# American Institute of Mathematical Sciences

April  2014, 10(2): 567-590. doi: 10.3934/jimo.2014.10.567

## Multimodal image registration by elastic matching of edge sketches via optimal control

 1 Otto-Hahn-Str. 15, D-30880 Laatzen, Germany 2 University of Leipzig, Department of Mathematics, P. O. B. 10 09 20, D-04009 Leipzig

Received  December 2012 Revised  August 2013 Published  October 2013

For the problem of multimodal image registration, an optimal control approach is presented. The geometrical information of the images will be transformed into weighted edge sketches, for which a linear-elastic or hyperelastic registration will be performed. For the numerical solution of this problem, we provide a direct method based on discretization methods and large-scale optimization techniques. A comparison of a separated and a joint access for the generation of the edge sketches and the determination of the matching deformation is made. The quality of the results obtained with the optimal control method competes well with those generated by a standard variational method.
Citation: Angel Angelov, Marcus Wagner. Multimodal image registration by elastic matching of edge sketches via optimal control. Journal of Industrial and Management Optimization, 2014, 10 (2) : 567-590. doi: 10.3934/jimo.2014.10.567
##### References:
 [1] A. Angelov, Multimodale Bildregistrierung durch elastisches Matching von Kantenskizzen, Diploma thesis, University of Münster, 2011. [2] B. Bourdin, Image segmentation with a finite element method, M2AN Mathematical Modelling and Numerical Analysis, 33 (1999), 229-244. doi: 10.1051/m2an:1999114. [3] C. Brune, H. Maurer and M. Wagner, Detection of intensity and motion edges within optical flow via multidimensional control, SIAM J. Imaging Sci., 2 (2009), 1190-1210. doi: 10.1137/080725064. [4] M. Burger, J. Modersitzki and L. Ruthotto, A hyperelastic regularization energy for image registration, SIAM J. Sci. Comput., 35 (2013), B132-B148. doi: 10.1137/110835955. [5] C. Clason, B. Jin and K. Kunisch, A semismooth Newton method for $L^1$ data fitting with automatic choice of regularization parameters and noise calibration, SIAM J. Imaging Sci., 3 (2010), 199-231. doi: 10.1137/090758003. [6] B. Dacorogna, Direct Methods in the Calculus of Variations, Second edition, Applied Mathematical Sciences, 78, Springer, New York, 2008. [7] M. Droske and M. Rumpf, A variational approach to nonrigid morphological image registration, SIAM J. Appl. Math., 64 (2004), 668-687. doi: 10.1137/S0036139902419528. [8] M. Droske and M. Rumpf, Multiscale joint segmentation and registration of image morphology, IEEE Trans. Pattern Recognition Machine Intelligence, 29 (2007), 2181-2194. [9] L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992. [10] B. Fischer and J. Modersitzki, Curvature based image registration, J. Math. Imaging Vision, 18 (2003), 81-85. doi: 10.1023/A:1021897212261. [11] R. Fourer, D. M. Gay and B. W. Kernighan, AMPL. A Modeling Language for Mathematical Programming, Second edition, Brooks/Cole-Thomson Learning, Pacific Grove, 2003. [12] L. Franek, M. Franek, H. Maurer and M. Wagner, A discretization method for the numerical solution of Dieudonné-Rashevsky type problems with application to edge detection within noisy image data, Opt. Control Appl. Meth., 33 (2012), 276-301. doi: 10.1002/oca.996. [13] L. A. Gallardo and M. A. Meju, Characterization of heterogeneous near-surface materials by joint 2D inversion of dc resistivity and seismic data, Geophysical Research Letters, 30 (2003). doi: 10.1029/2003GL017370. [14] H. Goering, H.-G. Roos and L. Tobiska, Finite-Element-Methode, Third edition, Akademie Verag, Berlin, 1993. [15] E. Haber and J. Modersitzki, Intensity gradient based registration and fusion of multi-modal images, Methods of Information in Medicine, 46 (2007), 292-299. [16] J. Han, B. Berkels, M. Rumpf, J. Hornegger, M. Droske, M. Fried, J. Scorzin and C. Schaller, A variational framework for joint image registration, denoising and edge detection, in Bildverarbeitung für die Medizin 2006 (eds. H. Handels, J. Ehrhardt, A. Horsch, H.-P. Meinzer and T. Tolxdorff) Springer, Berlin, 2006, 246-250. [17] S. Henn and K. Witsch, Iterative multigrid regularization techniques for image matching, SIAM J. Sci. Comput., 23 (2001), 1077-1093. [18] G. Hermosillo, C. Chefd'hotel and O. Faugeras, Variational methods for multimodal image matching, Int. J. Computer Vision, 50 (2002), 329-343. [19] M. Hintermüller and S. L. Keeling, Image registration and segmentation based on energy minimization, in Handbook of Optimization in Medicine (eds. P. M. Pardalos and H. E. Romeijn) Springer, New York, 2009, 213-252. [20] B. Jansen, Interior Point Techniques in Optimization, Kluwer, Dordrecht, 1997. [21] C. Laird and A. Wächter, Introduction to IPOPT: A Tutorial for Downloading, Installing, and Using IPOPT, Revision No. 1830. Available from: https://www.coin-or.org/Ipopt/~ex-port/2158/stable/3.9/Ipopt/doc/documentation.pdf (accessed 15.12.2012). [22] J. Min, M. Powell and K. W. Bowyer, Automated performance evaluation of range image segmentation algorithms, IEEE Trans. Systems, Man, and Cybernetics, Part B, 34 (2004), 263-271. [23] J. Modersitzki, Numerical Methods for Image Registration, Oxford University Press, Oxford, 2004. [24] J. Modersitzki, FAIR. Flexible Algorithms for Image Registration, SIAM, Philadelphia, 2009. [25] R. W. Ogden, Nonlinear elasticity, anisotropy, material stability and residual stresses in soft tissue, in Biomechanics of Soft Tissue in Cardiovascular Systems (eds. G. A. Holzapfel and R. W. Ogden), Springer, Wien, 2003, 65-108. [26] K. N. Plataniotis and A. N. Venetsanopoulos, Color Image Processing and Applications, Springer, Berlin, 2000. [27] H. Richter, Wahrscheinlichkeitstheorie, Second edition, Die Grundlehren der Mathematischen Wissenschaften, Band 86, Springer, Berlin-Heidelberg-New York, 1966. [28] O. Scherzer, M. Grasmair, H. Grossauer, M. Haltmeier and F. Lenzen, Variational Methods in Imaging, Springer, New York, 2009. [29] A. Wächter and L. T. Biegler, On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming, Math. Program. Ser. A, 106 (2006), 25-57. [30] M. Wagner, Elastic image registration in presence of polyconvex constraints, Karl-Franzens-Universität Graz, SFB-Report No. 2010-033. To appear in Proceedings of the International Workshop on Optimal Control in Image Processing, Heidelberg, Germany, May 31-June 1, 2010. [31] M. Wagner, A direct method for the solution of an optimal control problem arising from image registration, Numerical Algebra, Control and Optimization, 2 (2012), 487-510. [32] B. Zitová and J. Flusser, Image registration methods: A survey, Image and Vision Computing, 21 (2003), 977-1000.

show all references

##### References:
 [1] A. Angelov, Multimodale Bildregistrierung durch elastisches Matching von Kantenskizzen, Diploma thesis, University of Münster, 2011. [2] B. Bourdin, Image segmentation with a finite element method, M2AN Mathematical Modelling and Numerical Analysis, 33 (1999), 229-244. doi: 10.1051/m2an:1999114. [3] C. Brune, H. Maurer and M. Wagner, Detection of intensity and motion edges within optical flow via multidimensional control, SIAM J. Imaging Sci., 2 (2009), 1190-1210. doi: 10.1137/080725064. [4] M. Burger, J. Modersitzki and L. Ruthotto, A hyperelastic regularization energy for image registration, SIAM J. Sci. Comput., 35 (2013), B132-B148. doi: 10.1137/110835955. [5] C. Clason, B. Jin and K. Kunisch, A semismooth Newton method for $L^1$ data fitting with automatic choice of regularization parameters and noise calibration, SIAM J. Imaging Sci., 3 (2010), 199-231. doi: 10.1137/090758003. [6] B. Dacorogna, Direct Methods in the Calculus of Variations, Second edition, Applied Mathematical Sciences, 78, Springer, New York, 2008. [7] M. Droske and M. Rumpf, A variational approach to nonrigid morphological image registration, SIAM J. Appl. Math., 64 (2004), 668-687. doi: 10.1137/S0036139902419528. [8] M. Droske and M. Rumpf, Multiscale joint segmentation and registration of image morphology, IEEE Trans. Pattern Recognition Machine Intelligence, 29 (2007), 2181-2194. [9] L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992. [10] B. Fischer and J. Modersitzki, Curvature based image registration, J. Math. Imaging Vision, 18 (2003), 81-85. doi: 10.1023/A:1021897212261. [11] R. Fourer, D. M. Gay and B. W. Kernighan, AMPL. A Modeling Language for Mathematical Programming, Second edition, Brooks/Cole-Thomson Learning, Pacific Grove, 2003. [12] L. Franek, M. Franek, H. Maurer and M. Wagner, A discretization method for the numerical solution of Dieudonné-Rashevsky type problems with application to edge detection within noisy image data, Opt. Control Appl. Meth., 33 (2012), 276-301. doi: 10.1002/oca.996. [13] L. A. Gallardo and M. A. Meju, Characterization of heterogeneous near-surface materials by joint 2D inversion of dc resistivity and seismic data, Geophysical Research Letters, 30 (2003). doi: 10.1029/2003GL017370. [14] H. Goering, H.-G. Roos and L. Tobiska, Finite-Element-Methode, Third edition, Akademie Verag, Berlin, 1993. [15] E. Haber and J. Modersitzki, Intensity gradient based registration and fusion of multi-modal images, Methods of Information in Medicine, 46 (2007), 292-299. [16] J. Han, B. Berkels, M. Rumpf, J. Hornegger, M. Droske, M. Fried, J. Scorzin and C. Schaller, A variational framework for joint image registration, denoising and edge detection, in Bildverarbeitung für die Medizin 2006 (eds. H. Handels, J. Ehrhardt, A. Horsch, H.-P. Meinzer and T. Tolxdorff) Springer, Berlin, 2006, 246-250. [17] S. Henn and K. Witsch, Iterative multigrid regularization techniques for image matching, SIAM J. Sci. Comput., 23 (2001), 1077-1093. [18] G. Hermosillo, C. Chefd'hotel and O. Faugeras, Variational methods for multimodal image matching, Int. J. Computer Vision, 50 (2002), 329-343. [19] M. Hintermüller and S. L. Keeling, Image registration and segmentation based on energy minimization, in Handbook of Optimization in Medicine (eds. P. M. Pardalos and H. E. Romeijn) Springer, New York, 2009, 213-252. [20] B. Jansen, Interior Point Techniques in Optimization, Kluwer, Dordrecht, 1997. [21] C. Laird and A. Wächter, Introduction to IPOPT: A Tutorial for Downloading, Installing, and Using IPOPT, Revision No. 1830. Available from: https://www.coin-or.org/Ipopt/~ex-port/2158/stable/3.9/Ipopt/doc/documentation.pdf (accessed 15.12.2012). [22] J. Min, M. Powell and K. W. Bowyer, Automated performance evaluation of range image segmentation algorithms, IEEE Trans. Systems, Man, and Cybernetics, Part B, 34 (2004), 263-271. [23] J. Modersitzki, Numerical Methods for Image Registration, Oxford University Press, Oxford, 2004. [24] J. Modersitzki, FAIR. Flexible Algorithms for Image Registration, SIAM, Philadelphia, 2009. [25] R. W. Ogden, Nonlinear elasticity, anisotropy, material stability and residual stresses in soft tissue, in Biomechanics of Soft Tissue in Cardiovascular Systems (eds. G. A. Holzapfel and R. W. Ogden), Springer, Wien, 2003, 65-108. [26] K. N. Plataniotis and A. N. Venetsanopoulos, Color Image Processing and Applications, Springer, Berlin, 2000. [27] H. Richter, Wahrscheinlichkeitstheorie, Second edition, Die Grundlehren der Mathematischen Wissenschaften, Band 86, Springer, Berlin-Heidelberg-New York, 1966. [28] O. Scherzer, M. Grasmair, H. Grossauer, M. Haltmeier and F. Lenzen, Variational Methods in Imaging, Springer, New York, 2009. [29] A. Wächter and L. T. Biegler, On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming, Math. Program. Ser. A, 106 (2006), 25-57. [30] M. Wagner, Elastic image registration in presence of polyconvex constraints, Karl-Franzens-Universität Graz, SFB-Report No. 2010-033. To appear in Proceedings of the International Workshop on Optimal Control in Image Processing, Heidelberg, Germany, May 31-June 1, 2010. [31] M. Wagner, A direct method for the solution of an optimal control problem arising from image registration, Numerical Algebra, Control and Optimization, 2 (2012), 487-510. [32] B. Zitová and J. Flusser, Image registration methods: A survey, Image and Vision Computing, 21 (2003), 977-1000.
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