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81518.6 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) as816Chapter 18.Integral Equations and Inverse TheoryZub(x) =b x0 )u(x0 )dx0δ(x,(18.6.2)b x0 ). The Backusfor some so-called resolution function or averaging kernel δ(x,bGilbert method seeks to minimize the width or spread of δ (that is, maximize theresolving power). A is chosen to be some positive measure of the spread.While Backus-Gilbert’s philosophy is thus rather different from that of PhillipsTwomey and related methods, in practice the differences between the methods areless than one might think.
A stable solution is almost inevitably bound to besmooth: The wild, unstable oscillations that result from an unregularized solutionare always exquisitely sensitive to small changes in the data. Likewise, makingub(x) close to u(x) inevitably will bring error-free data into agreement with themodel. Thus A and B play roles closely analogous to their corresponding rolesin the previous two sections.The principal advantage of the Backus-Gilbert formulation is that it gives goodcontrol over just those properties that it seeks to measure, namely stability andresolving power.
Moreover, in the Backus-Gilbert method, the choice of λ (playingits usual role of compromise between A and B) is conventionally made, or at leastcan easily be made, before any actual data are processed. One’s uneasiness at makinga post hoc, and therefore potentially subjectively biased, choice of λ is thus removed.Backus-Gilbert is often recommended as the method of choice for designing, andpredicting the performance of, experiments that require data inversion.Let’s see how this all works. Starting with equation (18.4.5),Zci ≡ si + ni =ri (x)u(x)dx + ni(18.6.3)and building in linearity from the start, we seek a set of inverse response kernelsqi (x) such thatub(x) =Xqi (x)ci(18.6.4)iis the desired estimator of u(x).
It is useful to define the integrals of the responsekernels for each data point,ZRi ≡ri (x)dx(18.6.5)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).is used as a measure of how much the solution ub(x) varies as the data vary withintheir measurement errors.
Note that this variance is not the expected deviation ofub(x) from the true u(x) — that will be constrained by A — but rather measuresthe expected experiment-to-experiment scatter among estimates ub(x) if the wholeexperiment were to be repeated many times.For A the Backus-Gilbert method looks at the relationship between the solutionub(x) and the true function u(x), and seeks to make the mapping between these asclose to the identity map as possible in the limit of error-free data. The method islinear, so the relationship between ub(x) and u(x) can be written as81718.6 Backus-Gilbert MethodSubstituting equation (18.6.4) into equation (18.6.3), and comparing with equation(18.6.2), we see thatb x0 ) =δ(x,Xqi (x)ri (x0 )(18.6.6)iiiwhere q(x) and R are each vectors of length N , the number of measurements.Standard propagation of errors, and equation (18.6.1), giveB = Var[bu(x)] =XXiqi (x)Sij qj (x) = q(x) · S · q(x)(18.6.8)jwhere Sij is the covariance matrix (equation 18.4.6).
If one can neglect off-diagonalcovariances (as when the errors on the ci ’s are independent), then Sij = δij σi2is diagonal.b x0 ) at eachWe now need to define a measure of the width or spread of δ(x,value of x. While many choices are possible, Backus and Gilbert choose the secondmoment of its square. This measure becomes the functional A,Zb x0)]2 dx0A ≡ w(x) = (x0 − x)2 [δ(x,XX(18.6.9)=qi (x)Wij (x)qj (x) ≡ q(x) · W(x) · q(x)ijwhere we have here used equation (18.6.6) and defined the spread matrix W(x) byZ(18.6.10)Wij (x) ≡ (x0 − x)2 ri (x0 )rj (x0 )dx0The functions qi (x) are now determined by the minimization principleminimize: A + λB = q(x) · W(x) + λS · q(x)(18.6.11)subject to the constraint (18.6.7) that q(x) · R = 1.The solution of equation (18.6.11) isq(x) =[W(x) + λS]−1 · RR · [W(x) + λS]−1 · R(18.6.12)(Reference [4] gives an accessible proof.) For any particular data set c (set ofb(x) is thusmeasurements ci ), the solution uub(x) =c · [W(x) + λS]−1 · RR · [W(x) + λS]−1 · R(18.6.13)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).We can require this averaging kernel to have unit area at every x, givingZZXXb x0)dx0 =qi (x) ri (x0 )dx0 =qi (x)Ri ≡ q(x) · R (18.6.7)1 = δ(x,818Chapter 18.Integral Equations and Inverse TheoryCITED REFERENCES AND FURTHER READING:Backus, G.E., and Gilbert, F. 1968, Geophysical Journal of the Royal Astronomical Society,vol. 16, pp.
169–205. [1]Backus, G.E., and Gilbert, F. 1970, Philosophical Transactions of the Royal Society of LondonA, vol. 266, pp. 123–192. [2]Parker, R.L. 1977, Annual Review of Earth and Planetary Science, vol. 5, pp. 35–64. [3]Loredo, T.J., and Epstein, R.I. 1989, Astrophysical Journal, vol. 336, pp. 896–919. [4]18.7 Maximum Entropy Image RestorationAbove, we commented that the association of certain inversion methodswith Bayesian arguments is more historical accident than intellectual imperative.Maximum entropy methods, so-called, are notorious in this regard; to summarizethese methods without some, at least introductory, Bayesian invocations would beto serve a steak without the sizzle, or a sundae without the cherry.
We shouldSample 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).(Don’t let this notation mislead you into inverting the full matrix W(x) + λS.
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