, R.; Raydan, M. (1996) “Dykstra algorithm for a Constrained Least-squares Matrix Problem”, Num. Linear Algebra Appl. 3: 459–471.
  • Escalante, R.; Raydan, M. (1998) “Dykstra algorithm for a Constrained Least-squares Rectangular Matrix Problem”, Computers Math. Applic. 6:...
  • Garćıa-Palomares, U.M. (1999) “Relaxation in Projection Methods”, Preprint, Departamento de Procesos y Sistemas, Universidad Simón Bolívar,...
  • Garćıa-Palomares, U.M. (1999) “Preconditioning projection methods for solving algebraic linear systems”, Numerical Algorithms 21: 157–164.
  • Gilbert, P. (1972) “Iterative methods for the three-dimensional reconstruction of an object from projections”, J. Theoret. Biol. 36(1): 105–117.
  • González, R.C.; Woods, R.E. (1997) Tratamiento Digital de Imagenes. Adison-Wesley, New York.
  • Gordon, R.; Bender, R.; Herman, G.T. (1970) “Algebraic reconstruction techniques (ART) for three-dimensional electron microscopy and X-ray...
  • Gubin, L.G.; Polyak, B.T.; Raik, E.V. (1967) “The method of projections for finding the common point of convex sets”, USSR Comp. Math. Phys....
  • Halperin, I. (1962) “The product of projection operators”, Acta Sci. Math. (Szeged) 23: 96–99.
  • Herman, G.T. (1980) Image Reconstruction from Projections, the Fundamental of Computerized Tomography. Academic Press, New York.
  • Hocker, T.; Aranovich, G.L.; Donohue, M.D. (2001) “Adsorption-Energy Distribution of Heterogeneous Surfaces Predicted from Projections onto...
  • Iusem, A.; De Pierro, A. (1986) “Convergence results for an accelerated nonlinear Cimmino algorithm”, Numer. Math. 49(4): 367–378.
  • Katsaggelos, A.K. (1991) Digital Image Restoration. Springer-Verlag, New York.
  • Luenberger, D. G. (1969) Optimization by Vector Space Methods. John Wiley, New York.
  • Mandel, J. (1984) “Convergence of the cyclical relaxation method for linear inequalities”, Math. Programming 30: 218–228.
  • Moreau,J.J. (1963) “Fonctionnelles sous-différentiables”, C. R. Acad. Sci. Paris Sér. A 257(26): 4117–4119.
  • Ottavy, N. (1988) “Strong convergence of projection-like methods in Hilbert spaces”, J. Optim. Theory Appl. 56(3): 433–461.
  • Pierra, G. (1984) “Decomposition through formalization in a product space”, Math. Programing 28(1): 96–115.
  • Rockafellar, R.T. (1970) Convex Analysis. Princeton University Press, Princeton, NJ.
  • Sezan, M.I.; Stark, H. (1982) “Image restoration by the method of convex projections: Part 2-Applications and Numerical Results”, IEEE Trans....
  • Simard, P.Y.; Mailloux, G.E. (1988) “A projection operator for the restoration of divergence-free vector fields”, IEEE Trans. Patt. Anal....
  • Stark, H. (1987) Image Recovery: Theory and Application. Academic Press, San Diego, CA.
  • Trussell, H.J.; Civanlar, M.R. (1984) “The feasible solution in signal restoration”, IEEE Trans. Acoust., Speech, Signal Processing 32(2):...
  • Von Neumann, J. (1950) The Geometry of Orthogonal Spaces, Vol II. Princeton University Press, Princenton, New Jersey.
  • Yang, Y.; H. Stark, H. (2001) “Design of Self-Healing Arrays Using Vector-Space Projections”, IEEE Trans. on Antennas and Propagation 49(4):...
  • Youla, D.C. (1978) “Generalized image restoration by the method of alternating orthogonal projections”, IEEE Trans. Circuits Syst., CAS-25:...
  • Youla, D.C.; Webb, H. (1982) “Image restoration by the method of convex projections: Part 1-Theory”, IEEE Trans. Medical Imaging M1-1: 81–94.