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Journal of Inverse and Ill-posed Problems

Editor-in-Chief: Kabanikhin, Sergey I.

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Volume 25, Issue 3


Regularization and numerical solution of the inverse scattering problem using shearlet frames

Gitta Kutyniok / Volker Mehrmann / Philipp C. Petersen
Published Online: 2016-06-29 | DOI: https://doi.org/10.1515/jiip-2015-0048


Regularization techniques for the numerical solution of inverse scattering problems in two space dimensions are discussed. Assuming that the boundary of a scatterer is its most prominent feature, we exploit as model the class of cartoon-like functions. Since functions in this class are asymptotically optimally sparsely approximated by shearlet frames, we consider shearlets as a means for regularization. We analyze two approaches, namely solvers for the nonlinear problem and for the linearized problem obtained by the Born approximation. As example for the first class we study the acoustic inverse scattering problem, and for the second class, the inverse scattering problem of the Schrödinger equation. Whereas our emphasis for the linearized problem is more on the theoretical side due to the standardness of associated solvers, we provide numerical examples for the nonlinear problem that highlight the effectiveness of our algorithmic approach.

Keywords: Helmholtz equation; inverse medium scattering; regularization; Schrödinger equation, shearlets; sparse approximation

MSC 2010: 34L25; 35P25; 42C40; 42C15; 65J22; 65T60; 76B15; 78A46


  • [1]

    M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover, New York, 1992. Google Scholar

  • [2]

    R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975. Google Scholar

  • [3]

    B. Adcock, A. Hansen, C. Poon and B. Roman, Breaking the coherence barrier: Asymptotic incoherence and asymptotic sparsity in compressed sensing, preprint (2013), https://arxiv.org/abs/1302.0561; see also 10th International Conference on Sampling Theory and Applications (SampTA 2013), Bremen (2013), 1–4.

  • [4]

    B. Adcock, G. Kutyniok, A. C. Hansen and J. Ma, Linear stable sampling rate: Optimality of 2D wavelet reconstructions from Fourier measurements, SIAM J. Math. Anal. 47 (2015), no. 2, 1196–1233. Google Scholar

  • [5]

    S. W. Anzengruber and R. Ramlau, Morozov’s discrepancy principle for Tikhonov-type functionals with nonlinear operators, Inverse Problems 26 (2010), Article ID 025001. Google Scholar

  • [6]

    G. Bao and F. Triki, Error estimates for the recursive linearization of inverse medium problems, J. Comput. Math. 28 (2010), 725–744. Web of ScienceGoogle Scholar

  • [7]

    J. Barzilai and J. M. Borwein, Two-point step size gradient methods, IMA J. Numer. Anal. 8 (1988), 141–148. CrossrefGoogle Scholar

  • [8]

    M. Beals and M. Reed, Microlocal regularity theorems for nonsmooth pseudodifferential operators and applications to nonlinear problems, Trans. Amer. Math. Soc. 285 (1984), 159–184. Google Scholar

  • [9]

    E. J. Candès and D. L. Donoho, Recovering edges in ill-posed inverse problems: Optimality of curvelet frames, Ann. Statist. 30 (2002), 784–842. CrossrefGoogle Scholar

  • [10]

    E. J. Candès and D. L. Donoho, New tight frames of curvelets and optimal representation of objects with C2 singularities, Comm. Pure Appl. Math. 57 (2004), 219–266. Google Scholar

  • [11]

    O. Christensen, An Introduction to Frames and Riesz Bases, Birkhäuser, Basel, 2003. Google Scholar

  • [12]

    F. Colonna, G. Easley, K. Guo and D. Labate, Radon transform inversion using the shearlet representation, Appl. Comput. Harmon. Anal. 29 (2010), 232–250. Web of ScienceCrossrefGoogle Scholar

  • [13]

    D. Colton, J. Coyle and P. Monk, Recent developments in inverse acoustic scattering theory, SIAM Rev. 42 (2000), 369–414. CrossrefGoogle Scholar

  • [14]

    D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, Springer, New York, 2013. Google Scholar

  • [15]

    I. Daubechies, M. Defrise and C. De Mol, An iterative thresholding algorithm for linear inverse problems with a sparsity constraint, Comm. Pure Appl. Math. 57 (2004), 1413–1457. Google Scholar

  • [16]

    D. L. Donoho, Sparse components of images and optimal atomic decompositions, Constr. Approx. 17 (2001), 353–382. CrossrefGoogle Scholar

  • [17]

    D. L. Donoho and G. Kutyniok, Microlocal analysis of the geometric separation problem, Comm. Pure Appl. Math. 66 (2013), 1–47. Google Scholar

  • [18]

    I. M. Gelfand, M. I. Graev and N. Y. Vilenkin, Generalized Functions. Vol. 5: Integral Geometry and Representation Theory, Academic Press, New York, 1966. Google Scholar

  • [19]

    M. Genzel and G. Kutyniok, Asymptotic analysis of inpainting via universal shearlet systems, SIAM J. Imaging Sci. 7 (2014), no. 4, 2301–2339. Google Scholar

  • [20]

    M. Grasmair, M. Haltmeier and O. Scherzer, Sparse regularization with q penalty term, Inverse Problems 24 (2008), Article ID 055020. Google Scholar

  • [21]

    K. Guo, G. Kutyniok and D. Labate, Sparse multidimensional representations using anisotropic dilation and shear operators, Wavelets and Splines: Athens 2005, Nashboro Press, Brentwood (2006), 189–201. Google Scholar

  • [22]

    A. Hansen B. Roman and B. Adcock, On asymptotic structure in compressed sensing, preprint (2014), http://arxiv.org/abs/1406.4178.

  • [23]

    D. Jerison and C. E. Kenig, Unique continuation and absence of positive eigenvalues for Schrödinger operators, Ann. of Math. (2) 121 (1985), 463–494. Google Scholar

  • [24]

    B. Jin and P. Maass, Sparsity regularization for parameter identification problems, Inverse Problems 28 (2012), Article ID 123001. Google Scholar

  • [25]

    B. Kaltenbacher, A. Neubauer and O. Scherzer, Iterative Regularization Methods for Nonlinear Ill-Posed Problems, De Gruyter, Berlin, 2008. Google Scholar

  • [26]

    E. J. King, G. Kutyniok and X. Zhuang, Analysis of inpainting via clustered sparsity and microlocal analysis, J. Math. Imaging Vision 48 (2014), 205–234. Google Scholar

  • [27]

    P. Kittipoom, G. Kutyniok and W.-Q. Lim, Construction of compactly supported shearlet frames, Constr. Approx. 35 (2012), 21–72. Web of ScienceCrossrefGoogle Scholar

  • [28]

    G. Kutyniok and D. Labate, Introduction to shearlets, Shearlets: Multiscale Analysis for Multivariate Data, Birkhäuser, Boston (2012), 1–38. Google Scholar

  • [29]

    G. Kutyniok and W.-Q. Lim, Compactly supported shearlets are optimally sparse, J. Approx. Theory 163 (2011), 1564–1589. Web of ScienceGoogle Scholar

  • [30]

    G. Kutyniok and W.-Q. Lim, Image separation using wavelets and shearlets, Curves and Surfaces, Lecture Notes in Comput. Sci. 6920, Springer, Berlin (2012), 416–430. Google Scholar

  • [31]

    G. Kutyniok, W.-Q. Lim and R. Reisenhofer, ShearLab 3D: Faithful digital shearlet transform with compactly supported shearlets, ACM Trans. Math. Software 42 (2016), no. 1, 1–42. CrossrefGoogle Scholar

  • [32]

    A. Lechleiter, K. S. Kazimierski and M. Karamehmedovic, Tikhonov regularization in Lp applied to inverse medium scattering, Inverse Problems 29 (2013), Article ID 075003. Google Scholar

  • [33]

    J. Ma, Generalized sampling reconstruction from Fourier measurements using compactly supported shearlets, Appl. Comput. Harmon. Anal. (2015), 10.1016/j.acha.2015.07.006. Google Scholar

  • [34]

    S. Mallat, A wavelet tour of signal processing: The sparse way, Elsevier, Amsterdam, 2009. Google Scholar

  • [35]

    S. Moskow and J. C. Schotland, Convergence and stability of the inverse scattering series for diffuse waves, Inverse Problems 24 (2008), Article ID 065005. Google Scholar

  • [36]

    P. Ola, L. Päivärinta and V. Serov, Recovering singularities from backscattering in two dimensions, Comm. Partial Differential Equations 26 (2001), 697–715. Google Scholar

  • [37]

    V. M. Patel, G. R. Easley and D. M. Healy, Jr., Shearlet-based deconvolution, IEEE Trans. Image Process. 18 (2009), 2673–2685. Web of ScienceCrossrefGoogle Scholar

  • [38]

    R. Ramlau and G. Teschke, Tikhonov replacement functionals for iteratively solving nonlinear operator equations, Inverse Problems 21 (2005), 1571–1592. Google Scholar

  • [39]

    R. Ramlau and G. Teschke, A Tikhonov-based projection iteration for nonlinear ill-posed problems with sparsity constraints, Numer. Math. 104 (2006), 177–203. Google Scholar

  • [40]

    J. M. Reyes, Inverse backscattering for the Schrödinger equation in 2D, Inverse Problems 23 (2007), 625–643. Google Scholar

  • [41]

    O. Scherzer, M. Grasmair, H. Grossauer, M. Haltmeier and F. Lenzen, Variational Methods in Imaging, Springer, New York, 2009. Google Scholar

  • [42]

    V. Serov, Inverse backscattering Born approximation for a two-dimensional magnetic Schrödinger operator, Inverse Problems 29 (2013), Article ID 075015. Google Scholar

  • [43]

    J. Sylvester, An estimate for the free Helmholtz equation that scales, Inverse Probl. Imaging 3 (2009), 333–351. Google Scholar

  • [44]

    G. Teschke and C. Borries, Accelerated projected steepest descent method for nonlinear inverse problems with sparsity constraints, Inverse Problems 26 (2010), Article ID 025007. Google Scholar

  • [45]

    A. N. Tikhonov, On the regularization of ill-posed problems, Dokl. Akad. Nauk SSSR 153 (1963), 49–52. Google Scholar

  • [46]

    J. Weidmann, Linear Operators in Hilbert Spaces, Springer, New York, 1980. Google Scholar

About the article

Received: 2015-05-14

Revised: 2016-03-21

Accepted: 2016-05-04

Published Online: 2016-06-29

Published in Print: 2017-06-01

Funding Source: Deutsche Forschungsgemeinschaft

Award identifier / Grant number: KU 1446/14

Funding Source: Deutsche Forschungsgemeinschaft

Award identifier / Grant number: Collaborative Research Center TRR 109 “Discretization in Geometry and Dynamics”

Funding Source: Deutsche Forschungsgemeinschaft

Award identifier / Grant number: Research Center Matheon “Mathematics for key technologies”

The first author acknowledges support by the Einstein Foundation Berlin, by the Einstein Center for Mathematics Berlin (ECMath), by Deutsche Forschungsgemeinschaft (DFG) Grant KU 1446/14, by 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. The second author also acknowledges support by Matheon, and the third author thanks the DFG Collaborative Research Center TRR 109 “Discretization in Geometry and Dynamics” for its support.

Citation Information: Journal of Inverse and Ill-posed Problems, Volume 25, Issue 3, Pages 287–309, ISSN (Online) 1569-3945, ISSN (Print) 0928-0219, DOI: https://doi.org/10.1515/jiip-2015-0048.

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