Inverse Problems and Parameter Identification in Image Processing

  • Jens F. Acker
  • Benjamin Berkels
  • Kristian Bredies
  • Mamadou S. Diallo
  • Marc Droske
  • Christoph S. Garbe
  • Matthias Holschneider
  • Jaroslav Hron
  • Claudia Kondermann
  • Michail Kulesh
  • Peter Maass
  • Nadine Olischläger
  • Heinz-Otto Peitgen
  • Tobias Preusser
  • Martin Rumpf
  • Karl Schaller
  • Frank Scherbaum
  • Stefan Turek

Abstract

Many problems in imaging are actually inverse problems. One reason for this is that conditions and parameters of the physical processes underlying the actual image acquisition are usually not known. Examples for this are the inhomogeneities of the magnetic field in magnetic resonance imaging (MRI) leading to nonlinear deformations of the anatomic structures in the recorded images, material parameters in geological structures as unknown parameters for the simulation of seismic wave propagation with sparse measurement on the surface, or temporal changes in movie sequences given by intensity changes or moving image edges and resulting from deformation, growth and transport processes with unknown fluxes. The underlying physics is mathematically described in terms of variational problems or evolution processes. Hence, solutions of the forward problem are naturally described by partial differential equations. These forward models are reflected by the corresponding inverse problems as well. Beyond these concrete, direct modeling links to continuum mechanics abstract concepts from physical modeling are successfully picked up to solve general perceptual problems in imaging. Examples are visually intuitive methods to blend between images showing multiscale structures at different resolution or methods for the analysis of flow fields.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    J. F.Acker. PDF basierte Visualisierungsmethoden für instationäre Strömungen auf unstrukturierten Gittern. PhD thesis, Universität Dortmund, to appear in 2008.Google Scholar
  2. [2]
    L. Ambrosio and V.M. Tortorelli. On the approximation of free discontinuity problems. Bollettino de la Unione Matematica Italiana 6(7): 1 5-123, 1992.Google Scholar
  3. [3]
    L. Ambrosio, N. Fusco, and D. Pallara. Functions of Bounded Variation and Free Discontinuity Problems. Oxford Mathematical Monographs. The Clarendon Press, New York, 2000.MATHGoogle Scholar
  4. [4]
    M. Arredondo, K. Lebart, and D. Lane. Optical flow using textures. Pattern Recognition Letters, 25(4):4 9-457, 2004.Google Scholar
  5. [5]
    R. Bajcsy and C Broit. Matching of deformed images. In Proceedings of the 6th International Conference on Pattern Recognition, pages 351–353, 1982.Google Scholar
  6. [6]
    J.M. Ball. Global invertibility of sobolev functions and the interpenetration of matter. Proceedings of the Royal Society of Edinburgh, 88A: 3 5-328, 1981.Google Scholar
  7. [7]
    L. Bar, B. Berkels, M. Rumpf, and G. Sapiro. A variational framework for simulateneous motion estimation and restoration of motion-blurred video. In Proceedings ICCV, to appear, 2007.Google Scholar
  8. [8]
    J. Bigün, G.H. Granlund, and J. Wiklund. Multidimensional orientation estimation with applications to texture analysis and optical flow. IEEE Transactions on Pattern Analysis and Machine Intelligence 13(8): 7 5-790, 1991.Google Scholar
  9. [9]
    V. Blanz and T. Vetter. A morphable model for the synthesis of 3d faces. In Proceedings of SIGGRAPH 99, Computer Graphics Proceedings, Annual Conference Series, pages 187–194, 1999.Google Scholar
  10. [10]
    K. Bredies. Optimal control of degenerate parabolic equations in image processing. PhD thesis, University of Bremen, 2007.Google Scholar
  11. [11]
    K. Bredies, D.A. Lorenz, and P. Maass. A generalized conditional gradient method and its connection to an iterative shrinkage method. To appear in Computational Optimization and Applications, 2008.Google Scholar
  12. [12]
    L. Gottesfeld Brown. A survey of image registration techniques. ACM Computing Surveys, 24(4):325–376, 1992.CrossRefGoogle Scholar
  13. [13]
    A. Bruhn and J. Weickert. A confidence measure for variational optic flow methods. In R. Klette, R. Kozera, L. Noakes, J. Weickert, editors, Geometric Properties for Incomplete data Series: Computational Imaging and Vision, vol. 31, pages 2 3-298. Springer-Verlag, 2006.Google Scholar
  14. [14]
    A. Bruhn, J. Weickert, and C. Schnörr. Lucas/kanade meets horn/schunck: Combining local and global optic flow methods. International Journal of Computer Vision, 61(3):211–231, 2005.CrossRefGoogle Scholar
  15. [15]
    D. Bürkle, T. Preusser, and M. Rumpf. Transport and anisotropic diffusion in time-dependent flow visualization. In Proceedings Visualization’ 01, 2001.Google Scholar
  16. [16]
    B. Cabrai and L. Leedom. Imaging vector fields using line integral convolution. In James T. Kajiya, editor, Computer Graphics (SIGGRAPH’ 93 Proceedings), volume 27, pages 223–272, August 1993.Google Scholar
  17. [17]
    J. Capon. High-resolution frequency-wavenumber spectrum analysis. Proceedings of the IEEE, 57(8):1408–1418, 1969.CrossRefGoogle Scholar
  18. [18]
    F. Catté, P.-L. Lions, J.-M. Morel, and T. Coll. Image selective smoothing and edge detection by nonlinear diffusion. SIAM Journal On Numerical Analysis, 29(1):1 2-193, 1992.Google Scholar
  19. [19]
    M.S. Chong, A.E. Perry, and B.J. Cantwell. A general classification of three-dimensional flow fields. Physics of Fluids A, 2(5):7 5-777, 1990.Google Scholar
  20. [20]
    P.G. Ciarlet. Three-Dimensional Elasticity. Elsevier, New York, 1988.MATHGoogle Scholar
  21. [21]
    U. Clarenz, N. Litke, and M. Rumpf. Axioms and variational problems in surface parameterization. Computer Aided Geometric Design, 21(8): 7 7-749, 2004.Google Scholar
  22. [22]
    U. Clarenz, M. Droske, S. Henn, M. Rumpf, and K. Witsch. Computational methods for nonlinear image registration. In O. Scherzer, editor, Mathematical Models for Registration and Applications to Medical Imaging, Mathematics in Industry, volume 10, Springer, 2006.Google Scholar
  23. [23]
    A. Collignon et al. Automated multi-modality image registration based on information theory. In Y. Bizais, C. Barillot and R. Di Paola, editors, Proceedings of the XIVth international conference on information processing in medical imaging IPMI’95, computational imaging and vision, vol. 3, pp. 263–274, June 26–30, Ile de Berder, France, 1995. Kluwer Academic Publishers.Google Scholar
  24. [24]
    M.S. Diallo, M. Kulesh, M. Holschneider, and F. Scherbaum. Instantaneous polarization attributes in the time-frequency domain and wavefield separation. Geophysical Prospecting, 53(5):7 3-731, 2005.Google Scholar
  25. [25]
    M.S. Diallo, M. Kulesh, M. Holschneider, K. Kurennaya, and F. Scherbaum. Instantaneous polarization attributes based on an adaptive approximate covariance method. Geophysics, 71(5):V99–V104, 2006.CrossRefGoogle Scholar
  26. [26]
    M.S. Diallo, M. Kulesh, M. Holschneider, F. Scherbaum, and F. Adler. Characterization of polarization attributes of seismic waves using continuous wavelet transforms. Geophysics, 71(3):V67–V77, 2006.CrossRefGoogle Scholar
  27. [27]
    U. Diewald, T. Preusser, and M. Rumpf. Anisotropic diffusion in vector field visualization on euclidean domains and surfaces. IEEE Transactions on Visualization and Computer Graphics, 6(2):139–149, 2000.CrossRefGoogle Scholar
  28. [28]
    M. Droske and W. Ring. A Mumford-Shah level-set approach for geometric image registration. SIAM Journal on Applied Mathematics, 66(6):2127–2148, 2006.MATHCrossRefMathSciNetGoogle Scholar
  29. [29]
    M. Droske and M. Rumpf. A variational approach to non-rigid morphological registration. SIAM Applied Mathematics, 64(2):6 8-687, 2004.Google Scholar
  30. [30]
    M. Droske and M. Rumpf. Multi scale joint segmentation and registration of image morphology. IEEE Transaction on Pattern Recognition and Machine Intelligence, 29(12):2181–2194, December 2007.Google Scholar
  31. [31]
    M. Droske, C. Garbe, T. Preusser, M. Rumpf, and A. Telea. A phase field method for joint denoising, edge detection and motion estimation. SIAM Applied Mathematics, Revised Version Submitted, 2007.Google Scholar
  32. [32]
    L. Florack and A. Kuijper. The topological structure of scale-space images. Journal of Mathematical Imaging and Vision, 12(l):65–79, 2000. ISSN 0924-9907.MATHCrossRefMathSciNetGoogle Scholar
  33. [33]
    U. Grenander and M.I. Miller. Computational anatomy: An emerging discipline. Quarterly Applied Mathematics, 56(4):617–694, 1998.MATHMathSciNetGoogle Scholar
  34. [34]
    M. Griebel, T. Preusser, M. Rumpf, M.A. Schweitzer, and A. Telea. Flow field clustering via algebraic multigrid. In Proceedings IEEE Visualization, pages 35–42, 2004.Google Scholar
  35. [35]
    X. Gu and B.C. Vemuri. Matching 3D shapes using 2D conformai representations. In MICCAI 2004, LNCS 3216, pages 771–780, 2004.Google Scholar
  36. [36]
    D. Heeger. Model for the extraction of image flow. Journal of the Optical Society of America, 4(8):1455–1471, 1987.Google Scholar
  37. [37]
    M. Holschneider. Wavelets: An Analysis Tool. Clarendon Press, Oxford, 1995.MATHGoogle Scholar
  38. [38]
    M. Holschneider, M.S. Diallo, M. Kulesh, M. Ohrnberger, E. Lück, and F. Scherbaum. Characterization of dispersive surface waves using continuous wavelet transforms. Geophysical Journal International, 163(2): 4 3-478, 2005.Google Scholar
  39. [39]
    J.C. R. Hunt, A.A. Wray, and P. Moin. Eddies, stream and convergence zones in turbulent flow fields. Technical Report CTR-S88, Center for turbulence research, 1988.Google Scholar
  40. [40]
    V. Interrante and C. Grosch. Stragegies for effectively visualizing 3D flow with volume LIC. In Proceedings Visualization’ 97, pages 2 5-292, 1997.Google Scholar
  41. [41]
    J. Jeong and F. Hussain. On the identification of a vortex. Journal of Fluid Mechanics, 285:69–94, 1995.MATHCrossRefMathSciNetGoogle Scholar
  42. [42]
    S. Kalkan, D. Calow, M. Felsberg, F. Worgotter, M. Lappe, and N. Kruger. Optic flow statistics and intrinsic dimensionality, 2004.Google Scholar
  43. [43]
    E. R. Kanasewich. Time Sequence Analysis in Geophysics. University of Alberta Press, Edmonton, Alberta, 1981.Google Scholar
  44. [44]
    B. Kawohl and N. Kutev. Maximum and comparison principle for onedimensional anisotropic diffusion. Mathematische Annalen, 311(l):107–123, 1998.MATHCrossRefMathSciNetGoogle Scholar
  45. [45]
    C. Kondermann, D. Kondermann, B. Jähne, and C. Garbe. Comparison of confidence and situation measures and their optimality for optical flows, submitted to International Journal of Computer Vision, February 2007.Google Scholar
  46. [46]
    C. Kondermann, D. Kondermann, B. Jähne, and C. Garbe. An adaptive confidence measure for optical flows based on linear subspace projections. In Proceedings of the DAGM-Symposium, pages 132–141, 2007. http://dx.doi.org/10.1007/978-3-540-74936-3_14.Google Scholar
  47. [47]
    C. Kondermann, D. Kondermann, B. Jähne, and C. Garbe. Optical flow estimation via flow inpainting using surface situation measures. submitted, 2007.Google Scholar
  48. [48]
    A. Kufner. Weighted sobolev spaces, 1980. Teubner-Texte zur Mathematik, volume 31.Google Scholar
  49. [49]
    M. Kulesh, M. Holschneider, M. S. Diallo, Q. Xie, and F. Scherbaum. Modeling of wave dispersion using continuous wavelet transforms. Pure and Applied Geophysics, 162(5):843–855, 2005.CrossRefGoogle Scholar
  50. [50]
    M. Kulesh, M.S. Diallo, M. Holschneider, K. Kurennaya, F. Krüger, M. Ohrnberger, and F. Scherbaum. Polarization analysis in the wavelet domain based on the adaptive covariance method. Geophysical Journal International, 170(2):667–678, 2007.CrossRefGoogle Scholar
  51. [51]
    M. Kulesh, M. Holschneider, M. Ohrnberger, and E. Lück. Modeling of wave dispersion using continuous wavelet transforms II: wavelet based frequency-velocity analysis. Technical Report 154, Preprint series of the DFG priority program 1114 “Mathematical methods for time series analysis and digital image processing”, January 2007.Google Scholar
  52. [52]
    M.A. Kulesh, M.S. Diallo, and M. Holschneider. Wavelet analysis of ellipticity, dispersion, and dissipation properties of Rayleigh waves. Acoustical Physics, 51(4):425–434, 2005.CrossRefGoogle Scholar
  53. [53]
    D. Kuzmin and S. Turek. High-resolution FEM-TVD schemes based on a fully multidimensional flux limiter. Journal of Computational Physics, 198:131–158, 2004.MATHCrossRefMathSciNetGoogle Scholar
  54. [54]
    S. H. Lai and B.C. Vemuri. Robust and efficient algorithms for optical flow computation. In Proceedings of the International Symposium on Computer Vision, pages 455–460, November 1995.Google Scholar
  55. [55]
    R.S. Laramee, H. Hausser, H. Doleisch, B. Vrolijk, F.H. Post, and D. Weiskopf. The state of the art in flow visualization: Dense and texturebased techniques. Computer Graphics Forum, 23(2):203–221, 2004.CrossRefGoogle Scholar
  56. [56]
    A. Lee, D. Dobkin, W. Sweldens, and P. Schröder. Multiresolution mesh morphing. In Proceedings of SIGGRAPH 99, Computer Graphics Proceedings, Annual Conference Series, pages 343–350, August 1999.Google Scholar
  57. [57]
    N. Litke, M. Droske, M. Rumpf, and P. Schröder. An image processing approach to surface matching. In M. Desbrun and H. Pottmann, editors, Third Eurographics Symposium on Geometry Processing, Eurographics Association, pages 207–216, 2005.Google Scholar
  58. [58]
    S. Masnou and J. Morel. Level lines based disocclusion. In Proceedings of ICIP, volume 3, pages 259–263, 1998.Google Scholar
  59. [59]
    B. McCane, K. Novins, D. Crannitch, and B. Galvin. On benchmarking optical flow. http://of-eval.sourceforge.net/, 2001.Google Scholar
  60. [60]
    M.I. Miller, A. Trouvé, and L. Younes. On the metrics and euler-lagrange equations of computational anatomy. Annual Review of Biomedical Enginieering, 4:375–405, 2002.CrossRefGoogle Scholar
  61. [61]
    I.B. Morozov and S.B. Smithson. Instantaneous polarization attributes and directional filtering. Geophysics, 61(3):872–881, 1996.CrossRefGoogle Scholar
  62. [62]
    O.A. Oleinik and E.V. Radkevic. Second order equations with nonnegative characteristic form. American Mathematical Society, Providence, Rhode Island and Plenum Press, New York, 1973.Google Scholar
  63. [63]
    F. Pacor, D. Bindi, L. Luzi, S. Parolai, S. Marzorati, and G. Monachesi. Characteristics of strong ground motion data recorded in the Gubbio sedimentary basin (Central Italy). Bulletin of Earthquake Engineering, 5(l):2-43, 2007.Google Scholar
  64. [64]
    H.A. Pedersen, J.I. Mars, and P.-O. Amblard. Improving surface-wave group velocity measurements by energy reassignment. Geophysics, 68(2): 677–684, 2003.CrossRefGoogle Scholar
  65. [65]
    P. Perona and J. Malik. Scale space and edge detection using anisotropic diffusion. In IEEE Computer Society Workshop on Computer Vision, 1987.Google Scholar
  66. [66]
    P. Perona and J. Malik. Scale-space and edge detection using anisotropic diffusion. Technical Report UCB/CSD-88-483, EECS Department, University of California, Berkeley, December 1988.Google Scholar
  67. [67]
    C.R. Pinnegar. Polarization analysis and polarization filtering of threecomponent signals with the time-frequency S transform. Geophysical Journal International, 165(2):596–606, 2006.CrossRefGoogle Scholar
  68. [68]
    O. Pironneau. On the transport-diffusion algorithm and its applications to the Navier-Stokes equations. Numerische Mathematics, 38:309–332, 1982.MATHCrossRefMathSciNetGoogle Scholar
  69. [69]
    W.H. Press, S.A. Teukolsky, W.T. Vetterling, and B.P. Flannery. Numerical Recipe in C: The Art of Scientific Computing. Cambridge University Press, 1992.Google Scholar
  70. [70]
    T. Preusser and M. Rumpf. An adaptive finite element method for large scale image processing. Journal of Visual Communication and Image Representation, 11:183–195, 2000.CrossRefGoogle Scholar
  71. [71]
    M. Schimmel and J. Gallart. The inverse S-transform in filters with timefrequency localization. IEEE Transaction on Signal Processing, 53(11): 4417–4422, 2005.CrossRefMathSciNetGoogle Scholar
  72. [72]
    H.-W. Shen and D.L. Kao. Uflic: A line integral convolution algorithm for visualizing unsteady flows. In Proceedings Visualization’ 97, pages 317–322, 1997.Google Scholar
  73. [73]
    N. Soma, H. Niitsuma, and R. Baria. Reflection technique in timefrequency domain using multicomponent acoustic emission signals and application to geothermal reservoirs. Geophysics, 67(3):928–938, 2002.CrossRefGoogle Scholar
  74. [74]
    H. Spies and C. Garbe. Dense parameter fields from total least squares. In L. Van Gool, editor, Pattern Recognition, volume LNCS 2449 of Lecture Notes in Computer Science, pages 379–386, Zurich, CH, 2002. SpringerVerlag.Google Scholar
  75. [75]
    A. Telea, T. Preusser, C. Garbe, M. Droske, and M. Rumpf. A variational approach to joint denoising, edge detection and motion estimation. In Proceedings of DAGM 2006, pages 525–535, 2006.Google Scholar
  76. [76]
    M. Tobak and D.J. Peake. Topology of 3D separated flow. Annual Review of Fluid Mechanics, 14:61–85, 1982.CrossRefMathSciNetGoogle Scholar
  77. [77]
    S. Turek. Efficient Solvers for Incompressible Flow Problems: An Algorithmic and Computational Approach, volume 6 of LNCSE. Springer Verlag Berlin Heidelberg New York, 1999.Google Scholar
  78. [78]
    S. Turek, L. Rivkind, J. Hron, and R. Glowinski. Numerical analysis of a new time-stepping θ-scheme for incompressible flow simulations, Journal of Scientific Computing, 28(2–3):533–547, September 2006.MATHCrossRefMathSciNetGoogle Scholar
  79. [79]
    G. Turk and D. Banks. Image-guided streamline placement. In Proc. 23rd annual conference on Computer graphics, August 4–9, 1996, New Orleans, LA USA. ACM Press, 1996.Google Scholar
  80. [80]
    J.J. vanWijk. Spot noise-texture synthesis for data visualization. In T.W. Sederberg, editor, Computer Graphics (SIGGRAPH’ 91 Proceedings), volume 25, pages 309–318, Addison Wesley July 1991.Google Scholar
  81. [81]
    J.J. vanWijk. Flow visualization with surface particles. IEEE Computer Graphics and Applications, 13(4):18–24, July 1993.CrossRefGoogle Scholar
  82. [82]
    P. Viola and W.M. Wells. Alignment by maximization of mutual information. International Journal of Computer Vision, 24(2):137–154, 1997.CrossRefGoogle Scholar
  83. [83]
    J. Weickert. Anisotropic Diffusion in Image Processing. European Consortium for Mathematics in Industry. Teubner, Stuttgart, Leipzig, 1998.MATHGoogle Scholar
  84. [84]
    A.P. Witkin. Scale-space filtering. In Proceedings of the 8th IJCAI, pages 1019–1022, Karlsruhe, Germany, 1983.Google Scholar
  85. [85]
    Q. Xie, M. Holschneider, and M. Kulesh. Some remarks on linear diffeomorphisms in wavelet space. Technical Report 37, Preprint series of the DFG priority program 1114 “Mathematical methods for time series analysis and digital image processing”, July 2003.Google Scholar
  86. [86]
    C. Zetzsche and E. Barth. Fundamental limits of linear filters in the visual processing of two dimensional signals. Vision Research, 30(7):1111–1117, 1990.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Jens F. Acker
    • 5
  • Benjamin Berkels
    • 2
  • Kristian Bredies
    • 3
  • Mamadou S. Diallo
    • 6
  • Marc Droske
    • 2
  • Christoph S. Garbe
    • 4
  • Matthias Holschneider
    • 1
  • Jaroslav Hron
    • 5
  • Claudia Kondermann
    • 4
  • Michail Kulesh
    • 1
  • Peter Maass
    • 3
  • Nadine Olischläger
    • 2
  • Heinz-Otto Peitgen
    • 7
  • Tobias Preusser
    • 7
  • Martin Rumpf
    • 2
  • Karl Schaller
    • 9
  • Frank Scherbaum
    • 8
  • Stefan Turek
    • 5
  1. 1.Institute for MathematicsUniversity of PotsdamPotsdamGermany
  2. 2.Institute for Numerical SimulationUniversity of BonnBonnGermany
  3. 3.Center of Industrial Mathematics (ZeTeM)University of BremenBremenGermany
  4. 4.Interdisciplinary Center for Scientific ComputingUniversity of HeidelbergHeidelbergGermany
  5. 5.Institute for Applied MathematicsUniversity of DortmundDortmundGermany
  6. 6.Now at ExxonMobil Upstream Research companyHoustonUSA
  7. 7.Center for Complex Systems and VisualizationUniversity of BremenBremenGermany
  8. 8.Institute for GeosciencesUniversity of PotsdamPotsdamGermany
  9. 9.Hôpitaux Universitaires de GenèveGenèveSwitzerland

Personalised recommendations