# American Institute of Mathematical Sciences

2012, 2(3): 487-510. doi: 10.3934/naco.2012.2.487

## A direct method for the solution of an optimal control problem arising from image registration

 1 University of Leipzig, Department of Mathematics, P. O. B. 10 09 20, D-04009 Leipzig, Germany

Received  October 2011 Revised  February 2012 Published  August 2012

In the present paper, the the elastic/hyperelastic image registration problem is treated as a multidimensional control problem of Dieudonné-Rashevsky type. For its numerical solution, we describe a direct method based on discretization methods and large-scale optimization techniques. Selected numerical results will be presented and discussed. The quality of the results obtained with the optimal control method competes well with those generated from a standard variational method.
Citation: Marcus Wagner. A direct method for the solution of an optimal control problem arising from image registration. Numerical Algebra, Control and Optimization, 2012, 2 (3) : 487-510. doi: 10.3934/naco.2012.2.487
##### References:
 [1] A. Angelov, "Multimodale Bildregistrierung durch elastisches Matching von Kantenskizzen," Diploma thesis, University of Münster, 2011. [2] L. Alvarez, J. Weickert and J. Sánchez, Reliable estimation of dense optical flow fields with large displacements, Int. J. Computer Vision, 39 (2000), 41-56. doi: 10.1023/A:1008170101536. [3] J. M. Ball, Convexity conditions and existence theorems in nonlinear elasticity, Arch. Rat. Mech. Anal., 63 (1977), 337-403. [4] D. Balzani, P. Neff, J. Schröder and G. A. Holzapfel, A polyconvex framework for soft biological tissues. Adjustment to experimental data, Int. J. of Solids and Structures, 43 (2006), 6052-6070. doi: 10.1016/j.ijsolstr.2005.07.048. [5] S. Barbieri, M. Welk and J. Weickert, A variational approach to the registration of tensor-valued images, in "Tensors in Image Processing and Computer Vision" (eds. S. Aja-Fernández, R. de Luis-García, D. Tao and X. Li), Springer, London etc., 2009, 59-77. [6] D. Breitenreicher and C. Schnörr, Robust 3D object registration without explicit correspondence using geometric integration,, Machine Vis. and Appl., (): 00138. [7] C. Brune, "Berechnung des Optischen Flusses und Kantenerkennung mit Optimierungsmethoden," Diploma thesis, University of Münster, 2007. [8] 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. [9] F. Chmelka and E. Melan, "Einführung in die Festigkeitslehre," Springer, New York, 1976, 5th ed. [10] G. E. Christensen, R. D. Rabbitt and M. I. Miller, Deformable templates using large deformation kinematics, IEEE Trans. Image Processing, 5 (1996), 1435-1447. doi: 10.1109/83.536892. [11] B. Dacorogna, "Direct Methods in the Calculus of Variations," Springer, New York, 2008, 2nd ed. [12] M. Dawood, F. Büther, N. Lang, O. Schober and K. P. Schäfers, Respiratory gating in positron emission tomography: a quantitative comparision of different gating schemes, Med. Phys., 34 (2007), 3067-3076. doi: 10.1118/1.2748104. [13] 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. [14] M. Droske and M. Rumpf, Multiscale joint segmentation and registration of image morphology, IEEE Trans. Pattern Recognition Machine Intelligence, 29 (2007), 2181-2194. doi: 10.1109/TPAMI.2007.1120. [15] ,L. C. Evans and R. F. Gariepy, "Measure Theory and Fine Properties of Functions," CRC Press, Boca Raton etc., 1992. [16] O. Faugeras and G. Hermosillo, Well-posedness of two nonrigid multimodal image registration methods, SIAM J. Appl. Math., 64 (2004), 1550-1587. doi: 10.1137/S0036139903424904. [17] B. Fischer and J. Modersitzki, Curvature based image registration, J. Math. Imaging Vision, 18 (2003), 81-85. [18] R. Fourer, D. M. Gay and B. W. Kernighan, "AMPL. A Modeling Language for Mathematical Programming," Brooks/Cole - Thomson Learning, Pacific Grove, 2003, 2nd ed. [19] L. Franek, "Anwendung optimaler Steuerungsprobleme mit $L^\infty$-Steuerbeschrünkung auf ein Modell-problem der Bildverarbeitung," Diploma thesis, University of Münster, 2007. [20] M. Franek, "Bildentrauschung und Kantenerkennung mit $L^p$-Regularisierung und Gradienten-beschränkung bei Graustufenbildern," Diploma thesis, University of Münster, 2007. [21] 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. [22] 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) 13, 1658, 1 - 1 - 1 - 4. [23] T. C. Gasser and G. H. Holzapfel, A rate-independent elastoplastic constitutive model for biological fiber-reinforced composites at finite strains: continuum basis, algorithmic formulation and finite element implementation, Computational Mechanics, 29 (2002), 340-360. doi: 10.1007/s00466-002-0347-6. [24] H. Goering, H.-G. Roos and L. Tobiska, "Finite-Element-Method," Akademie-Verlag, Berlin, 1993, 3rd ed. [25] E. Haber and J. Modersitzki, Numerical methods for volume preserving image registration, Inverse Problems, 20 (2004), 1621-1638. doi: 10.1088/0266-5611/20/5/018. [26] E. Haber and J. Modersitzki, Intensity gradient based registration and fusion of multi-modal images, Methods of Information in Medicine, 46 (2007), 292-299. [27] S. Haker, L. Zhu, A. Tannenbaum and S. Angenent, Optimal mass transport for registration and warping, Int. J. Computer Vision, 60 (2004), 225-240. doi: 10.1023/B:VISI.0000036836.66311.97. [28] S. Henn and K. Witsch, A multigrid approach for minimizing a nonlinear functional for digital image matching, Computing, 64 (2000), 339-348. doi: 10.1007/s006070070029. [29] S. Henn and K. Witsch, Iterative multigrid regularization techniques for image matching, SIAM J. Sci. Comput., 23 (2001), 1077-1093. doi: 10.1137/S106482750037161X. [30] G. Hermosillo, C. Chefd'hotel and O. Faugeras, Variational methods for multimodal image matching, Int. J. Computer Vision, 50 (2002), 329-343. doi: 10.1023/A:1020830525823. [31] 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. [32] B. Jansen, "Interior Point Techniques in Optimization," Kluwer, Dordrecht, 1997. [33] T. Kaijser, Computing the Kantorovich distance for images, J. Math. Imaging Vision, 9 (1998), 173-191. doi: 10.1023/A:1008389726910. [34] S. L. Keeling and W. Ring, Medical image registration and interpolation by optical flow with maximal rigidity, J. Math. Imaging Vision, 23 (2005), 47-65. doi: 10.1007/s10851-005-4967-2. [35] C. Laird and A. Wächter, Introduction to IPOPT: A tutorial for downloading, installing, and using IPOPT, Revision No. 1863, electronically published: http://www.coin-or.org/Ipopt/documentation, (accessed at 12.10.2011). [36] C. Le Guyader and L. Vese, A combined segmentation and registration framework with a nonlinear elasticity smoother, in "Scale Space and Variational Methods in Computer Vision, Second International Conference, SSVM 2009, Voss, Norway, June 1-5, 2009. Proceedings" (eds. X.-C. Tai, K. Mørken, M. Lysaker and K.-A. Lie), Springer, Berlin - Heidelberg, 2009 (LNCS 5567), 600-611. [37] W.-H. Liao, C. L. Yu, M. Bergsneider, L. Vese and S.-C. Huang, A new framework of quantifying differences between images by matching gradient fields and its application to image blending, in "2002 Nuclear Science Symposium Conference Record, Vol. 2" (ed. S. Metzler), IEEE, Piscataway, 2003, 1092-1096. [38] G. Maess, "Vorlesungen über numerische Mathematik II," Akademie-Verlag, Berlin, 1988. [39] J. Modersitzki, "Numerical Methods for Image Registration," Oxford University Press, Oxford, 2004. [40] J. Modersitzki, "FAIR. Flexible Algorithms for Image Registration," SIAM, Philadelphia, 2009. [41] O. Museyko, M. Stiglmayr, K. Klamroth and G. Leugering, On the application of the Monge-Kantorovich problem to image registration, SIAM J. Imaging Sci., 2 (2009), 1068-1097. doi: 10.1137/080721522. [42] 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 etc., 2003, 65-108. [43] K. N. Plataniotis and A. N. Venetsanopoulos, "Color Image Processing and Applications," Springer, Berlin etc., 2000. [44] C. Pöschl, J. Modersitzki and O. Scherzer, A variational setting for volume constrained image registration, Inverse Probl. Imaging, 4 (2010), 505-522. doi: 10.3934/ipi.2010.4.505. [45] O. Scherzer, M. Grasmair, H. Grossauer, M. Haltmeier and F. Lenzen, "Variational Methods in Imaging," Springer, New York etc., 2009. [46] B. C. Vemuri, J. Ye, Y. Chen and C. M. Leonard, A level-set based approach to image registration, in "IEEE Workshop on Mathematical Methods in Biomedical Image Analysis (MMBIA'00)," IEEE Computer Society, Washington, 2000, 86-93. [47] 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. doi: 10.1007/s10107-004-0559-y. [48] M. Wagner, Pontryagin's maximum principle for multidimensional control problems in image processing, J. Optim. Theory Appl., 140 (2009), 543-576. doi: 10.1007/s10957-008-9460-9. [49] M. Wagner, Elastic image registration in presence of polyconvex constraints, submitted: Proceedings of the International Workshop on Optimal Control in Image Processing, Heidelberg, Germany, May 31 - June 1, 2010. [50] M. Wagner, Quasiconvex relaxation of multidimensional control problems with integrands $f(t,\xi,v)$, ESAIM: Control, Optimisation and Calculus of Variations 17 (2011), 190-221. doi: 10.1051/cocv/2010008. [51] A. Yezzi, L. Zollei and T. Kapur, A variational framework for joint segmentation and registration, in "Proceedings of the IEEE Workshop on Mathematical Methods in Biomedical Image Analysis (MMBIA'01)," IEEE Computer Society, Washington, 2001, 44-51.

show all references

##### References:
 [1] A. Angelov, "Multimodale Bildregistrierung durch elastisches Matching von Kantenskizzen," Diploma thesis, University of Münster, 2011. [2] L. Alvarez, J. Weickert and J. Sánchez, Reliable estimation of dense optical flow fields with large displacements, Int. J. Computer Vision, 39 (2000), 41-56. doi: 10.1023/A:1008170101536. [3] J. M. Ball, Convexity conditions and existence theorems in nonlinear elasticity, Arch. Rat. Mech. Anal., 63 (1977), 337-403. [4] D. Balzani, P. Neff, J. Schröder and G. A. Holzapfel, A polyconvex framework for soft biological tissues. Adjustment to experimental data, Int. J. of Solids and Structures, 43 (2006), 6052-6070. doi: 10.1016/j.ijsolstr.2005.07.048. [5] S. Barbieri, M. Welk and J. Weickert, A variational approach to the registration of tensor-valued images, in "Tensors in Image Processing and Computer Vision" (eds. S. Aja-Fernández, R. de Luis-García, D. Tao and X. Li), Springer, London etc., 2009, 59-77. [6] D. Breitenreicher and C. Schnörr, Robust 3D object registration without explicit correspondence using geometric integration,, Machine Vis. and Appl., (): 00138. [7] C. Brune, "Berechnung des Optischen Flusses und Kantenerkennung mit Optimierungsmethoden," Diploma thesis, University of Münster, 2007. [8] 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. [9] F. Chmelka and E. Melan, "Einführung in die Festigkeitslehre," Springer, New York, 1976, 5th ed. [10] G. E. Christensen, R. D. Rabbitt and M. I. Miller, Deformable templates using large deformation kinematics, IEEE Trans. Image Processing, 5 (1996), 1435-1447. doi: 10.1109/83.536892. [11] B. Dacorogna, "Direct Methods in the Calculus of Variations," Springer, New York, 2008, 2nd ed. [12] M. Dawood, F. Büther, N. Lang, O. Schober and K. P. Schäfers, Respiratory gating in positron emission tomography: a quantitative comparision of different gating schemes, Med. Phys., 34 (2007), 3067-3076. doi: 10.1118/1.2748104. [13] 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. [14] M. Droske and M. Rumpf, Multiscale joint segmentation and registration of image morphology, IEEE Trans. Pattern Recognition Machine Intelligence, 29 (2007), 2181-2194. doi: 10.1109/TPAMI.2007.1120. [15] ,L. C. Evans and R. F. Gariepy, "Measure Theory and Fine Properties of Functions," CRC Press, Boca Raton etc., 1992. [16] O. Faugeras and G. Hermosillo, Well-posedness of two nonrigid multimodal image registration methods, SIAM J. Appl. Math., 64 (2004), 1550-1587. doi: 10.1137/S0036139903424904. [17] B. Fischer and J. Modersitzki, Curvature based image registration, J. Math. Imaging Vision, 18 (2003), 81-85. [18] R. Fourer, D. M. Gay and B. W. Kernighan, "AMPL. A Modeling Language for Mathematical Programming," Brooks/Cole - Thomson Learning, Pacific Grove, 2003, 2nd ed. [19] L. Franek, "Anwendung optimaler Steuerungsprobleme mit $L^\infty$-Steuerbeschrünkung auf ein Modell-problem der Bildverarbeitung," Diploma thesis, University of Münster, 2007. [20] M. Franek, "Bildentrauschung und Kantenerkennung mit $L^p$-Regularisierung und Gradienten-beschränkung bei Graustufenbildern," Diploma thesis, University of Münster, 2007. [21] 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. [22] 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) 13, 1658, 1 - 1 - 1 - 4. [23] T. C. Gasser and G. H. Holzapfel, A rate-independent elastoplastic constitutive model for biological fiber-reinforced composites at finite strains: continuum basis, algorithmic formulation and finite element implementation, Computational Mechanics, 29 (2002), 340-360. doi: 10.1007/s00466-002-0347-6. [24] H. Goering, H.-G. Roos and L. Tobiska, "Finite-Element-Method," Akademie-Verlag, Berlin, 1993, 3rd ed. [25] E. Haber and J. Modersitzki, Numerical methods for volume preserving image registration, Inverse Problems, 20 (2004), 1621-1638. doi: 10.1088/0266-5611/20/5/018. [26] E. Haber and J. Modersitzki, Intensity gradient based registration and fusion of multi-modal images, Methods of Information in Medicine, 46 (2007), 292-299. [27] S. Haker, L. Zhu, A. Tannenbaum and S. Angenent, Optimal mass transport for registration and warping, Int. J. Computer Vision, 60 (2004), 225-240. doi: 10.1023/B:VISI.0000036836.66311.97. [28] S. Henn and K. Witsch, A multigrid approach for minimizing a nonlinear functional for digital image matching, Computing, 64 (2000), 339-348. doi: 10.1007/s006070070029. [29] S. Henn and K. Witsch, Iterative multigrid regularization techniques for image matching, SIAM J. Sci. Comput., 23 (2001), 1077-1093. doi: 10.1137/S106482750037161X. [30] G. Hermosillo, C. Chefd'hotel and O. Faugeras, Variational methods for multimodal image matching, Int. J. Computer Vision, 50 (2002), 329-343. doi: 10.1023/A:1020830525823. [31] 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. [32] B. Jansen, "Interior Point Techniques in Optimization," Kluwer, Dordrecht, 1997. [33] T. Kaijser, Computing the Kantorovich distance for images, J. Math. Imaging Vision, 9 (1998), 173-191. doi: 10.1023/A:1008389726910. [34] S. L. Keeling and W. Ring, Medical image registration and interpolation by optical flow with maximal rigidity, J. Math. Imaging Vision, 23 (2005), 47-65. doi: 10.1007/s10851-005-4967-2. [35] C. Laird and A. Wächter, Introduction to IPOPT: A tutorial for downloading, installing, and using IPOPT, Revision No. 1863, electronically published: http://www.coin-or.org/Ipopt/documentation, (accessed at 12.10.2011). [36] C. Le Guyader and L. Vese, A combined segmentation and registration framework with a nonlinear elasticity smoother, in "Scale Space and Variational Methods in Computer Vision, Second International Conference, SSVM 2009, Voss, Norway, June 1-5, 2009. Proceedings" (eds. X.-C. Tai, K. Mørken, M. Lysaker and K.-A. Lie), Springer, Berlin - Heidelberg, 2009 (LNCS 5567), 600-611. [37] W.-H. Liao, C. L. Yu, M. Bergsneider, L. Vese and S.-C. Huang, A new framework of quantifying differences between images by matching gradient fields and its application to image blending, in "2002 Nuclear Science Symposium Conference Record, Vol. 2" (ed. S. Metzler), IEEE, Piscataway, 2003, 1092-1096. [38] G. Maess, "Vorlesungen über numerische Mathematik II," Akademie-Verlag, Berlin, 1988. [39] J. Modersitzki, "Numerical Methods for Image Registration," Oxford University Press, Oxford, 2004. [40] J. Modersitzki, "FAIR. Flexible Algorithms for Image Registration," SIAM, Philadelphia, 2009. [41] O. Museyko, M. Stiglmayr, K. Klamroth and G. Leugering, On the application of the Monge-Kantorovich problem to image registration, SIAM J. Imaging Sci., 2 (2009), 1068-1097. doi: 10.1137/080721522. [42] 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 etc., 2003, 65-108. [43] K. N. Plataniotis and A. N. Venetsanopoulos, "Color Image Processing and Applications," Springer, Berlin etc., 2000. [44] C. Pöschl, J. Modersitzki and O. Scherzer, A variational setting for volume constrained image registration, Inverse Probl. Imaging, 4 (2010), 505-522. doi: 10.3934/ipi.2010.4.505. [45] O. Scherzer, M. Grasmair, H. Grossauer, M. Haltmeier and F. Lenzen, "Variational Methods in Imaging," Springer, New York etc., 2009. [46] B. C. Vemuri, J. Ye, Y. Chen and C. M. Leonard, A level-set based approach to image registration, in "IEEE Workshop on Mathematical Methods in Biomedical Image Analysis (MMBIA'00)," IEEE Computer Society, Washington, 2000, 86-93. [47] 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. doi: 10.1007/s10107-004-0559-y. [48] M. Wagner, Pontryagin's maximum principle for multidimensional control problems in image processing, J. Optim. Theory Appl., 140 (2009), 543-576. doi: 10.1007/s10957-008-9460-9. [49] M. Wagner, Elastic image registration in presence of polyconvex constraints, submitted: Proceedings of the International Workshop on Optimal Control in Image Processing, Heidelberg, Germany, May 31 - June 1, 2010. [50] M. Wagner, Quasiconvex relaxation of multidimensional control problems with integrands $f(t,\xi,v)$, ESAIM: Control, Optimisation and Calculus of Variations 17 (2011), 190-221. doi: 10.1051/cocv/2010008. [51] A. Yezzi, L. Zollei and T. Kapur, A variational framework for joint segmentation and registration, in "Proceedings of the IEEE Workshop on Mathematical Methods in Biomedical Image Analysis (MMBIA'01)," IEEE Computer Society, Washington, 2001, 44-51.
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