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Erschienen in:
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2015 | OriginalPaper | Buchkapitel

Linear Inverse Problems

verfasst von : Charles Groetsch

Erschienen in: Handbook of Mathematical Methods in Imaging

Verlag: Springer New York

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Abstract

This introductory treatment of linear inverse problems is aimed at students and neophytes. A historical survey of inverse problems and some examples of model inverse problems related to imaging are discussed to furnish context and texture to the mathematical theory that follows. The development takes place within the sphere of the theory of compact linear operators on Hilbert space, and the singular value decomposition plays an essential role. The primary concern is regularization theory: the construction of convergent well-posed approximations to ill-posed problems. For the most part, the discussion is limited to the familiar regularization method devised by Tikhonov and Phillips.

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Nächstes Kapitel Large-Scale Inverse Problems in Imaging
Literatur
1.
Ambarzumian, V.: On the derivation of the frequency function of space velocities of the stars from the observed radial velocities. Mon. Not. R. Astron. Soc. Lond. 96, 172–179 (1936)CrossRef
2.
Anderssen, R.S.: Inverse problems: a pragmatist’s approach to the recovery of information from indirect measurements. Aust. N.Z. Ind. Appl. Math. J. 46, 588–622 (2004)MathSciNet
3.
Aster, R., Borchers, B., Thurber, C.: Parameter Estimation and Inverse Problems. Elsevier, Boston (2005)MATH
4.
Bennett, A.: Inverse Modeling of the Ocean and Atmosphere. Cambridge University Press, Cambridge (2002)CrossRefMATH
5.
Ben-Israel, A.: The Moore of the Moore penrose inverse. Electron. J. Linear Algebr. 9, 150–157 (2002)MATHMathSciNet
6.
Bertero, M., Boccacci, P.: Introduction to Inverse Problems in Imaging. IOP, London (1998)CrossRefMATH
7.
Bonilla, L. (ed.): Inverse Problems and Imaging. LNM 1943. Springer, Berlin (2008)
8.
Carasso, A., Sanderson, J., Hyman, J.: Digital removal of random media image degradations by solving the diffusion equation backwards in time. SIAM J. Numer. Anal. 15, 344–367 (1978)CrossRefMATHMathSciNet
9.
Chalmond, B.: Modeling and Inverse Problems in Image Analysis. Springer, New York (2003)CrossRefMATH
10.
11.
Chen, Z., Xu, Y., Yang, H.: Fast collocation methods for solving ill-posed integral equations of the first kind. Inverse Probl. 24, 065007(21) (2008)CrossRefMathSciNet
12.
Cormack, A.: Representation of a function by its line integrals, with some radiological applications I. J. Appl. Phys. 34, 2722–2727 (1963)CrossRefMATH
13.
Cormack, A.: Representation of a function by its line integrals, with some radiological applications II. J. Appl. Phys. 35, 2908–2912 (1964)CrossRefMATH
14.
Cormack, A.: Computed tomography: some history and recent developments. In: Shepp, L.A. (ed.) Computed Tomography. Proceedings of Symposia in Applied Mathematics, vol. 27, pp. 35–42. American Mathematical Society, Providence (1983)
15.
Courant, R., Hilbert, D.: Methods of Mathematical Physics. Partial Differential Equations, vol. 2. Interscience, New York (1962)
16.
Craig, I., Brown, J.: Inverse Problems in Astronomy. Adam Hilger, Bristol (1986)MATH
17.
Deans, S.R.: The Radon Transform and Some of Its Applications. Wiley, New York (1983)MATH
18.
Deutsch, F.: Best Approximation in Inner Product Spaces. Springer, New York (2001)CrossRefMATH
19.
Engl, H.W., Hanke, M., Neubauer, A.: Regularization of Inverse Problems. Kluwer, Dordrecht (1996)CrossRefMATH
20.
Epstein, C.L.: Introduction to the Mathematics of Medical Imaging. Pearson Education, Upper Saddle River (2003)MATH
21.
Galilei, G.: Sidereus Nuncius (Trans.: van Helden, A.). University of Chicago Press, Chicago, 1989 (1610)CrossRef
22.
Gates, E.: Einstein’s Telescope. W.W. Norton, New York (2009)
23.
24.
Glasko, V.: Inverse Problems of Mathematical Physics (Trans.: Bincer, A. (Russian)). American Institute of Physics, New York (1984)
25.
Goldberg, R.R.: Fourier Transforms. Cambridge University Press, Cambridge (1961)MATH
26.
Groetsch, C.W.: Comments on Morozov’s Discrepancy Principle. In: Hämmerlin, G., Hoffmann, K.-H. (eds.) Improperly Posed Problems and Their Numerical Treatment, pp. 97–104. Birkhäuser, Basel (1983)CrossRef
27.
Groetsch, C.W.: On the asymptotic order of convergence of Tikhonov regularization. J. Optim. Theory Appl. 41, 293–298 (1983)CrossRefMATHMathSciNet
28.
Groetsch, C.W.: The Theory of Tikhonov Regularization for Fredholm Equations of the First Kind. Pitman, Boston (1984)MATH
29.
Groetsch, C.W.: Convergence analysis of a regularized degenerate kernel method for Fredholm integral equations of the first kind. Integr. Equ. Oper. Theory 13, 67–75 (1990)CrossRefMATHMathSciNet
30.
Groetsch, C.W.: Inverse Problems in the Mathematical Sciences. Vieweg, Braunschweig (1993)CrossRefMATH
31.
Groetsch, C.W.: The delayed emergence of regularization theory. Bollettino di Storia delle Scienze Matematiche 23, 105–120 (2003)MATHMathSciNet
32.
Groetsch, C.W.: Nascent function concepts in Nova Scientia. Int. J. Math. Educ. Sci. Technol. 35, 867–875 (2004)CrossRefMathSciNet
33.
34.
Groetsch, C.W., Neubauer, A.: Regularization of ill-posed problems: optimal parameter choice in finite dimensions. J. Approx. Theory 58, 184–200 (1989)CrossRefMATHMathSciNet
35.
Groetsch, C.W.: Stable Approximate Evaluation of Unbounded Operators. LNM 1894. Springer, New York (2007)
36.
Grosser, M.: The Discovery of Neptune. Harvard University Press, Cambridge (1962)
37.
Hadamard, J.: Sur les problèmes aux dériveès partielles et leur signification physique. Princet. Univ. Bull. 13, 49–52 (1902)MathSciNet
38.
Hadamard, J.: Lectures on Cauchy’s Problems in Linear Partial Differential Equations. Yale University Press, New Haven (1923). (Reprinted by Dover, New York, 1952)
39.
Halley, E.: A discourse concerning gravity, and its properties, wherein the descent of heavy bodies, and the motion of projects is briey, but fully handled: together with the solution of a problem of great use in gunnery. Philos. Trans. R. Soc. Lond. 16, 3–21 (1686)
40.
Hanke, M.: Iterative regularization techniques in image reconstruction. In: Colton, D. et al. (eds.) Surveys on Solution Methods for Inverse Problems, pp. 35–52. Springer, Vienna (2000)CrossRef
41.
42.
Hanke, M., Neubauer, A., Scherzer, O.: A convergence analysis of Landweber iteration for nonlinear ill-posed problems. Numer. Math. 72, 21–37 (1995)CrossRefMATHMathSciNet
43.
Hansen, P.C.: Rank Deficient and Discrete Ill-Posed Problems. SIAM, Philadelphia (1997)MATH
44.
Hansen, P.C., Nagy, J., O’Leary, D.: Deblurring images: matrices, spectra, and filtering. SIAM, Philadelphia (2006)CrossRef
45.
Hensel, E.: Inverse Theory and Applications for Engineers. Prentice-Hall, Englewood Cliffs (1991)
46.
Hofmann, B.: Regularization for Applied Inverse and Ill-Posed Problems. Teubner, Leipzig (1986)CrossRefMATH
47.
Joachimstahl, F.: Über ein attractionsproblem. J. für die reine und angewandte Mathematik 58, 135–137 (1861)CrossRef
48.
Kaczmarz, S.: Angenäherte Auflösung von Systemen linearer Gleichungen. Bulletin International de l’Academie Polonaise des Sciences, Cl. d. Sc. Mathém. A, pp. 355–357 (1937)
49.
Kaltenbacher, B., Neubauer, A., Scherzer, O.: Iterative Regularization Methods for Nonlinear Ill-Posed Problems. Walter de Gruyter, Berlin (2008)CrossRefMATH
50.
Kirsch, A.: An Introduction to the Mathematical Theory of Inverse Problems. Springer, New York (1993)
51.
52.
Lewitt, R.M., Matej, S.: Overview of methods for image reconstruction from projections in emission computed tomography. Proc. IEEE 91, 1588–1611 (2003)CrossRef
53.
Morozov, V.A.: On the solution of functional equations by the method of regularization. Sov. Math. Dokl. 7, 414–417 (1966)MATH
54.
Nashed, M.Z. (ed.): Generalized Inverses and Applications. Academic, New York (1976)MATH
55.
Natterer, F., Wübberling, F.: Mathematical Methods in Image Reconstruction. SIAM, Philadelphia (2001)CrossRefMATH
56.
57.
58.
Parker, R.L.: Geophysical Inverse Theory. Princeton University Press, Princeton (1994)MATH
59.
Phillips, D.L.: A technique for the numerical solution of certain integral equations of the first kind. J. Assoc. Comput. Mach. 9, 84–97 (1962)CrossRefMATHMathSciNet
60.
Picard, E.: Sur un théorème général relatif aux équations intégrales de premiére espéce et sur quelques probl_emes de physique mathématique. Rendiconti del Cicolo Matematico di Palermo 29, 79–97 (1910)CrossRefMATH
61.
Radon, J.: Über die Bestimmung von Funktionen durch ihre Integralwerte längs gewisser Mannigfaltigkeiten. Berichte über die Verhandlungen der Königlich Sächsischen Gesellshaft der Wissenschaften zur Leipzig 69, 262–277 (1917)
62.
Scherzer, O., Grasmair, M., Grossauer, H., Haltmeier, M., Lenzen, F.: Variational Methods in Imaging. Springer, New York (2009)MATH
63.
Sheehan, W., Kollerstrom, N., Waff, C.: The case of the pilfered planet: did the British steal Neptune? Sci. Am. 291(6), 92–99 (2004)CrossRef
64.
Shepp, L.A. (ed.): Computed Tomography. Proceedings of Symposia in Applied Mathematics, vol. 27. American Mathematical Society, Providence (1983)
65.
66.
Tikhonov, A.N.: On the stability of inverse problems. Dokl. Akad. Nau. SSSR 39, 176–179 (1943)MATHMathSciNet
67.
Tihonov (Tikhonov), A.N.: Solution of incorrectly formulated problems and the regularization method. Sov. Math. Dokl. 4, 1035–1038 (1963)
68.
Tikhonov, A.N., Arsenin, V.Y.: Solutions of Ill-Posed Problems. Winston & Sons, Washington, DC (1977)MATH
69.
Uhlmann, G. (ed.): Inside Out: Inverse Problems and Applications. Cambridge University Press, New York (2003)
70.
71.
Wing, G.M.: A Primer on Integral Equations of the First Kind: The Problem of Deconvolution and Unfolding. SIAM, Philadelphia (1992)
72.
Wrenn, F.R., Good, M.L., Handler, P.: The use of positron-emitting radioisotopes for the localization of brain tumors. Science 113, 525–527 (1951)CrossRef
73.
Wunsch, C.: The Ocean Circulation Inverse Problem. Cambridge University Press, Cambridge (1996)CrossRef
Metadaten
Titel
Linear Inverse Problems
verfasst von
Charles Groetsch
Copyright-Jahr
2015
Verlag
Springer New York
Buch
Handbook of Mathematical Methods in Imaging

Print ISBN: 978-1-4939-0789-2

Electronic ISBN: 978-1-4939-0790-8

Copyright-Jahr: 2015

DOI
https://doi.org/10.1007/978-1-4939-0790-8_1