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While thisalgorithm is non-expansive, and is frequently convergent in practice, it has not beenproved to converge in all cases [9]. In the phase-retrieval problem mentioned above,the algorithm often “gets stuck” on a plateau for many iterations before makingsudden, dramatic improvements. As many as 104 to 105 iterations are sometimesSample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5)Copyright (C) 1988-1992 by Cambridge University Press.Programs Copyright (C) 1988-1992 by Numerical Recipes Software.Permission is granted for internet users to make one paper copy for their own personal use.
Further reproduction, or any copying of machinereadable files (including this one) to any servercomputer, is strictly prohibited. To order Numerical Recipes books,diskettes, or CDROMsvisit website http://www.nr.com or call 1-800-872-7423 (North America only),or send email to trade@cup.cam.ac.uk (outside North America).• known bounds (i.e., uL (x) ≤ ub(x) ≤ uU (x) for specified functions uLand uU ).(In this last case, the bounds might be related to an initial estimate and its error bars,e.g., ub0 (x) ± γσ(x), where γ is of order 1 or 2.) Notice that these, and similar,constraints can be either in the image space, or in the Fourier transform space, or(in fact) in the space of any linear transformation of bu.If Ci is a convex set, then Pi is called a nonexpansive projection operator ontothat set if (i) Pi leaves unchanged any bu already in Ci , and (ii) Pi maps any bu outsideCi to the closest element of Ci , in the sense that81518.6 Backus-Gilbert Methodbu(k+1) = [1 − λH] · bu(k) + AT · (b − A · bu(k) )(18.5.27)If the iteration is modified by the insertion of projection operators at each stepbu(k) + AT · (b − A · bu(k))u(k+1) = (P1 P2 · · · Pm )[1 − λH] · b(18.5.28)(or, instead of Pi ’s, the Ti operators of equation 18.5.26), then it can be shown thatthe convergence condition (18.5.22) is unmodified, and the iteration will convergeto minimize the quadratic functional (18.5.6) subject to the desired nonlineardeterministic constraints.
See [7] for references to more sophisticated, and fasterconverging, iterations along these lines.CITED REFERENCES AND FURTHER READING:Phillips, D.L. 1962, Journal of the Association for Computing Machinery, vol. 9, pp. 84–97. [1]Twomey, S. 1963, Journal of the Association for Computing Machinery, vol. 10, pp. 97–101. [2]Twomey, S.
1977, Introduction to the Mathematics of Inversion in Remote Sensing and IndirectMeasurements (Amsterdam: Elsevier). [3]Craig, I.J.D., and Brown, J.C. 1986, Inverse Problems in Astronomy (Bristol, U.K.: Adam Hilger).[4]Tikhonov, A.N., and Arsenin, V.Y. 1977, Solutions of Ill-Posed Problems (New York: Wiley). [5]Tikhonov, A.N., and Goncharsky, A.V.
(eds.) 1987, Ill-Posed Problems in the Natural Sciences(Moscow: MIR).Miller, K. 1970, SIAM Journal on Mathematical Analysis, vol. 1, pp. 52–74. [6]Schafer, R.W., Mersereau, R.M., and Richards, M.A. 1981, Proceedings of the IEEE, vol. 69,pp. 432–450.Biemond, J., Lagendijk, R.L., and Mersereau, R.M.
1990, Proceedings of the IEEE, vol. 78,pp. 856–883. [7]Gerchberg, R.W., and Saxton, W.O. 1972, Optik, vol. 35, pp. 237–246. [8]Fienup, J.R. 1982, Applied Optics, vol. 15, pp. 2758–2769. [9]Fienup, J.R., and Wackerman, C.C. 1986, Journal of the Optical Society of America A, vol. 3,pp. 1897–1907. [10]18.6 Backus-Gilbert MethodThe Backus-Gilbert method [1,2] (see, e.g., [3] or [4] for summaries) differs fromother regularization methods in the nature of its functionals A and B. For B, themethod seeks to maximize the stability of the solution ub(x) rather than, in the firstinstance, its smoothness.
That is,B ≡ Var[bu(x)](18.6.1)Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5)Copyright (C) 1988-1992 by Cambridge University Press.Programs Copyright (C) 1988-1992 by Numerical Recipes Software.Permission is granted for internet users to make one paper copy for their own personal use.
Further reproduction, or any copying of machinereadable files (including this one) to any servercomputer, is strictly prohibited. To order Numerical Recipes books,diskettes, or CDROMsvisit website http://www.nr.com or call 1-800-872-7423 (North America only),or send email to trade@cup.cam.ac.uk (outside North America).necessary. (For “unsticking” procedures, see [10].) The uniqueness of the solutionis also not well understood, although for two-dimensional images of reasonablecomplexity it is believed to be unique.Deterministic constraints can be incorporated, via projection operators, intoiterative methods of linear regularization. In particular, rearranging terms somewhat,we can write the iteration (18.5.21) as.
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