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808Chapter 18.Integral Equations and Inverse TheoryThe single central idea in inverse theory is the prescriptionminimize:A + λB(18.4.12)CITED REFERENCES AND FURTHER READING:Craig, I.J.D., and Brown, J.C. 1986, Inverse Problems in Astronomy (Bristol, U.K.: Adam Hilger).Twomey, S. 1977, Introduction to the Mathematics of Inversion in Remote Sensing and IndirectMeasurements (Amsterdam: Elsevier).Tikhonov, A.N., and Arsenin, V.Y. 1977, Solutions of Ill-Posed Problems (New York: Wiley).Tikhonov, A.N., and Goncharsky, A.V. (eds.) 1987, Ill-Posed Problems in the Natural Sciences(Moscow: MIR).Parker, R.L.
1977, Annual Review of Earth and Planetary Science, vol. 5, pp. 35–64.Frieden, B.R. 1975, in Picture Processing and Digital Filtering, T.S. Huang, ed. (New York:Springer-Verlag).Tarantola, A. 1987, Inverse Problem Theory (Amsterdam: Elsevier).Baumeister, J. 1987, Stable Solution of Inverse Problems (Braunschweig, Germany: Friedr. Vieweg& Sohn) [mathematically oriented].Titterington, D.M. 1985, Astronomy and Astrophysics, vol. 144, pp.
381–387.Jeffrey, W., and Rosner, R. 1986, Astrophysical Journal, vol. 310, pp. 463–472.18.5 Linear Regularization MethodsWhat we will call linear regularization is also called the Phillips-Twomeymethod [1,2] , the constrained linear inversion method [3], the method of regularization [4], and Tikhonov-Miller regularization [5-7]. (It probably has other names also,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).for various values of 0 < λ < ∞ along the so-called trade-off curve (see Figure18.4.1), and then to settle on a “best” value of λ by one or another criterion, rangingfrom fairly objective (e.g., making χ2 = N ) to entirely subjective. Successfulmethods, several of which we will now describe, differ as to their choices of A andB, as to whether the prescription (18.4.12) yields linear or nonlinear equations, asto their recommended method for selecting a final λ, and as to their practicality forcomputer-intensive two-dimensional problems like image processing.They also differ as to the philosophical baggage that they (or rather, theirproponents) carry.
We have thus far avoided the word “Bayesian.” (Courts haveconsistently held that academic license does not extend to shouting “Bayesian” in acrowded lecture hall.) But it is hard, nor have we any wish, to disguise the fact thatB has something to do with a priori expectation, or knowledge, of a solution, whileA has something to do with a posteriori knowledge. The constant λ adjudicates adelicate compromise between the two.
Some inverse methods have acquired a moreBayesian stamp than others, but we think that this is purely an accident of history.An outsider looking only at the equations that are actually solved, and not at theaccompanying philosophical justifications, would have a difficult time separating theso-called Bayesian methods from the so-called empirical ones, we think.The next three sections discuss three different approaches to the problem ofinversion, which have had considerable success in different fields. All three fitwithin the general framework that we have outlined, but they are quite different indetail and in implementation.80918.5 Linear Regularization Methodsµ=1since it is nonnegative and equal to zero only when ub(x) is constant. Hereb(xµ ), and the second equality (proportionality) assumes that the xµ ’s areubµ ≡ uuniformly spaced.
We can write the second form of B asB = |B · bu|2 = bu · (BT · B) · bu≡bu·H·bu(18.5.2)b is the vector of components uwhere ubµ , µ = 1, . . . , M , B is the (M − 1) × Mfirst difference matrix−1100000 ···010000 ···0 0 −1 ... ....(18.5.3)B=.. 0 ···0000 −1100 ···00000 −11and H is the M × M matrix1 −102 −1 −12 0 −1 ...H = BT · B = 0 0 ··· 0 ···00 ···000−1000..000.000000000−120 −100000.. .−102 −1 −11·········(18.5.4)Note that B has one fewer row than column. It follows that the symmetric His degenerate; it has exactly one zero eigenvalue corresponding to the value of aconstant function, any one of which makes B exactly zero.If, just as in §15.4, we writeAiµ ≡ Riµ/σibi ≡ ci /σi(18.5.5)then, using equation (18.4.9), the minimization principle (18.4.12) isminimize:A + λB = |A · bu − b|2 + λbu·H·bu(18.5.6)This can readily be reduced to a linear set of normal equations, just as in §15.4: Thecomponents ubµ of the solution satisfy the set of M equations in M unknowns,"#XX Xbρ =Aiµ Aiρ + λHµρ uAiµ biµ = 1, 2, .
. . , M (18.5.7)ρiiSample 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).since it is so obviously a good idea.) In its simplest form, the method is an immediategeneralization of zeroth-order regularization (equation 18.4.11, above). As before,the functional A is taken to be the χ2 deviation, equation (18.4.9), but the functionalB is replaced by more sophisticated measures of smoothness that derive from firstor higher derivatives.For example, suppose that your a priori belief is that a credible u(x) is not toodifferent from a constant.
Then a reasonable functional to minimize isZM−1X02[buµ − ubµ+1 ]2(18.5.1)B ∝ [bu (x)] dx ∝810Chapter 18.Integral Equations and Inverse Theoryor, in vector notation,u = AT · b(AT · A + λH) · b(18.5.8)where the identity matrix in the form A · A−1 has been inserted. This is schematicnot only because the matrix inverse is fancifully written as a denominator, butalso because, in general, the inverse matrix A−1 does not exist. However, it isilluminating to compare equation (18.5.9) with equation (13.3.6) for optimal orWiener filtering, or with equation (13.6.6) for general linear prediction.
One seesthat AT · A plays the role of S 2 , the signal power or autocorrelation, while λHplays the role of N 2 , the noise power or autocorrelation. The term in parenthesesin equation (18.5.9) is something like an optimal filter, whose effect is to pass theill-posed inverse A−1 · b through unmodified when AT · A is sufficiently large, butto suppress it when AT · A is small.The above choices of B and H are only the simplest in an obvious sequence ofderivatives. If your a priori belief is that a linear function is a good approximationto u(x), then minimizeZM−2XB ∝ [bu00 (x)]2 dx ∝[−buµ + 2buµ+1 − ubµ+2 ]2(18.5.10)µ=1implying−12 −102 −1 0 −1 ..B= . 0 ···000 ···00and1 −2 −25 1 −41 0 ...H = BT · B = 0 ··· 0 ··· 0 ···0 ···00...0010−416 −4−46000010000000−120 −1001−4...0001······00..
.0(18.5.11)−12 −10000−46 −41 −4601 −40010000.. .10−415 −2 −21············(18.5.12)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).Equations (18.5.7) or (18.5.8) can be solved by the standard techniques ofChapter 2, e.g., LU decomposition.
The usual warnings about normal equationsbeing ill-conditioned do not apply, since the whole purpose of the λ term is to curethat same ill-conditioning. Note, however, that the λ term by itself is ill-conditioned,since it does not select a preferred constant value. You hope your data can atleast do that!Although inversion of the matrix (AT · A + λH) is not generally the best way tosolve for bu, let us digress to write the solution to equation (18.5.8) schematically as1Tbu=· A · A A−1 · b(schematic only!)(18.5.9)AT · A + λH81118.5 Linear Regularization MethodsThis H has two zero eigenvalues, corresponding to the two undetermined parametersof a linear function.If your a priori belief is that a quadratic function is preferable, then minimizeZB∝[bu000(x)]2 dx ∝M−3X[−buµ + 3buµ+1 − 3buµ+2 + ubµ+3 ]2(18.5.13)with−13 −3103 −31 0 −1 ....B = .. 0 ···00 −10 ···000and now00003 −3−13······1−300..
.0(18.5.14)100000.. .20 −156−10−1520 −156 −1 6 −1519 −123−16 −1210 −3 0−13−31(18.5.15)(We’ll leave the calculation of cubics and above to the compulsive reader.)Notice that you can regularize with “closeness to a differential equation,” ifyou want. Just pick B to be the appropriate sum of finite-difference operators (thecoefficients can depend on x), and calculate H = BT · B.
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