Inconsistent Signal Feasibility Problems: Least-Squares Solutions in a Product Space

IEEE Transactions on Signal Processing

Volumn 42, Issue 11, 1994, Pages 2955-2966

  Combettes, Patrick L ^a  

^a CITY COLLEGE OF NEW YORK   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

ALGORITHMS; CONSTRAINT THEORY; CONVERGENCE OF NUMERICAL METHODS; ESTIMATION; FREQUENCY DOMAIN ANALYSIS; LEAST SQUARES APPROXIMATIONS; SET THEORY; TIME DOMAIN ANALYSIS;

CARTESIAN PRODUCT; INCONSISTENT CONVEX SET THEORETIC SIGNAL SYNTHESIS PROBLEMS; PARALLEL PROJECTION METHOD; PRODUCT SPACE; PROJECTION ONTO CONVEX SETS ALGORITHMS; SIGNAL SYNTHESIS;

SIGNAL THEORY;

EID: 0028543165     PISSN: 1053587X     EISSN: 19410476     Source Type: Journal    
DOI: 10.1109/78.330356     Document Type: Article

References (39)

1. J. P. Aubin, Optima and Equilibria - An Introduction to Nonlinear Analysis. New York: Springer-Verlag, 1993.
2. A. Auslender, Optimisation - Methodés Numériques, Paris: Masson, 1976.
3. L. M. Bregman, “The method of successive projection for finding a common point of convex sets,” Soviet Math. Dokl., vol. 6, no. 3, pp. 688–692, May 1965.
4. Y. Censor, P. P. B. Eggermont, and D. Gordon, “Strong underrelaxation in Kaczmarz’s method for inconsistent systems,” Numer. Math., vol. 41, no. 1, pp. 83–92, Apr. 1983.
5. A. E. Cetin, “An iterative algorithm for signal reconstruction from bispectrum,’’ IEEE Trans. Signal Processing. vol. 39, no. 12, pp. 2621–2628, Dec. 1991.
6. W. Cheney and A. A. Goldstein, “Proximity maps for convex sets,” in Proc. Amer. Math. Soc., vol. 10, no. 3, pp. 448– 450, June 1959.
7. G. Cimmino, “Calcolo approssimato per le soluzioni dei sistemi di equazioni lineari,” La Ricerca Scientifica (Roma), vol. 1, pp. 326–333, 1938.
8. M. R. Civanlar and R. A. Nobakht, “Optimal pulse shape design using projections onto convex sets,” Proc. IEEE ICASSP Conf. (New York. NY), Apr. 11-14, 1988, pp. 1874–1877, See also R. A. Nobakht, M. R. Civanlar, and S. H. Ardalan, “Comments on ‘Design of a class of time-constrained FIR digital filters by successive projections’,” IEEE Trans. Circuits, Syst., vol. 37, no. 12, p. 1581, Dec. 1990.
9. P. L. Combettes, “The foundations of set theoretic estimation,” Proc. IEEE, vol. 81, no. 2, pp. 182–208, Feb. 1993.
10. P. L. Combettes, “Signal recovery by best feasible approximation,” IEEE Trans. Image Processing, vol. 2, no. 2, pp. 269–271, Apr. 1993.
11. P. L. Combettes and H. J. Trussell, “Methods for digital restoration of signals degraded by a stochastic impulse response,” IEEE Trans. Acoustics, Speech, Signal Processing, vol. 37, no. 3, pp. 393 – 401, Mar. 1989.
12. P. L. Combettes and H. J. Trussell, “The use of noise properties in set theoretic estimation,” IEEE Trans. Signal Processing, vol. 39, no. 7, pp. 1630–1641, July 1991.
13. A. R. De Pierro and A. N. Iusem, “A simultaneous projections method for linear inequalities,” Linear Algebra Applicat., vol. 64, pp. 243–253, Jan. 1985.
14. A. R. De Pierro and A. N. Iusem, “A parallel projection method for finding a common point of a family of convex sets,” Pesquisa Operational, vol. 5, no. 1, pp. 1–20, 1985.
15. J. A. Dieudonné, Foundations of Modern Analysis, 2nd ed. New York: Academic, 1969.
16. P. Gilbert, “Iterative methods for the three-dimensional reconstruction of an object from projections.” J. Theoret. Biol., vol. 36, no. 1, pp. 105–117, July 1972.
17. M. Goldburg and R. J. Marks II, “Signal synthesis in the presence of an inconsistent set of constraints,” IEEE Trans. Circuits, Syst., vol. 32, no. 7, pp. 647–663, July 1985.
18. L. G. Gubin, B. T. Polyak, and E. V. Raik, “The method of projections for finding the common point of convex sets,” USSR Comput. Math., Math. Phys., vol. 7, no. 6, pp. 1–24, 1967.
19. G. T. Herman, “A relaxation method for reconstructing objects from noisy X-rays,” Math. Programming, vol. 8, no. 1, pp. 1–19, Feb. 1975.
20. G. T. Herman, Image Reconstruction from Projections, the Fundamentals of Computerized Tomography. New York: Academic, 1980.
21. A. N. Iusem and A. R. De Pierro, “Convergence results for an accelerated nonlinear Cimmino algorithm,” Numer. Math, vol. 49, no. 4, pp. 367–378. Aug. 1986.
22. A. Levi and H. Stark, “Signal reconstruction from phase by projection onto convex sets,” J. Opt. Soc. Am. A, vol. 73, no. 6, pp. 810–822, June 1983.
23. P. L. Lions, “Approximation de points fixes de contractions,” C. R. Acad. Sci. Paris, vol. A284, no. 21, pp. 1357–1359, June 1977.
24. K. M. Nashold and B. E. A. Saleh, “Image construction through diffraction-limited high-contrast imaging systems: An iterative approach,” J. Opt. Soc. Am. A, vol. 2, no. 5, pp. 635–643, May 1985.
25. S. Oh, C. Ramon, R. J. Marks II, A. C. Nelson, and M. G. Meyer, “Resolution enhancement of biomagnetic images using the method of alternating projections,” IEEE Trans. Biomed. Eng., vol. 40, no. 4, pp. 323–328, Apr. 1993.
26. S. C. Pei and I. I. Yang, “Design of a class of time-constrained FIR digital filters by successive projections,” IEEE Trans. Circuits, Syst., vol. 36, no. 1, pp. 164–167, Jan. 1989.
27. G. Pierra, “Méthodes de projections paralleles extrapolées relatives a une intersection de convexes,” Rapport de Recherche, INPG, Grenoble, France, Sept. 1975.
28. G. Pierra, “Decomposition through formalization in a product space,” Math. Programming, vol. 28, no. 1, pp. 96–115, Jan. 1984.
29. S. V. Plotnikov, “Cyclic projection on a system of convex sets with empty intersection,” (in Russian) in Improper Optimization Problems, I. I. Eremin and V. D. Skarin, Eds. Sverdlovsk: Akademiia Nauk SSSR, 1982, pp. 60–66.
30. E. Polak, Computational Methods in Optimization: A Unified Approach. New York: Academic, 1971.
31. L. Schwartz, Analyse Hilbertienne. Paris: Hermann, 1979.
32. M. I. Sezan and H. Stark, “Image restoration by convex projections in the presence of noise,” Appl. Optics, vol. 22, no. 18, pp. 2781–2789, Sept. 1983.
33. H. J. Trussell and M. R. Civanlar, “The feasible solution in signal restoration,” IEEE Trans. Acoust., Speech. Signal Processing, vol. 32, no. 2. pp. 201–212, Apr. 1984.
34. P. Tseng, “On the convergence of products of firmly nonexpansive mappings,” SIAM J. Optim., vol. 2, no. 3, pp. 425–434, Aug. 1992.
35. D. C. Youla and V. Velasco, “Extensions of a result on the synthesis of signals in the presence of inconsistent constraints,” IEEE Trans. Circuits Syst., vol. 33, no. 4, pp. 465–468, Apr. 1986.
36. D. C. Youla and H. Webb, “Image restoration by the method of convex projections: Part I- Theory,” IEEE Trans. Med. Imaging, vol. 1, no. 2, pp. 81–94, Oct. 1982.
37. E. H. Zarantonello, “Projections on convex sets in Hilbert space and spectral theory,” in Contributions to Nonlinear Functional Analysis, E. H. Zarantonello, Ed. New York: Academic, 1971, pp. 237–424.
38. E. Zeidler, Nonlinear Functional Analysis and Its Applications I: Fixed-Point Theorems. New York: Springer-Verlag, 1986.
39. E. Zeidler, Nonlinear Functional Analysis and Its Applications III: Variational Methods and Optimization, New York: Springer-Verlag, 1985.