18.6 Backus-Gilbert Method 815 necessary.(For"unsticking"procedures,see [101.)The uniqueness of the solution is also not well understood,although for two-dimensional images of reasonable complexity it is believed to be unique. Deterministic constraints can be incorporated,via projection operators,into iterative methods of linear regularization.In particular,rearranging terms somewhat we can write the iteration (18.5.21)as +1)=(1-eH田）.)+AT.(b-A.) (18.5.27) If the iteration is modified by the insertion of projection operators at each step +1=(P1P2.Pm)[1-AHD·)+AT.(b-A.】 (18.5.28) (or,instead of Pi's,the Ti operators of equation 18.5.26),then it can be shown that 茶 the convergence condition(18.5.22)is unmodified,and the iteration will converge to minimize the quadratic functional (18.5.6)subject to the desired nonlinear deterministic constraints.See [7]for references to more sophisticated,and faster converging,iterations along these lines. 9 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] 9 Twomey,S.1977,Introduction to the Mathematics of Inversion in Remote Sensing and Indirect 9 Measurements (Amsterdam:Elsevier).[3] 是 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 Il-Posed Problems (New York:Wiley).[5] 6 Tikhonov,A.N.,and Goncharsky,A.V.(eds.)1987,IlI-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 /EEE.vol.69. Pp.432-450. Biemond,J.,Lagendijk,R.L.,and Mersereau,R.M.1990,Proceedings of the /EEE,vol.78. pp.856-883.7 Numerica 10621 Gerchberg,R.W.and Saxton,W.O.1972.Optik,vol.35.pp.237-246.[8] 431 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, (outside Recipes pp.1897-1907.[101 North Software. 18.6 Backus-Gilbert Method The Backus-Gilbert method [1.2](see,e.g.,[3]or [4]for summaries)differs from other regularization methods in the nature of its functionals A and B.For B,the method seeks to maximize the stability of the solution u(x)rather than,in the first instance,its smoothness.That is, B≡Var[a(x] (18.6.1)18.6 Backus-Gilbert Method 815 Permission is granted for internet users to make one paper copy for their own personal use. Further reproduction, or any copyin Copyright (C) 1988-1992 by Cambridge University Press. Programs Copyright (C) 1988-1992 by Numerical Recipes Software. Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5) g of machine￾readable files (including this one) to any server computer, is strictly prohibited. To order Numerical Recipes books or CDROMs, visit website http://www.nr.com or call 1-800-872-7423 (North America only), or send email to directcustserv@cambridge.org (outside North America). necessary. (For “unsticking” procedures, see [10].) The uniqueness of the solution is also not well understood, although for two-dimensional images of reasonable complexity it is believed to be unique. Deterministic constraints can be incorporated, via projection operators, into iterative methods of linear regularization. In particular, rearranging terms somewhat, we can write the iteration (18.5.21) as
u(k+1) = (1 −
λH) ·
u(k) +
AT · (b − A ·
u(k) ) (18.5.27) If the iteration is modified by the insertion of projection operators at each step
u(k+1) = (P1P2 ···Pm)[(1 −
λH) ·
u(k) +
AT · (b − A ·
u(k) )] (18.5.28) (or, instead of Pi’s, the Ti operators of equation 18.5.26), then it can be shown that the convergence condition (18.5.22) is unmodified, and the iteration will converge to minimize the quadratic functional (18.5.6) subject to the desired nonlinear deterministic constraints. See [7] for references to more sophisticated, and faster converging, 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 Indirect Measurements (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 Method The Backus-Gilbert method [1,2] (see, e.g., [3] or [4] for summaries) differs from other regularization methods in the nature of its functionals A and B. For B, the method seeks to maximize the stability of the solution u
(x) rather than, in the first instance, its smoothness. That is, B ≡ Var[u
(x)] (18.6.1)