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Mathematical Methods in Medical Image Processing

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Quantification of Biophysical Parameters in Medical Imaging

Abstract

Medical imaging problems, such as magnetic resonance imaging, can typically be modeled as inverse problems. A novel methodological approach which was already proven to be highly effective and widely applicable is based on the assumption that most real-life images are intrinsically of low-dimensional nature. This sparsity property can be revealed by representation systems from the area of applied harmonic analysis such as wavelets or shearlets. The inverse problem itself is then solved by sparse regularization, which in certain situations is referred to as compressed sensing. This chapter shall serve as an introduction to and a survey of mathematical methods for medical imaging problems with a specific focus on sparsity-based methods. The effectiveness of the presented methods is demonstrated with a small case study from sparse parallel magnetic resonance imaging.

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References

  1. Epstein CL. Introduction to the mathematics of medical imaging. Philadelphia: SIAM; 2008.

    Book  Google Scholar 

  2. Lustig M, Donoho D, Pauly JM. Sparse MRI: the application of compressed sensing for rapid MR imaging. Magn Reson Med. 2007;58:1182–95.

    Article  PubMed  Google Scholar 

  3. Hadamard J. Lectures on Cauchy’s problem in linear differential equations. New Haven: Yale University Press; 1923.

    Google Scholar 

  4. Grafakos L. Classical Fourier analysis. New York: Springer; 2008.

    Google Scholar 

  5. Foster J, Richards FB. The Gibbs phenomenon for piecewise-linear approximation. Am Math Mon. 1991;98:47–9.

    Article  Google Scholar 

  6. Bertero M, Boccacci P. Introduction to inverse problems in imaging. London: Institute of Physics Publishing; 1998.

    Book  Google Scholar 

  7. Mueller JL, Siltanen S. Linear and nonlinear inverse problems with practical applications. Philadelphia: SIAM; 2012.

    Book  Google Scholar 

  8. Tikhonov AN, Arsenin VY. Solutions of ill-posed problems. New York: Wiley; 1977.

    Google Scholar 

  9. Candès EJ, Tao T. Decoding by linear programming. IEEE Trans Inf Theory. 2005;51:4203–15.

    Article  Google Scholar 

  10. Donoho DL. Compressed sensing. IEEE Trans Inf Theory. 2006;52:1289–306.

    Article  Google Scholar 

  11. Boche H, Calderbank R, Kutyniok G, Vybiral J. A survey of compressed sensing. In: Boche H, et al., editors. Compressed sensing an its applications. Berlin: Springer; 2015.

    Chapter  Google Scholar 

  12. Tropp JA, Wright SJ. Computational methods for sparse solution of linear inverse problems. Proc IEEE. 2010;98:948–58.

    Article  Google Scholar 

  13. Candès EJ, Plan Y. A probabilistic and RIPless theory of compressed sensing. IEEE Trans Inf Theory. 2010;57:7235–54.

    Article  Google Scholar 

  14. Candès EJ, Tao T. The Dantzig selector: statistical estimation when p is much larger than n. Ann Statist. 2007;35:2313–51.

    Article  Google Scholar 

  15. Tibshirani R. Regression shrinkage and selection via the Lasso. J R Stat Soc Ser B Methodol. 1996;58:267–88.

    Google Scholar 

  16. Rudin L, Osher S, Fatemi E. Nonlinear total variation based noise removal algorithms. Physica D. 1992;60:256–86.

    Article  Google Scholar 

  17. Chambolle A, Lions P. Image recovery via total variation minimization and related problems. Numer Math. 1997;76:167–88.

    Article  Google Scholar 

  18. Mallat S. A wavelet tour of signal processing: the sparse way. 3rd ed. Amsterdam: Elsevier/Academic; 2009. With contributions from Gabriel Peyré

    Google Scholar 

  19. Daubechies I. Ten lectures on wavelets. Philadelphia: SIAM; 1992.

    Book  Google Scholar 

  20. Donoho DL. Sparse components of images and optimal atomic decomposition. Constr Approx. 2001;17:353–82.

    Article  Google Scholar 

  21. Kutyniok G, Labate D. Shearlets: multiscale analysis for multivariate data. Boston: Birkhäuser Basel; 2012.

    Book  Google Scholar 

  22. Guo K, Kutyniok G, Labate D. Sparse multidimensional representations using anisotropic dilation and shear operators. In:Wavelets and Splines (Athens, GA, 2005). Nashville: Nashboro Press; 2006. p. 189–201.

    Google Scholar 

  23. Labate D, Lim W-Q, Kutyniok G, Weiss G. Sparse multidimensional representation using shearlets. In: Papadakis M, Laine MF, Unser MA, editors. Wavelets XI, SPIE proc. 5914. Bellingham, WA: SPIE; 1974. p. 254–62.

    Google Scholar 

  24. Kittipoom P, Kutyniok G, Lim W-Q. Construction of compactly supported shearlet frames. Constr Approx. 2012;35:21–72.

    Article  Google Scholar 

  25. Christensen O. An introduction to frames and Riesz bases, Applied and numerical harmonic analysis. Boston: Birkhäuser Boston; 2003.

    Book  Google Scholar 

  26. Kutyniok G, Lemvig J, Lim W-Q. Optimally sparse approximations of 3D functions by compactly supported shearlet frames. SIAM J Math Anal. 2012;44:2962–3017.

    Article  Google Scholar 

  27. Pruessmann KP, Weiger M, Scheidegger MB, Boesiger P. SENSE: sensitivity encoding for fast MRI. Magn Reson Med. 1999;42:952–62.

    Article  CAS  PubMed  Google Scholar 

  28. Kutyniok G, Lim W-Q, Reisenhofer R. ShearLab 3D: faithful digital shearlet transforms based on compactly supported shearlets. ACM Trans Math Softw. 2016;42:1–42.

    Article  Google Scholar 

  29. Combettes PL, Pesquet J-C. Proximal splitting methods in signal processing. In:Fixed-point algorithms for inverse problems in science and engineering. New York: Springer; 2011. p. 185–212.

    Chapter  Google Scholar 

  30. Ma J, März M. A multilevel based reweighting algorithm with joint regularizers for sparse recovery (submitted).

    Google Scholar 

  31. Ma J, März M, Funk S, Schulz-Menger J, Kutyniok G, Schäffter T, Kolbitsch C. Shearlet-based compressed sensing for fast 3D MR imaging using iterative reweighting (submitted).

    Google Scholar 

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Acknowledgements

G. Kutyniok acknowledges partial support by the Einstein Foundation Berlin, the Einstein Center for Mathematics Berlin (ECMath), the European Commission-Project DEDALE (contract no. 665044) within the H2020 Framework Program, DFG Grant KU 1446/18, DFG-SPP 1798 Grants KU 1446/21 and KU 1446/23, the DFG Collaborative Research Center TRR 109 Discretization in Geometry and Dynamics, and by the DFG Research Center Matheon “Mathematics for Key Technologies” in Berlin. J. Ma acknowledges partial support by the DFG Collaborative Research Center TRR 109 Discretization in Geometry and Dynamics. M. März acknowledges partial support by the DFG SPP 1798 Compressed Sensing in Information Processing.

Author information

Authors and Affiliations

  1. Department of Mathematics, Technische Universität Berlin, Berlin, Germany

    Gitta Kutyniok & Maximilian März

  2. Image and Video Coding Group, Fraunhofer Institute for Telecommunications–Heinrich Hertz Institute, Berlin, Germany

    Jackie Ma

Authors
  1. Gitta Kutyniok
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  2. Jackie Ma
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  3. Maximilian März
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Corresponding author

Correspondence to Gitta Kutyniok .

Editor information

Editors and Affiliations

  1. Department of Radiology, Humboldt University of Berlin Charité University Hospital, Berlin, Germany

    Ingolf Sack

  2. Medical Physics and Metrological Information Technology, Physikalisch-Technische Bundesanstalt , Berlin, Germany

    Tobias Schaeffter

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Kutyniok, G., Ma, J., März, M. (2018). Mathematical Methods in Medical Image Processing. In: Sack, I., Schaeffter, T. (eds) Quantification of Biophysical Parameters in Medical Imaging. Springer, Cham. https://doi.org/10.1007/978-3-319-65924-4_7

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  • DOI: https://doi.org/10.1007/978-3-319-65924-4_7

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-65923-7

  • Online ISBN: 978-3-319-65924-4

  • eBook Packages: MedicineMedicine (R0)

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