Sparsity in Inverse Geophysical Problems
Many geophysical imaging problems are ill-posed in the sense that the solution does not depend continuously on the measured data. Therefore their solutions cannot be computed directly, but instead require the application of regularization. Standard regularization methods find approximate solutions with small L 2 norm. In contrast, sparsity regularization yields approximate solutions that have only a small number of nonvanishing coefficients with respect to a prescribed set of basis elements. Recent results demonstrate that these sparse solutions often much better represent real objects than solutions with small L 2 norm. In this survey, recent mathematical results for sparsity regularization are reviewed. As an application of the theoretical results, synthetic focusing in Ground Penetrating Radar is considered, which is a paradigm of inverse geophysical problem.
KeywordsGround Penetrate Radar Tikhonov Regularization Parameter Choice Residual Method Constrain Minimization Problem
This work has been supported by the Austrian Science Fund (FWF) within the national research networks Industrial Geometry, project 9203-N12, and Photoacoustic Imaging in Biology and Medicine, project S10505-N20. The authors thank Sylvia Leimgruber (alpS - Center for Natural Hazard Management in Innsbruck) and Harald Grossauer (University Innsbruck) for providing real life data sets.
- Bleistein N, Cohen JK, Stockwell Jr JW (2001) Mathematics of multidimensional seismic imaging, migration, and inversion. Interdisciplinary applied mathematics: Geophysics and planetary sciences, vol 13. Springer, New YorkGoogle Scholar
- Bredies K, Lorenz D (2009) Minimization of non-smooth, non-convex functionals by iterative thresholding. DFG-Schwerpunktprogramm 1324, Preprint 10, 2009Google Scholar
- Candès EJ, Romberg J (2005) ℓ 1-MAGIC: recovery of sparse signals via convex programming. Technical report, 2005. Available at http://www.acm.caltech.edu/l1magic
- Daniels D (2004) Ground penetrating radar. The Institution of Electrical Engineers, LondonGoogle Scholar
- Finch D, Rakesh (2007) The spherical mean value operator with centers on a sphere. Inverse Probl 23(6):37–49Google Scholar
- Grasmair M, Haltmeier M, Scherzer O (2009a) Necessary and sufficient conditions for linear convergence of ℓ 1-regularization. Reports of FSP S105—“Photoacoustic Imaging” 18, University of Innsbruck, Austria, August 2009 (submitted)Google Scholar
- Grasmair M, Haltmeier M, Scherzer O (2009b) The residual method for regularizing ill-posed problems. Reports of FSP S105—“Photoacoustic Imaging” 14, University of Innsbruck, Austria, May 2009 (submitted)Google Scholar
- Haltmeier M, Kowar R, Scherzer O (2005) Computer aided location of avalanche victims with ground penetrating radar mounted on a helicopter. In Lenzen F, Scherzer O, Vincze M (eds) Digital imaging and pattern recognition. Proceedings of the 30th workshop of the Austrian Association for Pattern Recognition, Obergugl, Austria, pp 1736–1744Google Scholar
- Haltmeier M, Scherzer O, Zangerl G (2009) Influence of detector bandwidth and detector size to the resolution of photoacoustic tomagraphy. In Breitenecker F, Troch I (eds) Argesim Report no. 35: Proceedings Mathmod 09, Vienna, pp 1736–1744Google Scholar
- Ivanov VK, Vasin VV, Tanana VP (2002) Theory of linear ill-posed problems and its applications 2nd edn. Inverse and ill-posed problems series. (Translated and revised from the 1978 Russian original). VSP, UtrechtGoogle Scholar
- Louis AK, Quinto ET (2000) Local tomographic methods in sonar. In Surveys on solution methods for inverse problems. Springer, Vienna, pp 147–154Google Scholar
- Scherzer O, Grasmair M, Grossauer H, Haltmeier M, Lenzen F (2009) Variational methods in imaging. Applied mathematical sciences vol 167. Springer, New YorkGoogle Scholar