Numerische Mathematik

, Volume 104, Issue 2, pp 177–203 | Cite as

A Tikhonov-based projection iteration for nonlinear Ill-posed problems with sparsity constraints



In this paper, we consider nonlinear inverse problems where the solution is assumed to have a sparse expansion with respect to a preassigned basis or frame. We develop a scheme which allows to minimize a Tikhonov functional where the usual quadratic regularization term is replaced by a one-homogeneous (typically weighted ℓ p ) penalty on the coefficients (or isometrically transformed coefficients) of such expansions. For (p < 2), the regularized solution will have a sparser expansion with respect to the basis or frame under consideration. The computation of the regularized solution amounts in our setting to a Landweber-fixed-point iteration with a projection applied in each fixed-point iteration step. The performance of the resulting numerical scheme is demonstrated by solving the nonlinear inverse single photon emission computerized tomography (SPECT) problem.


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  1. 1.
    Candès E.J., Guo F. (2001) New Multiscale Transforms, Minimum Total Variation Synthesis: Application, to Edge-Preserving Image Restoration. CalTech, PreprintGoogle Scholar
  2. 2.
    Censor Y., Gustafson D., Lent A., Tuy H. (1979) A new approach to the emission computerized tomography problem: simultaneous calculation of attenuation and activity coefficients. IEEE Trans. Nucl. Sci. 26, 2275–2279CrossRefGoogle Scholar
  3. 3.
    Ciarlet PG. (1995) Introduction to Numerical Linear Algebra and Optimisation. Cambrigde University Press, CambridgeGoogle Scholar
  4. 4.
    Cohen A., Dahmen W., Daubechies I., DeVore R. Harmonic analysis of the space BV. In: IGPM report no.,195, RWTH Aachen (2000)Google Scholar
  5. 5.
    Cohen A., DeVore R., Petrushev P., Xu H. (1999) Nonlinear approximation and the space BV( \(\mathbb{R}^{2}\)). Am. J. Math. 121, 587–628MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Daubechies I. (1992) Ten lectures on wavelets. SIAM, PhiladelphiaMATHGoogle Scholar
  7. 7.
    Daubechies I., Defrise M., De Mol C. (2004) An iterative thresholding algorithm for linear inverse problems with a sparsity constraint. Commun. Pure Appl. Math. 51, 1413–1541CrossRefMathSciNetGoogle Scholar
  8. 8.
    Daubechies I., Teschke G.: Wavelet–based image decomposition by variational functionals. In: Proceedings of SPIE vol. 5266, Wavelet Applications in Industrial Processing; Frederic Truchetet pp. 94–105, 2004Google Scholar
  9. 9.
    Daubechies I., Teschke G. (2005) Variational image restoration by means of wavelets: simultaneous decomposition, deblurring and denoising. Appl. Comput. Harmonic. Anal 19(1): 1–16MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Dicken V.: Simultaneous activity and attenuation reconstruction in single photon emission computed tomography, a nonlinear ill-posed problem. Ph.D. Thesis, Universität Potsdam, (1998)Google Scholar
  11. 11.
    Dicken V. (1999) A new approach towards simultaneous activity and attenuation reconstruction in emission tomography. Inverse Probl. 15(4): 931–960MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Manglos S.H., Young T.M.: Constrained intraSPECT reconstructions from SPECT projections. In: Conf. Rec. IEEE Nuclear Science Symp. and Medical Imaging Conference, San Francisco, CA, pp. 1605–1609 (1993)Google Scholar
  13. 13.
    Meyer Y.: Oscillating Patterns in Image Processing and Nonlinear Evolution Equations. University Lecture Series, vol. 22, AMS (2002)Google Scholar
  14. 14.
    Meyer Y.: Oscillating patterns in some nonlinear evolution equations. In: CIME report (2003)Google Scholar
  15. 15.
    Osher S., Vese L.: Modeling textures with total variation minimization and oscillating patterns in image processing. In: Technical report 02-19, University of California Los C.A.M. (2002)Google Scholar
  16. 16.
    Ramlau R. (2002) Morozov’s discrepancy principle for Tikhonov regularization of nonlinear . Numer. Funct. Anal. Optim. 23(1&2): 147–172MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Ramlau R. (2002) A steepest descent algorithm for the global minimization of the Tikhonov– functional. Inverse Probl. 18(2): 381–405MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Ramlau R. (2003) TIGRA–an iterative algorithm for regularizing nonlinear ill-posed problems. Inverse Probl. 19(2): 433–467MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Ramlau R. (2005) On the use of fixed point iterations for the regularization of nonlinear ill-posed problems. J. Inverse Ill-Posed Probl. 13(2): 175–200MATHMathSciNetGoogle Scholar
  20. 20.
    Ramlau R., Clackdoyle R., Noo F., Bal G. (2000) Accurate attenuation correction in SPECT imaging using optimization of bilinear functions and assuming an unknown spatially-varying attenuation distribution. Z. Angew. Math. Mech. 80(9): 613–621MATHCrossRefGoogle Scholar
  21. 21.
    Ramlau R., Teschke G. (2005) Tikhonov replacement functionals for iteratively solving nonlinear operator equations. Inverse Probl. 21, 1571–1592MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Rudin L., Osher S., Fatemi E. (1992) Nonlinear total variations based noise removal algorithms. Physica D 60, 259–268MATHCrossRefGoogle Scholar
  23. 23.
    Terry J.A., Tsui B.M.W., Perry J.R., Hendricks J.L., Gullberg G.T.: The design of a mathematical phantom of the upper human torso for use in 3-d spect imaging research. In: Proceedings of 1990 Fall Meeting Biomed. Eng. Soc. (Blacksburg, VA), pp. 1467–1474. New York University Press, Newyork (1990)Google Scholar
  24. 24.
    Welch A., Clack R., Christian P.E., Gullberg G.T. (1996) Toward accurate attenuation correction without transmission measurements. J. Nucl. Med. 37, 18PGoogle Scholar
  25. 25.
    Welch A., Clack R., Natterer F., Gullberg G.T. (1997) Toward accurate attenuation correction in SPECT without transmission measurements. IEEE Trans. Med. Imaging 16, 532–540CrossRefGoogle Scholar

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© Springer-Verlag 2006

Authors and Affiliations

  1. 1.Johann Radon Institute for Computational and Applied Mathematics (RICAM)Austrian Academy of SciencesLinzAustria
  2. 2.Konrad–Zuse–Zentrum für Informationstechnik Berlin (ZIB)Berlin-DahlemGermany

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