The convex feasibility problem in image recovery

Advances in Imaging and Electron Physics

Volumn 95, Issue C, 1996, Pages 155-270

  Combettes, P L ^a  

^a NEW YORK UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 77954607855     PISSN: 10765670     EISSN: None     Source Type: Book Series    
DOI: 10.1016/s1076-5670(08)70157-5     Document Type: Article

References (192)

1. I. S. Agmon, "Therelaxation method for linear inequalities," Canadiun Journal of Mathematics, vol. 6, no.3, pp. 382-392, 1954.
2. R. Aharoni, A. Berman, and Y. Censor, "An interior points algorithm for the convex feasibility problem," Advances in Applied Mathematics, vol. 4, no.4, pp. 479-489, December 1983.
3. R. Aharoni and Y. Censor, "Block-iterative methods for parallel computation of solutions to convex feasibility problems," Linear Algebra and Its Applications, vol. 120, pp. 165-175, August 1989.
4. I. Amemiya and T. Ando, "Convergence of random products of contractions in Hilbert space," Acta Scientiarum Mathematicarum (Szeged), vol. 26, no.3, pp. 239-244, 1965.
5. H. C. Andrews and B. R. Hunt, Digital Image Resforation. Englewood Cliffs, NJ: Prentice-Hall, 1977.
6. L. Armijo, "Minimization of functions having Lipschitz continuous first partial derivatives," Pacific Journal of Mathematics, vol. 16, no.1 , pp. 1-3, 1966.
7. J. Arsac, Transformation de Fourier et Théorie des Distributions. Paris: Dunod, 1961.
8. J. P. Aubin, Optima and Equilibria-An Introduction to Nonlinear Analysis. New York: Springer-Verlag, 1993.
9. R. Barakat and G. Newsam, "Algorithms for reconstruction of partially known, bandlimited Fourier transform pairs from noisy data," Journal of the Optical Society of America A , vol. 2, no. 11 , pp. 2027-2039, November 1985.
10. H. H. Bauschke, "A norm convergence result on random products of relaxed projections in Hilbert space," Transactions of the American Mathematical Society, vol. 347, no.4, pp. 1365-1373, April 1995.
11. H. H. Bauschke and J. M. Borwein, "On projection algorithms for solving convex feasibility problems," accepted for publication in SIAM Review.
12. H. H. Bauschke, J. M. Borwein, and A. S. Lewis, "On the method of cyclic projections for convex sets in Hilbert space." Research report, Simon Fraser University, 1994.
13. P. Billingsley, Convergence of Probability Measures. New York: Wiley, 1968.
14. J. M. Boone, B. A. Arnold, and J. A. Seibert, "Characterization of the point spread function and modulation transfer function of scattered radiation using a digital imaging system," Medical Physics, vol.13, pp. 254-256, 1986.
15. G. Bouligand, Introduction a la Géométrie Infinitésimale Directe. Paris: Vuibert, 1932.
16. N. Bourbaki, Elements de Mathématique-Espaces Vectoriels Topologiques, Chapitires 1 à 5. Pans: Masson, 1981.
17. D. Braess, Nonlinear Approximation Theory. New York: Springer-Verlag. 1986.
18. L. M. Brkgman, "The method of successive projection for finding a common point of convex sets," Soviet Mathematics-Doklady, vol. 6, no. 3, pp. 688-692, May 1965.
19. F. E. Browder, "Convergence theorems for sequences of nonlinear operators in Banach spaces," Mathematische Zeirschrift, vol.100, no. 3, pp. 201-225, July 1967.
20. R. E. Bruck, "Random products of contractions in metric and Banach spaces," Journal of Mathematical Analysis and Applications, vol.88, no.2, pp. 319-332, August 1982.
21. D. Butnariu and Y. Censor, "On the behavior of a block-iterative projection method for solving convex feasibility problems," International Journal of Computer Mathematics, vol. 34, nos. 1-2, pp. 79-94, 1990.
22. D. Butnariu and Y. Censor, "Strong convergence of almost simultaneous block-iterative projection methods in Hilbert spaces," Journal of Computational and Applied Mathematics, vol. 53, no. 1, pp. 33-42, July 1994.
23. C. L. Byme, "Iterative image reconstruction algorithms based on cross-entropy minimization," IEEE Transactions on Image Processing, vol. 2, no. 1 , pp. 96-103, January 1993.
24. ("Erratum and addendum," vol. 4, no. 2, pp. 226-227, February 1995.)
25. J. M. Carazo and J. L. Carrascosa, "Information recovery in missing angular data cases: An approach by the convex projections method in three dimensions," Journal of Microscopy, vol. 145, pt. 1 , pp. 23-43, January 1987.
26. Y. Censor, "Iterative methods for the convex feasibility problem," Annals of Discrete Mathematics, vol. 20, pp. 83-91, 1984.
27. Y . Censor, "Parallel application of block-iterative methods in medical imaging and radiation therapy," Mathematical Programming, vol. 42, no. 2, pp. 307-325, 1988.
28. Y. Censor, P. P. B. Eggermont, and D. Gordon, "Strongunderrelaxation in Kaczmarz's method for inconsistent systems," Numerische Mathematik, vol.41, no.1, pp. 83-92, April 1983.
29. Y. Censor and G. T. Herman, "On some optimization techniques in image reconstruction from projections," Applied Numerical Mathematics, vol. 3, pp. 365-391, 1987.
30. Y. Censor and A. Lent, "Cyclic subgradient projections," Mathematical Programming, vol.24, no. 2, pp. 233-235, 1982.
31. A. E. Cetin, "An iterative algorithm for signal reconstruction from bispectrum," IEEE Transactions on Signal Processing, vol. 39, no. 12, pp. 2621-2628, December 1991.
32. A. E. Cetin and R. Ansari, "Convolution-based framework for signal recovery and applications," Journal of the Optical Society of America A , vol. 5 , no.8, pp. 1193-1200, August 1988.
33. A. E. Cetin and R. Ansari, "Signal recovery from wavelet transform maxima,'' IEEE Transactions on Signal Processing, vol. 42, no. 1, pp. 194-196, January 1994.
34. W. Cheney and A. A. Goldstein, "Proximity maps for convex sets," Proceedings of the American Mathematical Society, vol.10, no. 3, pp. 448-450, June 1959.
35. R. T. Chin, C. L. Yeh, and W. S. Olson, "Restoration of multichannel microwave radiometric images," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 7, no. 4, pp. 475-484, July 1985.
36. G. Cimmino, "Calcolo approssimato per le soluzioni dei sistemi di equazioni lineari," La Ricerca Scientifica (Roma), vol. 1, pp. 326-333, 1938.
37. M. R. Civanlar and H. J. Trussell, "Digital signal restoration using fuzzy sets," IEEE Transactions on Acoustics, Speech, and Signal Processing, vol. 34, no. 4, pp. 919-936, August 1986.
38. D. Cochran, "Phase and magnitude in normalized images," IEEE Transactions on Image Processing, vol. 3, no. 6, pp. 858-862, November 1994.
39. P. L. Combettes, "The foundations of set theoretic estimation," Proceedings of the IEEE, vol. 81, no. 2, pp. 182-208, February 1993.
40. P. L. Combettes, "Signal recovery by best feasible approximation," IEEE Transactions on Image Processing, vol. 2, no. 2, pp. 269-271, April 1993.
41. P. L. Combettes, "Hilbertian convex feasibility problem: Convergence of projection methods," accepted for publication in Applied Mathematics and Optimization.
42. P. L. Combettes, "Convex set theoretic image recovery by extrapolated iterations of parallel subgradient projections," IEEE Transactions on Image Processing, submitted.
43. P. L. Combettes, "Inconsistent signal feasibility problems: Least-squares solutions in a product space," IEEE Transactions on Signal Processing, vol. 42, no. 11 , pp. 2955-2966, November 1994.
44. P. L. Combettes, "Construction d'un point fixe commun à uni e famille de contractions fermes," Comptes Rendus de I'Académie des Sciences de Paris, Série I , vol. 320, no. 11 , pp. 1385-1390, June 1995.
45. P. L. Combettes, M. Benidir, and B. Picinbono, "A general framework for the incorporation of uncertainty in set theoretic estimation," Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing. vol. 3, pp. 349-352. San Francisco, CA, March 23-26, 1992.
46. P. L. Combettes and H. Puh, "Extrapolated projection method for the euclidean convex feasibility problem." Technical report, City University of New York, 1993.
47. P. L. Combettes and H. Puh, "Iterations of parallel convex projections in Hilbert spaces," Numerical Functional Analysis and Optimization, vol. 15, nos. 3-4, pp. 225-243, 1994.
48. P. L. Combettes and H. J. Trussell, "Modèles et algorithmes en vue de la restauration numérique d'images rayons-X," Proceedings of MARI-Cognitiva Electronic Image, pp. 146-151. Paris, France, May 18-22, 1987.
49. P. L. Combettes and H. J. Trussell, "Methods for digital restoration of signals degraded by a stochastic impulse response," IEEE Transactions on Acoustics, Speech, and Signal Processing, vol. 37, no. 3, pp. 393-401, March 1989.
50. P. L. Combettes and H. J. Trussell, "Method of successive projections for finding a common point of sets in metric spaces," Journal of Optimization Theory arid Applications, vol. 67, no. 3, pp. 487-507, December 1990.
51. P. L. Combettes and H. J. Trussell, "The use of noise properties in set theoretic estimation," IEEE Transactions on Signal Processing, vol. 39, no. 7, pp. 1630-1641, July 1991.
52. P. L. Combettes and H. J. Trussell, "Deconvolution with bounded uncertainty," International Journal of Adaptive Control and Signal Processing, vol. 9, no. 1, pp. 3-17, January 1995.
53. I. Csiszfir, "Why least squares and maximum entropy? An axiomatic approach to inference for linear inverse problems," The Annals of Statistics, vol. 19, no. 4, pp. 2032-2066, December 1991.
54. G. Demoment "Image reconstruction and restoration: Overview of common estimation structures and problems," IEEE Transactions on Acoustics, Speech, and Signal Processing, vol. 37, no. 12, pp. 2024-2036, December 1989.
55. A. R. De Pierro and A. N. Iusem, "A simultaneous projections method for linear inequalities," Linear Algebra and Its Applications, vol. 64, pp. 243-253, January 1985.
56. A. R. De Pierro and A. N. Iusem, "A parallel projection method for finding a common point of a family of convex sets," Pesquisa Operacional, vol. 5 , no.1, pp. 1-20, July 1985.
57. F. Deutsch, "The method of alternating orthogonal projections," in Approximation Theory, Spline Functions and Applications (S. P. Singh, Ed.), pp. 105-121. Dordrecht, The Netherlands: Kluwer, 1992.
58. J. A. Dieudonnk, Foundations of Modern Analysis, 2nd ed. New York: Academic Press, 1969.
59. J. L. Doob, Stochastic Processes. New York: Wiley, 1953.
60. J. L. Doob, Measure Theory. New York: Springer-Verlag, 1994.
61. L. T. Dos Santos, "A parallel subgradient projections method for the convex feasibility problem," Journal of Computational and Applied Mathematics, vol. 18, no.3, pp. 307-320, June 1987.
62. J. M. Dye and S. Reich, "Unrestricted iterations of nonexpansive mappings in Hilbert space," Nonlinear Analysis-Theory, Methods, and Applications, vol. 18, no. 2, pp. 199-207, January 1992.
63. S. Ebstein, "Stellar speckle interferometry energy spectrum recovery by convex projections," Applied Optics, vol. 26, no.8, pp. 1530-1536, April 1987.
64. B. Efron, "Controversies in the foundations of statistics," American Mathematical Monthly, vol. 85, no.4, pp. 231-246, April 1978.
65. B. Efron, "Why isn't everyone a Bayesian?" American Statistician, vol. 40, no.1 , pp. 1-5, February 1986.
66. I. Ekeland and R. Temam, Analyse Convexe et Problèmes Variationnels. Paris: Dunod, 1974.
67. P. Erdos, "Some remarks on the measurability of certain sets," Bulletin of the American Mathematical Society, vol. 51, no.10, pp. 728-731, October 1945.
68. M. Fisz, Probability Theory and Mathematical Statistics, 3rd ed. New York: Wiley, 1963.
69. S. D. Flgm and J. Zowe, "Relaxed outer projections, weighted averages, and convex feasibility," BIT, vol. 30, no.2, pp. 289-300, 1990.
70. D. L. Fried, "Optical resolution through a randomly inhomogeneous medium for very long and very short exposures," Journal of the Optical Society of America, vol.56, no.10, pp 1372-1379, October 1966.
71. B. R. Frieden, "Restoring with maximum likelihood and maximum entropy," Journal of the Optical Society of America, vol. 62, no.4, pp. 511-518, April 1972.
72. S. Geman and D. Geman, "Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 6, no. 6, pp. 721-741, November 1984.
73. R. W. Gerchberg, "Super-resolution through error energy reduction," Optica Acta, vol. 21, no.9, pp. 709-720, September 1974.
74. R. W. Gerchberg and W. O. Saxton, "A practical algorithm for the determination of phase from image and diffraction plane pictures," Optik, vol. 35, no. 2, pp. 237-246, April 1972.
75. P. Gilbert, "Iterative methods for the three-dimensional reconstruction of an object fromprojections," Journal of Theoretical Biology, vol. 36, no. 1, pp. 105-1 17, July 1972
76. K. Goebel and W. A. Kirk, Topics in Metric Fixed Point Theory. Cambridge: Cambridge University Press, 1990.
77. M. Goldburg and R. J. Marks 11, "Signal synthesis in the presence of an inconsistent set of constraints," IEEE Transactions on Circuits and Systems, vol. 32, no.7, pp. 647-663, July 1985.
78. J. W. Goodman, Statistical Optics. New York: Wiley Interscience, 1985.
79. 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, vol. 29, no.3, pp. 471-481, December 1970.
80. L. G. Gubin, B. T. Polyak, and E. V. Raik, "The method of projections for finding the common point of convex sets," USSR Computational Mathematical and Mathematical Physics, vol. 7, no.6, pp. 1-24, 1967.
81. S. S. Gupta and J. O. Berger (Eds.), Statistical Decision Theory and Related Topics IV, vol. 1. New York: Springer-Verlag. 1988.
82. I. Halperin, "The product of projection operators," Acta Scientiarum Mathematicarum (Szeged), vol. 23, no.1, pp. 96-99, 1962.
83. M. H. Hayes, "The reconstruction of a multidimensional sequence from the phase or magnitude of its Fourier transform," IEEE Transactions on Acoustics, Speech, and Signal Processing, vol. 30, no.2, pp. 140-154, April 1982.
84. S. Hein and A. Zakhor, "Halftone to continuous-tone conversion of error-diffusion coded images," IEEE Transactions on Image Processing, vol. 4, no. 2, pp. 208-216, February 1995.
85. G. T. Herman, "A relaxation method for reconstructing objects from noisy X-rays," Mathematical Programming, vol.8, no. 1, pp. 1-19, February 1975.
86. G. T. Herman, Image Reconstruction from Projections, the Fundamentals of Computerized Tomography. New York: Academic Press, 1980.
87. G. T. Herman, "Mathematical optimization versus practical performance: A case study based on the maximum entropy criterion in image reconstruction," Mathematical Programming Study, vol. 20, pp. 96-112, October 1982.
88. G. T. Herman, H. Hunvitz, A. Lent, and H. P. Lung, "On the Bayesian approach to image reconstruction," Information and Control, vol. 42, no. 1, pp. 60-71, July 1979.
89. G. T. Herman, A. Lent, and P. H. Lutz, "Relaxation methods for image reconstruction," Communications of the ACM, vol. 21, no.2, pp. 152-158, February 1978.
90. B. R. Hunt, "The application of constrained least-squares estimation to image restoration by digital computer,"IEEE Transactionson Computers, vol. 22, no.9, pp. 805-812, September 1973.
91. B. R. Hunt, "Bayesian methods in nonlinear digital image restoration," IEEE Transactions on Computers, vol. 26, no.3, pp. 219-229, March 1977.
92. N. E. Hurt, "Signal enhancement and the method of successive projections," Acta Applicandae Mathematicae, vol. 23, no.2, pp. 145-162, May 1991.
93. A. N. Iusem and A. R. De Pierro, "Convergence results for an accelerated nonlinear Cimmino algorithm," Numerische Mathematik, vol. 49, no.4, pp. 367-378, August 1986.
94. G. M. Jenkins and D. G. Watts, Spectral Analysis and Its Applications. Oakland, CA: Holden Day, 1968.
95. B. Jessen, "To saetninger om konvekse punktmaengder," Matematisk Tiddskrift B, vol. 1940, pp. 66-70, 1940.
96. M. Jiang, "On Johnson's example of a nonconvex Chebyshev set," Journal of Approximation Theory, vol. 74, no. 2, pp. 152-158, August 1993.
97. G. G. Johnson, "A nonconvex set which has the unique nearest point property," Journal of Approximation Theory, vol. 51, no. 4, pp. 289-332, December 1987.
98. S. Kaczmarz, "Angenäherte Auflösung von Systemen hearer Gleichungen," Bulletin de I'Academie des Sciences de Pologne, vol. A35, pp. 355-357, 1937.
99. A. K. Katsaggelos (Ed.), Digital Image Restoration. New York: Springer-Verlag, 1991.
100. A. K. Katsaggelos, J. Biemond, R. W. Schafer, and R. M. Mersereau, "A regularized iterative image restoration algorithm," IEEE Transactions on Signal Processing, vol. 39, no. 4, pp. 914-929, April 1991.
101. H. Kudo and T. Saito, "Sinogram recovery with the method of convex projections for limited-data reconstruction in computed tomography," Journal of the Optical Society of America A , vol. 8 , no. 7, pp. 1148-1160, July 1991.
102. 1 norm," IEEE Transactions on Signal Processing, vol. 40, no. 1, pp. 159-168, January 1992.
103. H. J. Landau and W. L. Miranker, "The recovery of distorted band-limited signals," Journal of Mathematical Analysis and Applications, vol. 2, no.1, pp. 97-104, February 1961.
104. A. M. Landraud, "Image restoration and enhancement of characters, using convex projection methods," CVGIP: Graphical Models and Image Processing, vol. 53, no. I , pp. 85-92, January 1991.
105. K. Lange, M. Bahn, and R. Little, "A theoretical study of some maximum likelihood algorithms for emission and transmission tomography," IEEE Tramactions on Medical Imaging, vol. 6, no. 2, pp. 106-114, June 1987.
106. R. M. Leahy and C. E. Goutis, "An optimal technique for constraint-based image restoration and reconstruction," IEEE Transactions on Acoustics, Speech, and Signal Processing, vol. 34, no. 6, pp. 1626-1642, December 1986.
107. A. Lent and H. Tuy, "An iterative method for the extrapolation of band-limited functions," Journal of Mathematical Analysis and Applications, vol. 83, no. 2, pp. 554-565, October 1981.
108. D. G. Luenberger, Linear and Nonlinear Programming. 2nd ed. Redwood City, CA: Addison-Wesley, 1984.
109. A. Levi and H. Stark, "Signal reconstruction from phase by projection onto convex sets," Journal of the Optical Society of America, vol. 73, no. 6, pp. 810-822, June 1983. (Pubitemid 13556024)
110. A. Levi and H. Stark, "Image restoration by the method of generalized projections with application to restoration from magnitude," Journal of the Optical Society of America A , vol.1, no. 9, pp. 932-943, September 1984. (Pubitemid 14623326)
111. E. S. Levitin and B. T. Polyak, "Convergence of minimizing sequences in conditional extremum problems," Soviet Mathematics-Doklady, vol. 7, no. 3, pp. 764-767, May 1966.
112. J. L. Lions, Quelqws Methodes de Resolution des Problèmes aux Limites Non Lineaires. Paris: Dunod, 1969.
113. P. L. Lions, "Approximation de points fixes de contractions," Comptes Rendus de I'Academie des Sciences de Paris. vol. A284, no. 21, pp. 1357-1359, June 1977.
114. G. E. Mailloux, F. Langlois, P. Y. Simard , and M. Bertrand, "Restoration of the velocity field of the heart from two-dimensional echograms," IEEE Transactions on Medical Imaging, vol. 8 , no.2, pp. 143-153, June 1989.
115. S. Mallat and S. Zhong, "Characterization of signals from multiscale edges," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 14, no. 7, pp. 710-732, July 1992
116. J. Mandel, "Convergence of the cyclical relaxation method for linear inequalities," Mathematical Programming, vol. 30, no. 2, pp. 218-228, 1984.
117. C. P. Mariadassou and B. Yegnanarayana, "Image reconstruction from noisy digital holograms," IEE Proceedings-F, vol. 137, no. 5 , pp. 351-356, October 1990.
118. C. L. Matson, "Fourier spectrum extrapolation and enhancement using support constraints," IEEE Transactions on Signal Proceuing, vol. 42, no. 1 , pp. 156-163, January 1994.
119. R. G. Miller, Jr., Simultaneous Sfatisticaf Inference. 2nd ed. New York: Springer-Verlag, 1981.
120. A. Mohammad-Djafari and G. Demoment, "Maximum entropy image reconstruction in x-ray and diffraction tomography," IEEE Transactions on Medical Imaging, vol.7, no. 4, pp. 345-354, December 1988.
121. W. D. Montgomery, "Optical applications on Von Neumann's alternating-projection theorem," Optics Letters, vol. 7, no. I , pp. 1-3, January 1982.
122. J. J. Moreau, "Un cas de convergence des itkrkes d'une contraction d'un espace hilbertien," Comptes Rendus de I'Acadkmie des Sciences de Paris, vol. A286, no. 3, pp. 143-144, January 1978.
123. T. S. Motzkin and I. J. Schoenberg, "The relaxation method for linear inequalities," Canadian Journal of Mathematics, vol. 6, no. 3, pp. 393-404, 1954.
124. S. Oh, C. Ramon, R. J. Marks 11, A. C. Nelson, and M. G. Meyer, "Resolution enhancement of biomagnetic images using the method of alternating projections," IEEE Transactions on Biomedical Engineering, vol. 40, no. 4, pp. 323-328, April 1993.
125. P. Oskoui-Fard and H. Stark, "Tomographic image reconstruction using the theory of convex projections," IEEE Transactions on Medical Imaging, vol. 7, no. 1 , pp. 45-58, March 1988. (Pubitemid 18638884)
126. P. Oskoui and H. Stark, "A comparative study of three reconstruction methods for a limited-view computer tomography problem," IEEE Transactions on Medical Imaging, vol. 8, no. 1, pp. 43-49, March 1989.
127. N. Ottavy, "Strong convergence of projection-like methods in Hilbert spaces," Journal of Optimization Theory and Applications, vol. 56, no. 3, pp, 433-461, March 1988.
128. M. K. Ozkan, A. M. Tekalp, and M. I. Sezan, "POCS-based restoration of spacevarying blurred images," IEEE Transactions on Image Processing, vol. 3, no. 4, pp. 450-454, July 1994.
129. A. Papoulis, "A new algorithm in spectral analysis and band-limited extrapolation," IEEE Transactions on Circuits and Systems, vol. 22, no. 9, pp. 735-742, September 1975.
130. H. Peng and H. Stark, "Signal recovery with similarity constraints," Journal of the Optical Society of American A , vol. 6, no. 6, pp. 844-851, June 1989.
131. H. Peng and H. Stark, "Image recovery in computer tomography from partial fanbeam data by convex projections," IEEE Transactions on Medical Imaging, vol. 11 , no. 4, pp. 470-478, December 1992.
132. W. V. Petryshyn, "Construction of fixed points of demicompact mappings in Hilbert space," Journal of Mathematical Analysis and dpplications, vol. 14, no. 2, pp. 276-284, May 1966.
133. G. Pierra, "Méthodes de projections parallèles extrapolées relatives à une intersection de convexes." Rapport de Recherche, INPG, Grenoble, France, September 1975.
134. See also "Decomposition through formalization in a product space," Mathematical Programming, vol. 28, no. I , pp. 96-115, January 1984.
135. 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.), pp. 60-66. Sverdlovsk: Akademiia Nauk SSSR, 1982.
136. H. Poincart, "Sur les équations aux derivées partielles de la physique Mathematique," American Journal of Mathematics, vol. 12, pp. 211-294, 1890.
137. E. Polak, Computational Methods in Optimization: A Uni3ed Approach. New York: Academic Press, 1971.
138. B. T. Polyak, "Minimization of unsmooth functionals," USSR Computational Mathematics and Mathematical Physics, vol. 9, no.3, pp. 14-29, 1969.
139. B. T. Polyak, Introduction to Optimization. New York: Optimization Software Inc., 1987.
140. W. K. Pratt, Digital Image Processing, 2nd ed. New York: Wiley, 1991.
141. R. Ramaseshan and B. Yegnanarayana, "Image reconstruction from multiple frames of sparse data," Multidimensional Systems and Signal Processing, vol. 4, no.2, pp. 167-179, April 1993.
142. R. Rangayyan, A. P. Dhawan, and R. Gordon, "Algorithms for limited-view computed tomography: An annoted bibliography and a challenge," Applied Optics, vol. 24, no.23, pp. 4000-4012, December 1985.
143. R. T. Rockafellar, "Monotone operators and the proximal point algorithm," SIAM Journal on Control and Optimization, vol. 14, no.5, pp. 877-898, August 1976.
144. A. J. Rockmore and A. Macovski, "A maximum likelihood approach to emission image reconstruction from projections," IEEE Transactions on Nuclear Science, vol. 23, no. 4, pp. 1428-1432, August 1976.
145. M. Rosenblatt, Stationary Sequences and Random Fields. Boston, MA: Birkhauser, 1985.
146. C. Sálnchez-Avila, "An adaptive regularized method for deconvolution of signals with edges by convex projections," IEEE Transactions on Signal Processing, vol. 42, no.7, pp. 1849-1851, July 1994.
147. J. L. C. Sanz and T. S. Huang, "Unified Hilbert space approach to iterative leastsquares linear signal restoration," Journal of the Optical Society of America, vol. 73, no. 11, pp. 1455-1465, November 1983.
148. Sasaki and T. Yamagami, "Image restoration in singular vector space by the method of convex projections," Applied Optics, vol. 26, no. 7, pp. 1216-1221, April 1987.
149. K. D. Sauer and J. P. Allebach, "Iterative reconstruction of band-limited images from nonuniformly spaced samples," IEEE Transactions on Circuits and Systems, vol. 34, no. 12, pp. 1497-1506, December 1987.
150. R. W. Schafer, R. M. Mersereau, and M. A. Richards, "Constrained iterative restoration algorithms," Proceedings of the IEEE, vol. 69, no. 4, pp. 432-450, April 1981.
151. L. Schwartz, Analyse IIZ-Calcul Intégral. Paris: Hermann, 1993.
152. F. C. Schweppe, "Recursive state estimation: Unknown but bounded errors and system inputs," ZEEE Transactions on Automatic Control, vol. 13, no.1, pp. 22-28, February 1968.
153. M. I. Sezan and H. Stark, "Image restoration by the method of convex projections: Part 2-Applications and numerical results," IEEE Transactions on Medical Imaging, vol. 1, no.2, pp. 95-101, October 1982. ("Correction," vol. 1, no. 3, p. 204, December 1982.)
154. M. I. Sezan and H. Stark, "Image restoration by convex projections in the presence of noise," Applied Optics, vol. 22, no.18, pp. 2781-2789, September 1983. (Pubitemid 14469212)
155. M. I. Sezan and H. Stark, "Tomographic image reconstruction from incomplete view data by convex projections and direct Fourier inversion," IEEE Transactions on Medicallmaging, vol. 3, no. 2, pp. 91-98, June 1984
156. M. I. Sezan and H. Stark, "Incorporation of a priori moment information into signal recovery and synthesis problems," Journal ofMathematica1 Analysis and Applications, vol. 122, no.1, pp. 172-186, February 1987.
157. M. I. Sezan and A. M. Tekalp, "Adaptive image restoration with artifact suppression using the theory of convex projections," IEEE Transactions on Acoustics, Speech, and Signal Processing, vol. 38, no.1 , pp. 181-185, January 1990.
158. M. I. Sezan and A. M. Tekalp, "Survey of recent developments in digital image restoration," Optical Engineering, vol. 29, no. 5 , pp. 393-404, May 1990.
159. M. I. Sezan and H. J. Trussell, "Prototype image constraints for set-theoretic image restoration," IEEE Transactions on Signal Processing, vol. 39, no. 10, pp. 2275-2285, October 1991.
160. P. Y. Simard and G. E. Mailloux, "Vector field restoration by the method of convex projections," Computer Vision, Graphics, and Image Processing, vol. 52, no. 3, pp. 360-385, December 1990.
161. H. Stark (Ed.), Image Recouery: Theory and Application. San Diego, CA: Academic Press, 1987.
162. H. Stark, D. Cahana, and H. Webb, "Restoration of arbitrary finite energy optical objects from limited spatial and spectral information," Journal of the Optical Society of America, vol. 71, no.6, pp. 635-642, June 1981.
163. H. Stark and E. T. Olsen, "Projection-based image restoration," Journal of the Optical Society of America A , vol. 9, no. 1 1 , pp. 1914-1919, November 1992.
164. H. Stark and P. Oskoui, "High-resolution image recovery from image-plane arrays, using convex projections," Journal of the Optical Society of America A , vol. 6, no.1 1 , pp. 1715-1726, November 1989.
165. W. J. Stiles, "Closest point maps and their product 11," Nieuw Archiefuoor Wiskunde, vol. 13, no. 3, pp. 212-225, November 1965.
166. A. M. Tekalp and H. J. Trussell, "Comparative study of some statistical and settheoretic methods for image restoration," CVGIP: Graphical Models and Image Processing, vol. 53, no. 2, pp. 108-120, March 1991.
167. V. T. Tom, T. F. Quatieri, M. H. Hayes, andJ. H. McClellan, "Convergence ofiterative nonexpansive signal reconstruction algorithms," IEEE Transactions on Acoustics, Speech, and Signal Processing, vol. 29, no. 5, pp. 1052-1058, October 1981.
168. H. J. Trussell, "Notes on linear image restoration by maximizing the a posteriori probability," IEEE Transactions on Acoustics, Speech, and Signal Processing, vol. 26, no. 2, pp. 174-176, April 1978.
169. H. J. Trussell, "The relationship between image restoration by maximum a posteriori and maximum entropy methods," IEEE Transactions on Acoustics, Speech, and Signal Processing, vol. 28, no.1 , pp. 114-117, February 1980.
170. H. J. Trussell, "Maximum power signal restoration," IEEE Transactions on Acoustics, Speech, and Signal Processing, vol. 29, no. 5, pp. 1059-1061, October 1981.
171. H. J. Trussell, "Convergence criteria for iterative restoration methods," IEEE Transactions on Acoustics, Speech, and Signal Processing, vol. 31, no.1, pp. 129-136, February 1983.
172. H. J. Trussell, "A priori knowledge in algebraic reconstruction methods," in Advances in Computer Vision and Image Processing (T. S . Huang, Ed.), vol. 1, pp. 265-316. Greenwich, CT: JAI Press, 1984.
173. H. J. Trussell and M. R. Civanlar, "The feasible solution in signal restoration," IEEE Transactions on Acoustics, Speech, and SignalProcessing, vol. 32, no. 2, pp. 201-212, April 1984.
174. H. J . Trussell, H. Orun-Ozturk, and M. R. Civanlar, "Errors in reprojection methods in computerized tomography," IEEE Transactions on Medical Imaging, vol. 6, no, 3, pp. 220-227, September 1987.
175. H. J. Trussell and P. L. Vora, "Bounds on restoration quality using a priori information," Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing, pp. 1758-1761. New York, NY, April 11-14, 1988.
176. P. Tseng, "On the convergence of products of firmly nonexpansive mappings," SIAM Journal on Optimization, vol. 2, no. 3, pp. 425-434, August 1992.
177. P. Tseng and D. P. Bertsekas, "Relaxation methods for problems with strictly convex separable costs and linear constraints," Mathematical Programming, vol. 38, no. 3, pp. 303-321, 1987.
178. J. Von Neumann, "On rings of operators. Reduction theory," Annals of Mathematics, vol. 50, no. 2, pp. 401-485, April 1949 (the result of interest first appeared in 1933 in lecture notes)
179. S. J. Wernecke and L. R. D'Addario, "Maximum entropy image reconstruction," IEEE Transactions on Computers, vol.26, no. 4, pp. 351-364, April 1977.
180. M. N. Wernick and C. T. Chen, "Superresolved tomography by convex projections and detector motion," Journal of the Optical Society of America A , vol. 9, no. 9, pp. 1547-1553, September 1992.
181. R. Wittmann, "Approximation of fixed points of nonexpansive mappings," Archiu der Mathematik, vol. 58, no. 5, pp. 486-491, May 1992.
182. S. J. Yeh and H. Stark, "Iterative and one-step reconstruction from nonuniform samples by convex projections," Journal of the Optical Society of America A , vol.7, no. 3, pp. 491-499, March 1990.
183. D. C. Youla, "Generalized image restoration by the method of alternating orthogonal projections," IEEE Transactions on Circuits and Systems, vol. 25, no. 9, pp. 694-702, September 1978.
184. D. C. Youla, "On deterministic convergence of iterations of relaxed projection operators," Journal of Visual Communication and Image Representation, vol. 1 , no. 1, pp. 12-20, September 1990.
185. D. C. Youla and V. Velasco, "Extensions of a result on the synthesis of signals in the presence of inconsistent constraints," IEEE Transactions on Circuits and Systems, vol. 33, no. 4, pp. 465-468, April 1986.
186. D. C. Youla and H. Webb, "Image restoration by the method of convex projections: Part I-Theory," IEEE Transactions on Medical Imaging, vol. 1 , no. 2, pp. 81-94, October 1982.
187. H. T. Yura and S. G. Hanson, "Second-order statistics for wave propagation through complex optical systems," Journal ofthe Optical Society of America A , vol. 6, no, 4, pp. 564-575, April 1989.
188. L. A. Zadeh, "What is optimal?," IRE Transactions on Information Theory, vol. 4, no. 1, p. 3, March 1958.
189. E. H. Zarantonello, "Projections on convex sets in Hilbert space and spectral theory," in Contributions to Nonlinear Functional Analysis (E. H. Zarantonello, Ed.), pp. 237-424. New York: Academic Press, 1971.
190. E. Zeidler, Nonlinear Functional Analysis and Its Applications I : Fixed-Point Theorems, 2nd ed. New York: Springer-Verlag, 1993.
191. E. Zeidler, Nonlinear Functional Analysis and Its Applications IIIA: Linear Monotone Operators. New York: Springer-Verlag. 1990.
192. E. Zeidler, Nonlinear Functional Analysis and Its Applications III : Variational Methods and Optimization. New York: Springer-Verlag, 198