Robert Ackroyd introduces some handy phrasing for the connections between statistical estimation theory and inverse problem solving.

Photogrammetry, MRIs, SLAM, volumetric reconstruction and X-ray crystallography are all examples of inverse problems. any of these can be constructed as classical belief propagaation, especially Gaussian BP, or in a basic case least squares.

I happen to think that this is a case where it is much easier to explain in terms of Bayesian inference, so my attempt at an actual explanation is under Bayesian inverse problems.

We can do it in terms of *frequentist* methods, but it does not add much in the way of explanatory value; we end up considering regularizers instead of priors, but the working in between is pretty much the same. (IMO)
However, in doing so we focus on point estimates rather than entire densities, which encourages us to solve the problem by optimisation rather than integration, which is a useful insight, for example, when we consider Laplace approximations.

## Interesting specific techniques

Leaning to reconstruct introduces partly-learned, partly designed reconstruction operator trick. ποΈ

## Radiance fields

A fun way of reconstructing objects from photos; differentiable photogrammetry.

## References

*IEEE Transactions on Medical Imaging*37 (6): 1322β32.

*arXiv:2106.06513 [Cs, Math, Stat]*, June.

*Parameter Estimation and Inverse Problems*. Third. Elsevier.

*Inverse Problems*20 (6): 1773β89.

*Communications on Pure and Applied Mathematics*58 (9): 1231β79.

*arXiv:1612.01183 [Cs, Math]*, December.

*Proceedings of the National Academy of Sciences*117 (10): 5242β49.

*Proceedings of the National Academy of Sciences*, May.

*Communications on Pure and Applied Mathematics*57 (11): 1413β57.

*Inverse Problems*21 (1): 399β412.

*Journal of Mathematical Analysis and Applications*83 (2): 582β610.

*Journal of Applied Geophysics*98 (November): 62β72.

*arXiv:1610.08035 [Stat]*.

*Statistical and Computational Inverse Problems*. Applied Mathematical Sciences. New York: Springer-Verlag.

*Journal of Computational and Applied Mathematics*198 (2): 493β504.

*Inverse Problems*5 (4): 599β612.

*Zeitschrift FΓΌr Wahrscheinlichkeitstheorie Und Verwandte Gebiete*65 (3): 385β97.

*Journal of Geophysical Research: Solid Earth*100 (B7): 12431β47.

*International Geophysics*, 81:237β65. Elsevier.

*Deterministic and Statistical Methods in Machine Learning*, edited by Joab Winkler, Mahesan Niranjan, and Neil Lawrence, 110β23. Lecture Notes in Computer Science. Springer Berlin Heidelberg.

*Twenty-Fifth AAAI Conference on Artificial Intelligence*.

*Statistical Science*1 (4): 502β18.

*Inverse Problems*22 (1): 365β79.

*Journal of the American Statistical Association*112 (519): 1274β85.

*arXiv:1706.04008 [Cs]*, June.

*arXiv:2203.10131 [Physics]*, March.

*Inverse Problems*28 (4): 045003.

*Acta Numerica*19: 451β559.

*arXiv:2006.15641 [Cs, Stat]*, June.

*Inverse Problem Theory and Methods for Model Parameter Estimation*. SIAM.

*Journal of Machine Learning Research*21 (179): 1β46.

*Proceedings of the IEEE*98 (6): 948β58.

*arXiv:1709.01841 [Cs]*, September.

*Environmental Modelling & Software*85 (November): 217β28.

*Geoscientific Model Development Discussions*, July, 1β51.

*Journal of Computational Physics*397 (November): 108850.

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