Book on inverse problems

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  • #1
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Hi there. I'm starting to work on diffuse optical tomography, and I need to introduce my self to the theory of inverse problems, and the different techniques to solve inverse problems, specially in the area I'm going to work, or things related to the inverse problems in electromagnetic theory, which I think should be closely related. I thought perhaps someone here is familiar with this issues, and could help me to find some introductory textbook, and something advanced too for the future.

Thanks in advance.
 
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  • #2
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Unfortunately, there are very few
... introductory textbook...
.
in the field of electromagnetic inverse problems. The only one which I am aware of is An Introduction to Electromagnetic Inverse Scattering by Hopcraft and Smith which is suprisingly very readable.

Of the other literature are research-level treatises, monograms, and of course research papers. Some that you might find reasonable, written from a slightly different perspective, are

(geophysics/remote sensing):
Parameter Estimation and Inverse Problems by Aster, Borchers and Thurber
Inverse Problem Theory and Methods for Model Parameter Estimation by Tarantola

(mathematicians):
Inverse Acoustic and Electromagnetic Scattering Theory Colton and Kress
An Introduction to the Mathematical Theory of Inverse Problems Kirsch

and a massive compilation (2 volumes, 1800 pages) which has a little of the inverse electromagnetic problem written by and for mathematicians, physicists, engineers and others is:

Scattering edited by Pike and Sabatier, but I don't think you will find it of use.
 
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  • #3
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The Radon Transform and Some of Its Applications by Deans is a Dover book now. I read the original, more expensive revised edition back in the 90s and found it illuminating. It's quite a bit more specific than you're asking for, but the Radon transform is useful enough in computed tomography that it deserves an extra reference or few on one's shelf, I think.
 
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  • #4
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Thank you verymuch to both of you. Is Bayesian statistics needed to work on inverse theory? I've heard that it is used somehow. I also took a fast course last year on EMG and EEG. The course was oriented mostly on a program that was developed to work on the field of EEG, it wasn't "physics" oriented (they din't have time to do it that way). But there is something that I remember from that course, and is that they've said that there is an infinite set of solutions when one works with inverse problems. For example, in the case of EEG, one measures the electric field at some place in the head, and then hopes to find the sources from the values of the electric field at those points. In that way it is easy to me to imagine that there is actually an infinity of possible configuration of currents and charges to give those values of the electric and magnetic fields at some specific points or regions in space, so one has to use some statistics to determine which solutions are the appropriate for a given situation. Anyway, it was said that the programs to do the inverse problem solution are already written and there are libraries for that, but I think it will be useful to introduce my self on these things. So, do you know if Bayesian statistics is also needed? because I'm not totally sure.
 
  • #5
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You'll need a good grounding in applied statistics. A good understanding of Bayes' theorem is a part of that.

Bayesian inference is sometimes explicitly used in tomography research I've seen, though I'm not sure how necessary or pervasive it is.

But yes, you'll need to understand what an intro "math stat" or "statistics for scientists" class covers on Bayesian inference.
 
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I'm not familiar with either of these, but Amazon just recommended two books to me based on the searches I did for this thread:

The Mathematics of Medical Imaging: A Beginner's Guide by Timothy G. Feeman.

Introduction to the Mathematics of Medical Imaging, Second Edition by Charles L. Epstein.

I have no opinion on these, but they might be worth borrowing via interlibrary loan to assess their potential.
 
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  • #7
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Great, and for the statistical part what would you recommend? thank you verymuch.
 
  • #8
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I learned statistics out of Probability and Statistics for Engineers & Scientists by Walpole, Myers, Myers, and Ye. It's not a bad text, and it covers the necessary bases adequately. Looking on Amazon, the 2006 edition is available used for a few dollars. You can probably find a better text, but that one's not bad.
 
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  • #9
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Thanks, I'll see what I find at the library.
 
  • #10
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But there is something that I remember from that course, and is that they've said that there is an infinite set of solutions when one works with inverse problems.
Yes this is so. One difficulty is that multiple source configurations can give identical scattered data (as you mentioned above) and a second is that the parameter space over which you are searching is vast. Mathematically this is know as an ill-posed problem (the opposite of Hadamard's well-posed problem) which unfortunately as you are aware rarely has a unique solution.
 
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  • #11
StatGuy2000
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I know this is almost a year old, but here is a technical paper by Stark (2009) on frequentist and Bayesian methods specifically in the context of inverse problems.

https://www.stat.berkeley.edu/~stark/Preprints/freqBayes09.pdf

The article also provides various references to books or articles on inverse problems.
 
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  • #12
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Another book which is both more complete and modern in comparison to the fine text by Hopcraft and Smith is the equally fine book:

Mathematical Foundations of Imaging, Tomography, and Wavefield Inversion
by Devaney
 
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  • #13
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When I created this thread I had just a few days of working. Now I'm much more involved in this thing, and actually what I am going to need to do is to find inverse solutions of the radiative transfer equation (linearized Boltzmann equation) and the diffusion equation. If you know any book that treats in particular the inverse problem on this equations, it would be of great help.

Thank you very much.
 
  • #14
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I think An Introduction to Invariant Imbedding by Bellman & Wang might be appropriate
 
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