# 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 & 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, (2011). Google Scholar [2] B. Bourdin, Image segmentation with a finite element method,, M2AN Mathematical Modelling and Numerical Analysis, 33 (1999), 229. doi: 10.1051/m2an:1999114. Google Scholar [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. doi: 10.1137/080725064. Google Scholar [4] M. Burger, J. Modersitzki and L. Ruthotto, A hyperelastic regularization energy for image registration,, SIAM J. Sci. Comput., 35 (2013). doi: 10.1137/110835955. Google Scholar [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. doi: 10.1137/090758003. Google Scholar [6] B. Dacorogna, Direct Methods in the Calculus of Variations,, Second edition, (2008). Google Scholar [7] M. Droske and M. Rumpf, A variational approach to nonrigid morphological image registration,, SIAM J. Appl. Math., 64 (2004), 668. doi: 10.1137/S0036139902419528. Google Scholar [8] M. Droske and M. Rumpf, Multiscale joint segmentation and registration of image morphology,, IEEE Trans. Pattern Recognition Machine Intelligence, 29 (2007), 2181. Google Scholar [9] L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions,, Studies in Advanced Mathematics, (1992). Google Scholar [10] B. Fischer and J. Modersitzki, Curvature based image registration,, J. Math. Imaging Vision, 18 (2003), 81. doi: 10.1023/A:1021897212261. Google Scholar [11] R. Fourer, D. M. Gay and B. W. Kernighan, AMPL. A Modeling Language for Mathematical Programming,, Second edition, (2003). Google Scholar [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. doi: 10.1002/oca.996. Google Scholar [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. Google Scholar [14] H. Goering, H.-G. Roos and L. Tobiska, Finite-Element-Methode,, Third edition, (1993). Google Scholar [15] E. Haber and J. Modersitzki, Intensity gradient based registration and fusion of multi-modal images,, Methods of Information in Medicine, 46 (2007), 292. Google Scholar [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, (2006), 246. Google Scholar [17] S. Henn and K. Witsch, Iterative multigrid regularization techniques for image matching,, SIAM J. Sci. Comput., 23 (2001), 1077. Google Scholar [18] G. Hermosillo, C. Chefd'hotel and O. Faugeras, Variational methods for multimodal image matching,, Int. J. Computer Vision, 50 (2002), 329. Google Scholar [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, (2009), 213. Google Scholar [20] B. Jansen, Interior Point Techniques in Optimization,, Kluwer, (1997). Google Scholar [21] C. Laird and A. Wächter, Introduction to IPOPT: A Tutorial for Downloading, Installing, and Using IPOPT,, Revision No. 1830. Available from: , (1830). Google Scholar [22] J. Min, M. Powell and K. W. Bowyer, Automated performance evaluation of range image segmentation algorithms,, IEEE Trans. Systems, 34 (2004), 263. Google Scholar [23] J. Modersitzki, Numerical Methods for Image Registration,, Oxford University Press, (2004). Google Scholar [24] J. Modersitzki, FAIR. Flexible Algorithms for Image Registration,, SIAM, (2009). Google Scholar [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), (2003), 65. Google Scholar [26] K. N. Plataniotis and A. N. Venetsanopoulos, Color Image Processing and Applications,, Springer, (2000). Google Scholar [27] H. Richter, Wahrscheinlichkeitstheorie,, Second edition, (1966). Google Scholar [28] O. Scherzer, M. Grasmair, H. Grossauer, M. Haltmeier and F. Lenzen, Variational Methods in Imaging,, Springer, (2009). Google Scholar [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. Google Scholar [30] M. Wagner, Elastic image registration in presence of polyconvex constraints,, Karl-Franzens-Universität Graz, (2010), 2010. Google Scholar [31] M. Wagner, A direct method for the solution of an optimal control problem arising from image registration,, Numerical Algebra, 2 (2012), 487. Google Scholar [32] B. Zitová and J. Flusser, Image registration methods: A survey,, Image and Vision Computing, 21 (2003), 977. Google Scholar

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##### References:
 [1] A. Angelov, Multimodale Bildregistrierung durch elastisches Matching von Kantenskizzen,, Diploma thesis, (2011). Google Scholar [2] B. Bourdin, Image segmentation with a finite element method,, M2AN Mathematical Modelling and Numerical Analysis, 33 (1999), 229. doi: 10.1051/m2an:1999114. Google Scholar [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. doi: 10.1137/080725064. Google Scholar [4] M. Burger, J. Modersitzki and L. Ruthotto, A hyperelastic regularization energy for image registration,, SIAM J. Sci. Comput., 35 (2013). doi: 10.1137/110835955. Google Scholar [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. doi: 10.1137/090758003. Google Scholar [6] B. Dacorogna, Direct Methods in the Calculus of Variations,, Second edition, (2008). Google Scholar [7] M. Droske and M. Rumpf, A variational approach to nonrigid morphological image registration,, SIAM J. Appl. Math., 64 (2004), 668. doi: 10.1137/S0036139902419528. Google Scholar [8] M. Droske and M. Rumpf, Multiscale joint segmentation and registration of image morphology,, IEEE Trans. Pattern Recognition Machine Intelligence, 29 (2007), 2181. Google Scholar [9] L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions,, Studies in Advanced Mathematics, (1992). Google Scholar [10] B. Fischer and J. Modersitzki, Curvature based image registration,, J. Math. Imaging Vision, 18 (2003), 81. doi: 10.1023/A:1021897212261. Google Scholar [11] R. Fourer, D. M. Gay and B. W. Kernighan, AMPL. A Modeling Language for Mathematical Programming,, Second edition, (2003). Google Scholar [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. doi: 10.1002/oca.996. Google Scholar [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. Google Scholar [14] H. Goering, H.-G. Roos and L. Tobiska, Finite-Element-Methode,, Third edition, (1993). Google Scholar [15] E. Haber and J. Modersitzki, Intensity gradient based registration and fusion of multi-modal images,, Methods of Information in Medicine, 46 (2007), 292. Google Scholar [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, (2006), 246. Google Scholar [17] S. Henn and K. Witsch, Iterative multigrid regularization techniques for image matching,, SIAM J. Sci. Comput., 23 (2001), 1077. Google Scholar [18] G. Hermosillo, C. Chefd'hotel and O. Faugeras, Variational methods for multimodal image matching,, Int. J. Computer Vision, 50 (2002), 329. Google Scholar [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, (2009), 213. Google Scholar [20] B. Jansen, Interior Point Techniques in Optimization,, Kluwer, (1997). Google Scholar [21] C. Laird and A. Wächter, Introduction to IPOPT: A Tutorial for Downloading, Installing, and Using IPOPT,, Revision No. 1830. Available from: , (1830). Google Scholar [22] J. Min, M. Powell and K. W. Bowyer, Automated performance evaluation of range image segmentation algorithms,, IEEE Trans. Systems, 34 (2004), 263. Google Scholar [23] J. Modersitzki, Numerical Methods for Image Registration,, Oxford University Press, (2004). Google Scholar [24] J. Modersitzki, FAIR. Flexible Algorithms for Image Registration,, SIAM, (2009). Google Scholar [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), (2003), 65. Google Scholar [26] K. N. Plataniotis and A. N. Venetsanopoulos, Color Image Processing and Applications,, Springer, (2000). Google Scholar [27] H. Richter, Wahrscheinlichkeitstheorie,, Second edition, (1966). Google Scholar [28] O. Scherzer, M. Grasmair, H. Grossauer, M. Haltmeier and F. Lenzen, Variational Methods in Imaging,, Springer, (2009). Google Scholar [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. Google Scholar [30] M. Wagner, Elastic image registration in presence of polyconvex constraints,, Karl-Franzens-Universität Graz, (2010), 2010. Google Scholar [31] M. Wagner, A direct method for the solution of an optimal control problem arising from image registration,, Numerical Algebra, 2 (2012), 487. Google Scholar [32] B. Zitová and J. Flusser, Image registration methods: A survey,, Image and Vision Computing, 21 (2003), 977. Google Scholar
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