Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Journal of Inverse and Ill-posed Problems

Editor-in-Chief: Kabanikhin, Sergey I.


IMPACT FACTOR 2017: 0.941
5-year IMPACT FACTOR: 0.953

CiteScore 2017: 0.91

SCImago Journal Rank (SJR) 2017: 0.461
Source Normalized Impact per Paper (SNIP) 2017: 1.022

Mathematical Citation Quotient (MCQ) 2017: 0.49

Online
ISSN
1569-3945
See all formats and pricing
More options …
Volume 25, Issue 4

Issues

A Mumford–Shah-type approach to simultaneous reconstruction and segmentation for emission tomography problems with Poisson statistics

Esther Klann
  • Corresponding author
  • Technical University Berlin, Straße des 17. Juni 136, 10623 Berlin, Germany;and Industrial Mathematics Institute, Johannes Kepler University Linz, Altenbergerstraße 69, 4040, Linz, Austria
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Ronny Ramlau
  • Radon Institute for Computational and Applied Mathematics (RICAM) and Johannes Kepler University Linz, Industrial Mathematics Institute and Doctoral Program “Computational Mathematics”, Altenbergerstraße 69, 4040 Linz, Austria
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Peng Sun
  • Johannes Kepler University Linz, Doctoral Program “Computational Mathematics”, Altenbergerstraße 69,4040 Linz, Austria
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
Published Online: 2017-01-12 | DOI: https://doi.org/10.1515/jiip-2016-0077

Abstract

We propose a variational model to simultaneous reconstruction and segmentation in emission tomography. As in the original Mumford–Shah model [27] we use the contour length as penalty term to preserve edge information whereas a different data fidelity term is used to measure the information discrimination between the computed tomography data of the reconstructed object and the observed (or simulated) data. As data fidelity term we use the Kullback–Leibler divergence which originates from the Poisson distribution present in emission tomography. In this paper we focus on piecewise constant reconstructions which is a reasonable assumption in medical imaging. The segmenting contour as well as the corresponding reconstructions are found as minimizers of a Mumford–Shah-type functional over the space of piecewise constant functions. The numerical scheme is implemented by evolving the level-set surface according to the shape derivative of the functional. The method is validated for simulated data with different levels of noise.

Keywords: Ill-posed problem; regularization method; Mumford–Shah; segmentation; level-set method; tomography; Poisson statistics

MSC 2010: 44A12; 47A52; 65K10; 92C55; 94A08

References

  • [1]

    G. Alessandrini and S. Vessella, Lipschitz stability for the inverse conductivity problem, Adv. in Appl. Math. 35 (2005), no. 2, 207–241. CrossrefGoogle Scholar

  • [2]

    L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Clarendon Press, Oxford, 2000. Google Scholar

  • [3]

    J. M. Bardsley and C. R. Vogel, A nonnegatively constrained convex programming method for image reconstruction, SIAM J. Sci. Comput. 25 (2004), no. 4, 1326–1343. Google Scholar

  • [4]

    M. Benning, T. Kosters, F. Wubbeling, K. Schafers and M. Burger, A nonlinear variational method for improved quantification of myocardial blood flow using dynamic H2O15 PET, Nuclear Science Symposium Conference Record (NSS ’08), IEEE Press, Piscataway (2008), 4472–4477. Google Scholar

  • [5]

    T. Brox and D. Cremers, On the statistical interpretation of the piecewise smooth Mumford–Shah functional, Scale Space and Variational Methods in Computer Vision, Lecture Notes in Comput. Sci. 4485, Springer, Berlin (2007), 203–213. Google Scholar

  • [6]

    M. Burger, J. Müller, E. Papoutsellis and C. B. Schönlieb, Total variation regularization in measurement and image space for PET reconstruction, Inverse Problems 30 (2014), no. 10, Article ID 105003. Google Scholar

  • [7]

    F. Cannizzaro, G. Greco, S. Rizzo and E. Sinagra, Results of the measurements carried out in order to verify the validity of the poisson-exponential distribution in radioactive decay events, Int. J. Appl. Radiation Isotopes 29 (1978), 10.1016/0020-708X(78)90101-1. Google Scholar

  • [8]

    M. Delfour and J. Zolésio, Shapes and Geometries, 2nd ed., SIAM, Philadelphia, 2011. Google Scholar

  • [9]

    A. P. Dempster, N. M. Laird and D. B. Rubin, Maximum likelihood from incomplete data via the em algorithm, J. R. Stat. Soc. Ser. B. 39 (1977), no. 1, 1–38. Google Scholar

  • [10]

    M. Droske and W. Ring, A Mumford–Shah level-set approach for geometric image registration, SIAM J. Appl. Math. 66 (2006), no. 6, 2127–2148. Google Scholar

  • [11]

    I. Ekeland and R. Témam, Convex Analysis and Variational Problems, Class. Appl. Math. 28, SIAM, Philadelphia, 1999.Google Scholar

  • [12]

    H. W. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems, Math. Appl. (Dordrecht) 375, Springer, Dordrecht, 1996. Google Scholar

  • [13]

    S. Geman and D. McClure, Bayesian image analysis: An application to single photon emission tomography, Proc. Statist. Comput. Sec. Amer. Statist. Assoc. (1985), 12–18. Google Scholar

  • [14]

    S. A. Geman and D. E. McClure, Statistical Methods for Tomographic Image Reconstruction, MIT, Cambridge, 1987. Google Scholar

  • [15]

    G. Gerig, O. Kubler, R. Kikinis and F. A. Jolesz, Nonlinear anisotropic filtering of mri data, IEEE Trans. Med. Imaging 11 (1992), no. 2, 221–232. CrossrefGoogle Scholar

  • [16]

    M. Hintermüller and W. Ring, An inexact newton-cg-type active contour approach for the minimization of the Mumford–Shah functional, J. Math. Imaging Vision 20 (2004), no. 1–2, 19–42. Google Scholar

  • [17]

    E. Jonsson, S.-C. Huang and T. Chan, Total-variation regularization in positron emission tomography, preprint (1998). Google Scholar

  • [18]

    J. Kay, Statistical models for PET and SPECT data, Stat. Methods Med. Res. 3 (1994), no. 1, 5–21. CrossrefGoogle Scholar

  • [19]

    C. T. Kelley, Iterative Methods for Optimization, Front. Appl. Math., SIAM, Philadelphia, 1999. Google Scholar

  • [20]

    E. Klann, A Mumford–Shah-like method for limited data tomography with an application to electron tomography, SIAM J. Imaging Sci. 4 (2011), no. 4, 1029–1048. Google Scholar

  • [21]

    E. Klann and R. Ramlau, Regularization properties of Mumford–Shah-type functionals with perimeter and norm constraints for linear ill-posed problems, SIAM J. Imaging Sci. 6 (2013), no. 1, 413–436. Google Scholar

  • [22]

    E. Klann, R. Ramlau and W. Ring, A Mumford–Shah level-set approach for the inversion and segmentation of SPECT/CT data, Inverse Probl. Imaging 5 (2011), no. 1, 137–166. CrossrefGoogle Scholar

  • [23]

    T. Le, R. Chartrand and T. J. Asaki, A variational approach to reconstructing images corrupted by Poisson noise, J. Math. Imaging Vision 27 (2007), no. 3, 257–263. Google Scholar

  • [24]

    S.-J. Lee, A. Rangarajan and G. Gindi, Bayesian image reconstruction in spect using higher order mechanical models as priors, IEEE Trans. Med. Imaging 14 (1995), no. 4, 669–680. CrossrefGoogle Scholar

  • [25]

    J. J. Moré and G. Toraldo, On the solution of large quadratic programming problems with bound constraints, SIAM J. Optim. 1 (1991), no. 1, 93–113. Google Scholar

  • [26]

    H. N. Mülthei, B. Schorr and W. Törnig, On an iterative method for a class of integral equations of the first kind, Math. Methods Appl. Sci. 9 (1987), no. 1, 137–168. CrossrefGoogle Scholar

  • [27]

    D. Mumford and J. Shah, Optimal approximations by piecewise smooth functions and associated variational problems, Comm. Pure Appl. Math. 42 (1989), no. 5, 577–685. CrossrefGoogle Scholar

  • [28]

    F. Natterer, The Mathematics of Computerized Tomography, Class. Appl. Math., SIAM, Philadelphia, 2001. Google Scholar

  • [29]

    J. Nocedal and S. Wright, Numerical Optimization, Springer, New York, 2006. Google Scholar

  • [30]

    S. Osher and R. Fedkiw, Level Set Methods and Dynamic Implicit Surfaces, Appl. Math. Sci. 153, Springer, New York, 2003. Google Scholar

  • [31]

    S. Osher and J. A. Sethian, Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton–Jacobi formulations, J. Comput. Phys. 79 (1988), no. 1, 12–49. CrossrefGoogle Scholar

  • [32]

    E. T. Quinto, An introduction to X-ray tomography and Radon transforms, The Radon Transform, Inverse Problems, and Tomography (Atlanta 2005), Proc. Sympos. Appl. Math. 63, American Mathematical Society, Providence (2006), 1–23. Google Scholar

  • [33]

    R. Ramlau and W. Ring, A Mumford–Shah level-set approach for the inversion and segmentation of X-ray tomography data, J. Comput. Phys. 221 (2007), no. 2, 539–557. CrossrefWeb of ScienceGoogle Scholar

  • [34]

    R. Ramlau and W. Ring, Regularization of ill-posed Mumford–Shah models with perimeter penalization, Inverse Problems 26 (2010), no. 11, Article ID 115001. Google Scholar

  • [35]

    E. Resmerita, H. W. Engl and A. N. Iusem, The expectation-maximization algorithm for ill-posed integral equations: A convergence analysis, Inverse Problems 23 (2007), no. 6, Article ID 2575. Google Scholar

  • [36]

    N. Roé-Vellvé, F. Pino, C. Falcon, A. Cot, J. D. Gispert, C. Marin, J. Pavía and D. Ros, Quantification of rat brain spect with 123i-ioflupane: Evaluation of different reconstruction methods and image degradation compensations using Monte Carlo simulation, Phys. Med. Biol. 59 (2014), no. 16, Article ID 4567. Google Scholar

  • [37]

    L. I. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms, Phys. D 60 (1992), no. 1–4, 259–268. Google Scholar

  • [38]

    A. Sawatzky, C. Brune, F. Wubbeling, T. Kosters, K. Schafers and M. Burger, Accurate EM-TV algorithm in PET with low SNR, Nuclear Science Symposium Conference Record (NSS ’08), IEEE Press, Piscataway (2008), 5133–5137. Google Scholar

  • [39]

    O. Scherzer, Handbook of Mathematical Methods in Imaging, Springer, Berlin, 2011. Google Scholar

  • [40]

    L. A. Shepp and Y. Vardi, Maximum likelihood reconstruction for emission tomography, IEEE Trans. Med. Imaging 1 (1982), no. 2, 113–122. CrossrefGoogle Scholar

  • [41]

    J. Sokolowski and J. P. Zolesio, Introduction to Shape Optimization: Shape Sensitivity Analysis, Springer Ser. Comput. Math., Springer, Berlin, 2012. Google Scholar

  • [42]

    J. A. Terry, B. M. W. Tsui, J. R. Perry, J. L. Hendricks and G. T. Gullberg, The design of a mathematical phantom of the upper human torso for use in 3-d spect imaging research, Biomedical Engineering: Opening New Doors, New York University Press, New York (1990), 185–190. Google Scholar

  • [43]

    Z. Wang, A. C. Bovik, H. R. Sheikh and E. P. Simoncelli, Image quality assessment: From error visibility to structural similarity, IEEE Trans. Image Process. 13 (2004), no. 4, 600–612.CrossrefGoogle Scholar

  • [44]

    M. N. Wernick and J. N. Aarsvold, Emission Tomography: The Fundamentals of PET and SPECT, Elsevier Science, Amsterdam, 2004. Google Scholar

  • [45]

    S. Winkler and P. Mohandas, The evolution of video quality measurement: From psnr to hybrid metrics, IEEE Trans. Broadcasting 54 (2008), no. 3, 660–668. Google Scholar

  • [46]

    M. Yan, A. A. T. Bui, J. Cong and L. A. Vese, General convergent expectation maximization (EM)-type algorithms for image reconstruction, Inverse Probl. Imaging 7 (2013), no. 3, 1007–1029. CrossrefGoogle Scholar

About the article

Received: 2016-11-03

Accepted: 2016-12-17

Published Online: 2017-01-12

Published in Print: 2017-08-01


The work of E. Klann has received funding from the People Programme (Marie Curie Actions) of the European Union’s Seventh Framework Programme (FP7/2007-2013) under REA grant agreement no. 600209 (TU Berlin – IPODI) as well as from the Austrian Science Funds (FWF) under grant T 529-N18 and the MPNS COST Action MP1207. The work of R. Ramlau and P. Sun is supported by DK (Doktoratskolleg) program “Computational Mathematics” (W1214) granted by Austrian Science Funds (FWF). The work of P. Sun was also supported by CSC-FWF joint program.


Citation Information: Journal of Inverse and Ill-posed Problems, Volume 25, Issue 4, Pages 521–542, ISSN (Online) 1569-3945, ISSN (Print) 0928-0219, DOI: https://doi.org/10.1515/jiip-2016-0077.

Export Citation

© 2017 Walter de Gruyter GmbH, Berlin/Boston.Get Permission

Comments (0)

Please log in or register to comment.
Log in