Abstract
In this chapter, we study the mathematical imaging problem of diffraction tomography (DT), which is an inverse scattering technique used to find material properties of an object by illuminating it with probing waves and recording the scattered waves. Conventional DT relies on the Fourier diffraction theorem, which is applicable under the condition of weak scattering. However, if the object has high contrasts or is too large compared to the wavelength, it tends to produce multiple scattering, which complicates the reconstruction. In this chapter, we give a survey on diffraction tomography and compare the reconstruction of low- and high-contrast objects. We also implement and compare the reconstruction using the full waveform inversion method which, contrary to the Born and Rytov approximations, works with the total field and is more robust to multiple scattering.
Similar content being viewed by others
References
Amestoy, P.R., Buttari, A., L’excellent, J.-Y., Mary, T.: Performance and scalability of the block low-rank multifrontal factorization on multicore architectures. ACM Trans. Math. Softw. (TOMS) 45(1), 1–26 (2019). https://doi.org/10.1145/3242094
Bamberger, A., Chavent, G., Lailly, P.: About the stability of the inverse problem in the 1-d wave equation. J. Appl. Math. Optim. 5, 1–47 (1979)
Barucq, H., Chavent, G., Faucher, F.: A priori estimates of attraction basins for nonlinear least squares, with application to Helmholtz seismic inverse problem. Inverse Probl. 35(11), 115004 (2019). https://doi.org/10.1088/1361-6420
Bednar, J.B., Shin, C., Pyun, S.: Comparison of waveform inversion, part 2: phase approach. Geophys. Prospect. 55(4), 465–475 (2007). ISSN: 1365-2478. https://doi.org/10.1111/j.1365-2478.2007.00618.x
Beinert, R., Quellmalz, M.: Total variation-based reconstruction and phase retrieval for diffraction tomography SIAM J. Imaging Sci. 15(3), 1373–1399 (2022). ISSN: 1936-4954. https://doi.org/10.1137/22M1474382
Bunks, C., Saleck, F.M., Zaleski, S., Chavent, G.: Multiscale seismic waveform inversion. Geophysics 60(5), 1457–1473 (1995). https://doi.org/10.1190/1.1443880
Chen, B., Stamnes, J.J.: Validity of diffraction tomography based on the first Born and the first Rytov approximations. Appl. Opt. 37(14), 2996 (1998). https://doi.org/10.1364/ao.37.002996
Clément, F., Chavent, G., Gómez, S.: Migration-based traveltime wave-form inversion of 2-D simple structures: a synthetic example. Geophysics 66(3), 845–860 (2001). https://doi.org/10.1190/1.1444974
Cockburn, B., Gopalakrishnan, J., Lazarov R.: Unified hybridization of discontinuous Galerkin, mixed, and continuous Galerkin methods for second order elliptic problems. SIAM J. Numer. Anal. 47(2), 1319–1365 (2009). https://doi.org/10.1137/070706616
Colton, D., Kress, R.: Inverse Acoustic and Electromagnetic Scattering Theory. Applied Mathematical Sciences, vol. 93, 3rd edn. Springer, Berlin (2013). ISBN: 978-1-4614-4941-6. https://doi.org/10.1007/978-1-4614-4942-3
Devaney, A.: A filtered backpropagation algorithm for diffraction tomography. Ultrason. Imaging 4(4), 336–350 (1982). https://doi.org/10.1016/0161-7346(82)90017-7
Devaney, A.: Mathematical Foundations of Imaging, Tomography and Wave-Field Inversion. Cambridge University Press (2012). https://doi.org/10.1017/CBO9781139047838
Engquist, B., Majda, A.: Absorbing boundary conditions for numerical simulation of waves. Proc. Natl. Acad. Sci. 74(5), 1765–1766 (1977)
Fan, S., Smith-Dryden, S., Li, G., Saleh, B.E.A.: An iterative reconstruction algorithm for optical diffraction tomography. In: IEEE Photonics Conference (IPC), pp. 671–672 (2017). https://doi.org/10.1109/ipcon.2017.8116276
Faucher, F.: Contributions to seismic full waveform inversion for time harmonic wave equations: Stability estimates, convergence analysis, numerical experiments involving large scale optimization algorithms. PhD thesis. Université de Pau et Pays de l’Ardour, pp. 1–400 (2017)
Faucher, F.: Hawen: time-harmonic wave modeling and inversion using hybridizable discontinuous Galerkin discretization. J. Open Source Softw. 6(57) (2021). https://doi.org/10.21105/joss.02699
Faucher, F., Scherzer, O.: Adjoint-state method for Hybridizable Discontinuous Galerkin discretization, application to the inverse acoustic wave problem. Comput. Methods Appl. Mech. Eng. 372, 113406 (2020). ISSN: 0045-7825. https://doi.org/10.1016/j.cma.2020.113406
Faucher, F., Alessandrini, G., Barucq, H., de Hoop, M., Gaburro, R., Sincich, E.: Full Reciprocity-Gap Waveform Inversion, enabling sparse-source acquisition. Geophysics 85(6), R461–R476 (2020a). https://doi.org/10.1190/geo2019-0527.1
Faucher, F., Chavent, G., Barucq, H., Calandra, H.: A priori estimates of attraction basins for velocity model reconstruction by time-harmonic Full Waveform Inversion and Data-Space Reflectivity formulation. Geophysics 85(3), R223–R241 (2020b). https://doi.org/10.1190/geo2019-0251.1
Faucher, F., Scherzer O., Barucq, H.: Eigenvector models for solving the seismic inverse problem for the Helmholtz equation. Geophys. J. Int. (2020c). ISSN: 0956-540X. https://doi.org/10.1093/gji/ggaa009
Faucher, F., de Hoop, M.V., Scherzer, O.: Reciprocitygap misfit functional for Distributed Acoustic Sensing, combining data from passive and active sources. Geophysics 86(2), R211–R220 (2021). ISSN: 0016-8033. https://doi.org/10.119/geo2020-0305.1
Fichtner, A., Kennett, B.L., Igel, H., Bunge, H.-P.: Theoretical back ground for continental- and global-scale full-waveform inversion in the time–frequency domain. Geophys. J. Int. 175(2), 665–685 (2008). https://doi.org/10.1111/j.1365-246X.2008.03923.x
Gbur, G., Wolf, E.: Hybrid diffraction tomography without phase information. J. Opt. Soc. Am. A 19(11), 2194–2202 (2002). https://doi.org/10.1364/OL27.001890
Hanke, M.: Conjugate Gradient Type Methods for Ill-Posed Problems. Pitman Research Notes in Mathematics Series, vol. 327. Longman Scientific & Technical, Harlow (1995)
Hesthaven, J.S., Warburton, T.: Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications. Springer Science & Business Media (2007). https://doi.org/10.1007/978-0-387-72067-8
Hielscher, R., Potts, D., Quellmalz, M.: An SVD in spherical surface wave tomography. In: Hofmann, B., Leitao, A., Zubelli, J.P. (eds.) New Trends in Parameter Identification for Mathematical Models. Trends in Mathematics, pp. 121–144. Birkhäuser, Basel (2018). ISBN: 978-3-319-70823-2. https://doi.org/10.1007/978-3-319-70824-9_7
Hielscher, R., Quellmalz, M.: Optimal mollifiers for spherical de-convolution. Inverse Probl. 31(8), 085001 (2015). https://doi.org/10.1088/02.665611/31/8/085001
Hielscher, R., Quellmalz, M.: Reconstructing a function on the sphere from its means along vertical slices. Inverse Probl. Imaging 10(3), 711–739 (2016). ISSN: 1930-8337. https://doi.org/10.3934/ipi.2016018
Horstmeyer, R., Chung, J., Ou, X., Zheng, G., Yang, C.: Diffraction tomography with Fourier ptychography. Optica 3(8), 827–835 (2016). https://doi.org/10.1364/OPTICA.3.000827
Huynh-Thu, Q., Ghanbari, M.: The accuracy of PSNR in predicting video quality for different video scenes and frame rates. Telecommun. Syst. 49(1), 35–48 (2010). https://doi.org/10.1007/s112350109351x
Iwata, K., Nagata, R.: Calculation of refractive index distribution from interferograms using the Born and Rytov’s approximation. Jpn. J. Appl. Phys. 14(S1), 379–383 (1975). https://doi.org/10.7567/jjaps.14s1.379
Kak, A.C., Slaney M.: Principles of Computerized Tomographic Imaging. Classics in Applied Mathematics, vol. 33. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2001)
Kaltenbacher, B.: Minimization based formulations of inverse problems and their regularization. SIAM J. Optim. 28(1), 620–645 (2018). https://doi.org/10.1137/17M1124036
Keiner, J., Kunis, S., Potts, D.: Using NFFT3 – a software library for various nonequispaced fast Fourier transforms. ACM Trans. Math. Softw. 36, Article 19, 1–30 (2009). https://doi.org/10.1145/1555386.1555388
Keiner, J., Kunis, S., Potts, D.: NFFT 3.5, C subroutine library (n.d.). https://www.tu-chemnitz.de/~potts/nfft
Kirby, R.M., Sherwin, S.J., Cockburn, B.: To CG or to HDG: a comparative study. J. Sci. Comput. 51(1), 183–212 (2012). https://doi.org/10.1007/s10915-011-9501-7
Kirisits, C., Quellmalz, M., Ritsch-Marte, M., Scherzer, O., Setterqvist, E., Steidl, G.: Fourier reconstruction for diffraction tomography of an object rotated into arbitrary orientations. Inverse Probl. (2021). ISSN: 0266-5611. https://doi.org/10.1088/1361-6420/ac2749
Knopp, T., Kunis, S., Potts, D.: A note on the iterative MRI reconstruction from nonuniform k-space data. Int. J. Biomed. Imag. (2007). https://doi.org/10.1155/2007/24727
Kunis, S., Potts, D.: Stability results for scattered data interpolation by trigonometric polynomials. SIAM J. Sci. Comput. 29, 1403–1419 (2007). https://doi.org/10.1137/060665075
Lailly, P.: The seismic inverse problem as a sequence of before stack migrations. In: Bednar, J.B. (ed.) Conference on Inverse Scattering: Theory and Application, pp. 206–220. Society for Industrial and Applied Mathematics (1983)
Luo, Y., Schuster, G.T.: Wave-equation traveltime inversion. Geophysics 56(5), 645–653 (1991). https://doi.org/10.1190/1.1443081
Maleki, M.H., Devaney, A.: Phase-retrieval and intensity-only recon-struction algorithms for optical diffraction tomography. J. Opt. Soc. Am. A 10(5), 1086 (1993). https://doi.org/10.1364/josaa.10.001086
Métivier, L., Brossier, R., Mérigot, Q., Oudet, E., Virieux, J.: Measuring the misfit between seismograms using an optimal transport distance: application to full waveform inversion. Geophys. J. Int. 205(1), 345–377 (2016). https://doi.org/10.1093/gji/ggw014
Monk, P.: Finite Element Methods for Maxwell’s Equations. Oxford University Press, Oxford (2003)
Müller, P., Schürmann, M., Guck, J.: ODTbrain: a Python library for full-view, dense diffraction tomography. BMC Bioinform. 16(367) (2015). https://doi.org/10.1186/s12859-015-0764-0
Müller, P., Schürmann, M., Guck, J.: The Theory of Diffraction Tomography (2016). arXiv: 1507.00466 [q-bio.QM]
Natterer, F.: The Mathematics of Computerized Tomography, x+ 222. B. G. Teubner, Stuttgart (1986). ISSN: 3-519-02103-X
Natterer, F., Wübbeling, F.: Mathematical Methods in Image Reconstruction. Monographs on Mathematical Modeling and Computation, vol. 5. SIAM, Philadelphia (2001)
Nocedal, J., Wright, S.J.: Numerical Optimization. Springer Series in Operations Research, 2nd edn. Springer, Berlin (2006)
Plessix, R.-E.: A review of the adjoint-state method for computing the gradient of a functional with geophysical applications. Geophys. J. Int. 167(2), 495–503 (2006). https://doi.org/10.1111/j.1365-246X.2006.02978.x
Plonka, G., Potts, D., Steidl, G., Tasche, M.: Numerical Fourier Analysis. Applied and Numerical Harmonic Analysis. Birkhäuser (2018). ISSN: 978-3-030-04305-6. https://doi.org/10.1007/978-3-030-04306-3
Potts, D., Steidl, G.: A new linogram algorithm for computerized tomography. IMA J. Numer. Anal. 21, 769–782 (2001). https://doi.org/10.1093/imanum/21.3.769
Pratt, R.G., Shin, C., Hick, G.J.: Gauss–Newton and full Newton methods in frequency–space seismic waveform inversion. Geophys. J. Int. 133(2), 341–362 (1998). https://doi.org/10.1046/j.1365-246X.1998.00498.x
Pyun, S., Shin, C., Bednar, J.B.: Comparison of waveform inversion, part 3: amplitude approach. Geophys. Prospect. 55(4), 477–485 (2007). ISSN: 1365-2478. https://doi.org/10.1111/j.1365-2478.2007.00619.x
Shin, C., Pyun, S., Bednar, J.B.: Comparison of waveform inversion, part 1: conventional wavefield vs logarithmic wavefield. Geophys. Prospect. 55(4), 449–464 (2007). ISSN: 1365-2478. https://doi.org/10.1111/j.1365-2478-2007.00617.x
Slaney, M., Kak, A.C., Larsen, L.E.: Limitations of imaging with first-order diffraction tomography. IEEE Trans. Microw. Theory Techn. 32(8), 860–874 (1984). https://doi.org/10.1109/TMTT.1984.1132783
Sung, Y., Choi, W., FangYen, C., Badizadegan, K., Dasari, R.R., Feld, M.S.: Optical diffraction tomography for high resolution live cell imaging. Opt. Express 17(1), 266–277 (2009)
Tarantola, A.: Inversion of seismic reflection data in the acoustic approximation. Geophysics 49, 1259–1266 (1984). https://doi.org/10.1190/1.1441754
Van Leeuwen, T., Mulder, W.: A correlation-based misfit criterion for wave-equation traveltime tomography. Geophys. J. Int. 182(3), 1383–1394 (2010)
Virieux, J.: SH-wave propagation in heterogeneous media: velocity-stress finite-difference method. Geophysics 49(11), 1933–1942 (1984)
Virieux, J., Operto, S.: An overview of full-waveform inversion in exploration geophysics. Geophysics 74(6), WCC1–WCC26 (2009). https://doi.org/10.1190/1.3238367
Wedberg, T.C., Stamnes, J.J.: Comparison of phase retrieval methods for optical diffraction tomography. Pure Appl. Opt. 4, 39–54 (1995). https://doi.org/10.1088/0963-9659/4/1/005
Wolf, E.: Three-dimensional structure determination of semi-transparent objects from holographic data. Opt. Commun. 1, 153–156 (1969)
Acknowledgements
We thank the anonymous reviewer for carefully reading the manuscript and making various suggestions for its improvement. This work is supported by the Austrian Science Fund (FWF) within SFB F68 (“Tomography across the Scales”), Projects F68-06 and F68-07. FF is funded by the Austrian Science Fund (FWF) under the Lise Meitner fellowship M 2791-N. Funding by the DFG under Germany’s Excellence Strategy – The Berlin Mathematics Research Center MATH+ (EXC-2046/1, Projektnummer: 390685689) as well as by the DFG project STE 571/19 (Projektnummer: 495365311) is gratefully acknowledged. For the numerical experiments, we acknowledge the use of the Vienna Scientific Cluster VSC-4 (https://vsc.ac.at/).
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Section Editor information
Rights and permissions
Copyright information
© 2022 Springer Nature Switzerland AG
About this entry
Cite this entry
Faucher, F., Kirisits, C., Quellmalz, M., Scherzer, O., Setterqvist, E. (2022). Diffraction Tomography, Fourier Reconstruction, and Full Waveform Inversion. In: Chen, K., Schönlieb, CB., Tai, XC., Younces, L. (eds) Handbook of Mathematical Models and Algorithms in Computer Vision and Imaging. Springer, Cham. https://doi.org/10.1007/978-3-030-03009-4_115-1
Download citation
DOI: https://doi.org/10.1007/978-3-030-03009-4_115-1
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-03009-4
Online ISBN: 978-3-030-03009-4
eBook Packages: Springer Reference MathematicsReference Module Computer Science and Engineering