Carleton Engineering SCE Faculty A. Adler Courses Directed Studies |

## Directed Studies: Inverse Problems## DescriptionInverse problems techniques are used to solve problems in image processing, medial imaging, geophysics, and many other fields in which we need to "reconstruct" the parameters we're interested in, based on measurements which relate to those parameters in some complicated way. The Hadamard (inverse) definition of inverse problem is: ## Instructor
## Marks
## ScheduleOffice ## Assignments, Presentations, and Project
## Texts and MaterialsThe following list represents a selection of inverse problem litterature. Students should read at least 1 book and/or 5-10 papers from this list during the completion of the project. - Web
- Books
- M Beretro and P Boccacci, (1998)
*Introduction to Inverse Problems in Imaging,*Institute of Physics Publishing ISBN 0750304391 - Albert Tarantola, (2004)
*Inverse Problem Theory,*Elsevier Science. ISBN 0444427651 - Albert Tarantola, (2004) Inverse Problem Theory and Methods for Model Parameter Estimation SIAM
- Per Christian Hansen, (1998)
*Rank-Deficient and Discrete Ill-Posed Problems: Numerical Aspects of Linear Inversion,*Society for Industrial and Applied Mathematics. ISBN 0898714036 - Heinz W. Engl, Martin Hanke, Andreas Neubauer, (1996)
*Regularization of Inverse Problems,*Kluwer Academic Publishers. ISBN 0792341570 - Curtis Vogel, (2002)
*Computational methods for inverse problems*Society for Industrial and Applied Mathematics. ISBN 0898715075 - Richard Aster, Brian Borchers, and Cliff Thurber, (2004)
*Parameter Estimation and Inverse Problems*Academic Press, ISBN 0120656043 - Frank Natterer and Frank Wubbeling, (2001)
*Mathematical Methods in Image Reconstruction,*Society for Industrial and Applied Mathematics. ISBN 0898714729
- M Beretro and P Boccacci, (1998)
- Journals
- Papers of interest
- Tikhonov, A.N. and Arsenin, V.A. (1977)
*Solutions of Ill-posed Problems.*Washington: Winston & Sons. -
Engl, H.W., Hanke, M., Neubauer, A.,
*Regularization of Inverse Problems,*Kluwer, Dordrecht, 1996. (referred by these documents) - Andia BI, Sauer KD, Bouman CA. Nonlinear backprojection for tomographic reconstruction. [Journal Paper] IEEE Transactions on Nuclear Science, vol.49, no.1, pt.1, Feb. 2002, pp.61-8. Publisher: IEEE, USA.
- Zheng J, Saquib SS, Sauer K, Bouman CA. Parallelizable Bayesian tomography algorithms with rapid, guaranteed convergence. [Journal Paper] IEEE Transactions of Image Processing, vol.9, no.10, Oct. 2000, pp.1745-59. Publisher: IEEE, USA.
- Bouman C, Sauer K. A generalized Gaussian image model for edge-preserving MAP estimation. [Journal Paper] IEEE Transactions of Image Processing, vol.2, no.3, July 1993, pp.296-310. USA.
- Chan T, Marquina A, Mulet P. High-order total variation-based image restoration. [Journal Paper] SIAM Journal on Scientific Computing, vol.22, no.2, 2000, pp.503-16. Publisher: SIAM, USA.
- A.Dalaney and Y.Bresler. Globally convergent edge-preserving regularization: An application to limited-angle tomography. IEEE Trans. on Image Processing, 7(2):204--221, 1998.
- S. Geman and D. Geman, "Stochastic relaxation, Gibbs distribution and the Bayesian restoration of images," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. PAMI-6, no. 6, pp. 721-741, November 1984.
- D. Geman and C. Yang, Nonlinear image recovery with half-quadratic regularization, IEEE Trans. Image Processing, 4(7):932--946, July 1995.
- Tikhonov, A.N. and Arsenin, V.A. (1977)
- Material from Per Christian Hansen, Dept. of Mathematical Modelling, Technical Univ. of Denmark.
- Regularization of Discrete Ill-Posed Problems
- Regularization Tools for Matlab
(Regularization Toolbox
*Num Alg.*6, 1994) - P. C. Hansen,
*Rank-Deficient and Discrete Ill-Posed Problems: Numerical Aspects of Linear Inversion,*SIAM, Philadelphia, 1998.
- A Reading List in Inverse Problems, Brian Borchers
- Johann Baumeister. Stable Solution of Inverse Problems. Vieweg,
Braunschweig, 1987.
M3. Theory of ill-posed problems, singular value decomposition. Tikhonov regularization, least squares solution of systems of linear equations, convolution equations, final value problems, parameter identification. No exercises. - James G. Berryman. Lecture notes on
nonlinear inversion and tomography: I. borehole siesmic tomography.
http://sepwww.stanford.edu/sep/berryman/NOTES/lecture notes.html, October 1991. M2. Seismic inversion, traveltime inversion, tomography. Exercises. - G. Backus and F. Gilbert. Uniqueness in the inversion of inaccurate
gross earth data. Philosophical Transactions of the Royal Society A,
266:123-192, 1970.
M2.5. This is a classic paper introducing the method of Backus and Gilbert for linear inverse problems. - Philip Carrion. Inverse Problems and Tomography in Acoustics and
Seismology. Penn Publishing Company, Atlanta, Georgia, 1987.
M2.5. Seismic inversion and tomography. No exercises. - Jon F. Claebourt. Imaging the Earth's Interior. Blackwell Scientific
Publications, Palo Alto, CA, 1985. Out of print, but available on the
web. See http://sepwww.stanford.edu/sep/prof/index.html.
M2. Seismic Inversion. No exercises. Claerbout has several other books available from the same web site. - Steven C. Constable, Robert L. Parker, and Catherine G. Constable.
Occam's inversion: A practical algorithm for generating smooth models
from electromagnetic sounding data. Geophysics, 52(3):289-300,
1987.
M2. This paper describes an iterative method for solving a nonlinear discrete inverse problem by linearizing the problem and using the discrepancy principle. - Heinz W. Engl, Martin Hanke, and Andreas Neubauer. Regularization
of Inverse Problems. Kluwer Academic Publishers, Dordrecht, 1996.
M3. Examples, ill-posed linear operator equations, Tikhonov regularization, iterative regularization methods, the conjugate gradient method, numerical implementation, nonlinear problems. No exercises. - Heinz W. Engl. Regularization methods for the stable solution of inverse
problems. Surveys on Mathematics for Industry, 3:71-143, 1993.
M2.5. A broad survey of regularization methods with lots of examples. - Charles W. Groetsch. Inverse Problems in the Mathematical Sciences.
Vieweg, Bruanschweig; Wiesbaden, 1993.
M2. An introduction to the mathematics of inverse problems, requiring only undergraduate mathematics. Integral equations of the first kind, parameter estimation in differentional equations, regularization, iterative methods, the maximum entropy method, the Backus-Gilbert method. Exercises. - Per Christian Hansen. Analysis of discrete ill-posed problems by
means of the L-curve. SIAM Review, 34(4):561-580, December 1992.
M2. This paper discusses the use of the L-curve criteria for selecting a regularization parameter and compares it with generalized cross validation. - Per Christian Hansen. Regularization tools: A MATLAB package for
analysis and solution of discrete ill{posed problems. Numerical
Algorithms, 6:1-35, 1994.
http://www.imm.dtu.dk/documents/users/pch/Regutools/regutools.html. M2. The regularization toolbox provides a variety of functions for solving inverse problems, including the SVD and generalized SVD, truncated SVD solutions, Tikhonov regularization, maximum entropy regularization, and a variety of examples. - Per Christian Hansen. Rank-Deficient and Discrete Ill-Posed
Problems. SIAM, Philadelphia, 1998.
M2. Numerical methods for discretized inverse problems, including basic theory, direct and iterative methods for regularization, and methods for picking the regularization parameter. Examples drawn from the author's regularization toolbox. No Exercises. - Martin Hanke and Per Christian Hansen. Regularization methods for
large-scale problems. Surveys on Mathematics for Industry, 3:253-
315, 1993.
M2.5. Another survey paper, with special emphasis on iterative methods for large scale problems. Contains a good discussion of methods for selecting the regularization parameter. - Andreas Kirsch. An Introduction to the Mathematical Theory of Inverse Problems. Springer Verlag, New York, 1996.
M3. Regularization theory, Tikhonov regularization, regularization by discretization, the method of Backus and Gilbert, inverse eigenvalue problems, inverse scattering problems. Includes exercises. - C. L. Lawson and R. J. Hanson. Solving Least Squares Problems.
Prentice-Hall, Englewood Cliffs, New Jersey, 1974.
M2. A classic book on the solution of linear least squares problems. Contains an early discussion of the L-curve. This book has recently been republished by SIAM. Exercises. - Laurence R. Lines, editor. Inversion of Geophysical Data. Society of
Exploration Geophysicists, 1988.
M2. A collection of reprints of tutorial/survey papers on inversion. Linearized inversion techniques, seismic inversion, inversion of electromagnetic and potential field data. No exercises - William Menke. Geophysical Data Analysis: Revised Edition,
volume 45 of International Geophysics Series. Academic Press, San
Diego, 1989.
M2. Generalized inverses, maximum likelihood methods, the method of Backus and Gilbert, nonlinear inverse problems, numerical algorithms, applications. No exercises. - V. A. Morozov. Methods for Solving Incorrectly Posed Problems.
Springer-Verlag, New York, 1984.
M3. Pseudoinverses, regularization, methods for picking the regularization parameter, nonlinear problems. No exercises. - Robert L. Parker. Geophysical Inverse Theory. Princeton University
Press, Princeton, NJ, 1994.
M2. Mathematical background, examples from geophsyics, Tikhonov regularization, resolution, nonlinear problems. Exercises. - Dilip N. Ghosh Roy. Methods of Inverse Problems in Physics. CRC
Press, Boca Raton, 1991.
M2.5. Inverse problems in physics, the Povzner-Levitan transform, the Gelfand-Levitan Equation, Jost Functions, the Marcenko integral equation, the Radon transform. No exercises. - P.C. Sabatier, editor. Inverse Problems: An Interdisciplinary Study.
Academic Press, London, 1987.
M2.5. A collection of papers, mostly applications in tomography, electromagnetic inverse scattering, quantum mechanics, and other areas. No exercises. - John Scales. Theory of seismic imaging.
http://landau.Mines.EDU/samizdat/imaging/index.html, 1997. M2. Seismic inversion, Kirchoff migration, ray tracing, finite difference methods. Exercises. - John Scales and Martin Smith. DRAFT: Geophysical inverse theory.
http://landau.Mines.EDU/ samizdat/inverse theory/, 1997. M2. Presents a Bayesian approach to inverse theory. Exercises. - Gilbert Strang. Linear Algebra and its Applications. Harcourt Brace
Jovanovich Inc., Fort Worth, third edition, 1988.
M1. This is a textbook on linear algebra at a slightly higher level than the typical introductory course. Contains a good discussion of the SVD. Exercises. - A. N. Tikhonov and V. Y. Arsenin. Solutions of Ill-Posed Problems.
John Wiley, New York, 1977.
M3. The definitive treatment of the theory of Tikhonov regularization. - Albert Tarantola. Inverse Problem Theory: Methods for Data Fitting
and Model Parameter Estimation. Elsevier, New York, 1987. M3. This book describes a Bayesian approach to discrete and continuous inverse problems. - S. Twomey. Introduction to the Mathematics of Inversion in Remote
Sensing and Indirect Measurements. Elsevier, Amsterdam, 1977. M1.5. Mathematical background, quadrature, Tikhonov regularization, examples. No exercises. This book has recently been republished by Dover. - J. M. Varah. A practical examination of some numerical methods for
linear discrete ill-posed problems. SIAM Review, 21:100-111, 1979. M2. This paper compares a variety of methods for solving discrete linear inverse problems. - Grace Wahba. Spline Models for Observational Data. SIAM, Philadelphia, 1990.
M3. Although this book is largely about fitting splines to data, it also contains a very useful discuss of cross validation. No exercises. - G. Milton Wing. A Primer on Integeral Equations of the First Kind.
SIAM, Philadelphia, 1991.
M2. This book provides an introduction to theory and methods for the practical solution of integral equations of the first kind.
- Johann Baumeister. Stable Solution of Inverse Problems. Vieweg,
Braunschweig, 1987.
- A Reading list in inverse problems
- Material from
Numerical Recipes in C : The Art of Scientific Computing (1993)
by William H. Press , Brian P. Flannery , Saul A. Teukolsky , William T. Vetterling
- 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 Indirect
- Measurements (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 Methods
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