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 & 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, (2011).   Google Scholar

[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.  doi: 10.1023/A:1008170101536.  Google Scholar

[3]

J. M. Ball, Convexity conditions and existence theorems in nonlinear elasticity,, Arch. Rat. Mech. Anal., 63 (1977), 337.   Google Scholar

[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.  doi: 10.1016/j.ijsolstr.2005.07.048.  Google Scholar

[5]

S. Barbieri, M. Welk and J. Weickert, A variational approach to the registration of tensor-valued images,, in, (2009), 59.   Google Scholar

[6]

D. Breitenreicher and C. Schnörr, Robust 3D object registration without explicit correspondence using geometric integration,, Machine Vis. and Appl., (): 00138.   Google Scholar

[7]

C. Brune, "Berechnung des Optischen Flusses und Kantenerkennung mit Optimierungsmethoden,", Diploma thesis, (2007).   Google Scholar

[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.  doi: 10.1137/080725064.  Google Scholar

[9]

F. Chmelka and E. Melan, "Einführung in die Festigkeitslehre,", Springer, (1976).   Google Scholar

[10]

G. E. Christensen, R. D. Rabbitt and M. I. Miller, Deformable templates using large deformation kinematics,, IEEE Trans. Image Processing, 5 (1996), 1435.  doi: 10.1109/83.536892.  Google Scholar

[11]

B. Dacorogna, "Direct Methods in the Calculus of Variations,", Springer, (2008).   Google Scholar

[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.  doi: 10.1118/1.2748104.  Google Scholar

[13]

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

[14]

M. Droske and M. Rumpf, Multiscale joint segmentation and registration of image morphology,, IEEE Trans. Pattern Recognition Machine Intelligence, 29 (2007), 2181.  doi: 10.1109/TPAMI.2007.1120.  Google Scholar

[15]

,L. C. Evans and R. F. Gariepy, "Measure Theory and Fine Properties of Functions,", CRC Press, (1992).   Google Scholar

[16]

O. Faugeras and G. Hermosillo, Well-posedness of two nonrigid multimodal image registration methods,, SIAM J. Appl. Math., 64 (2004), 1550.  doi: 10.1137/S0036139903424904.  Google Scholar

[17]

B. Fischer and J. Modersitzki, Curvature based image registration,, J. Math. Imaging Vision, 18 (2003), 81.   Google Scholar

[18]

R. Fourer, D. M. Gay and B. W. Kernighan, "AMPL. A Modeling Language for Mathematical Programming,", Brooks/Cole - Thomson Learning, (2003).   Google Scholar

[19]

L. Franek, "Anwendung optimaler Steuerungsprobleme mit $L^\infty$-Steuerbeschrünkung auf ein Modell-problem der Bildverarbeitung,", Diploma thesis, (2007).   Google Scholar

[20]

M. Franek, "Bildentrauschung und Kantenerkennung mit $L^p$-Regularisierung und Gradienten-beschränkung bei Graustufenbildern,", Diploma thesis, (2007).   Google Scholar

[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.  doi: 10.1002/oca.996.  Google Scholar

[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).   Google Scholar

[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.  doi: 10.1007/s00466-002-0347-6.  Google Scholar

[24]

H. Goering, H.-G. Roos and L. Tobiska, "Finite-Element-Method,", Akademie-Verlag, (1993).   Google Scholar

[25]

E. Haber and J. Modersitzki, Numerical methods for volume preserving image registration,, Inverse Problems, 20 (2004), 1621.  doi: 10.1088/0266-5611/20/5/018.  Google Scholar

[26]

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

[27]

S. Haker, L. Zhu, A. Tannenbaum and S. Angenent, Optimal mass transport for registration and warping,, Int. J. Computer Vision, 60 (2004), 225.  doi: 10.1023/B:VISI.0000036836.66311.97.  Google Scholar

[28]

S. Henn and K. Witsch, A multigrid approach for minimizing a nonlinear functional for digital image matching,, Computing, 64 (2000), 339.  doi: 10.1007/s006070070029.  Google Scholar

[29]

S. Henn and K. Witsch, Iterative multigrid regularization techniques for image matching,, SIAM J. Sci. Comput., 23 (2001), 1077.  doi: 10.1137/S106482750037161X.  Google Scholar

[30]

G. Hermosillo, C. Chefd'hotel and O. Faugeras, Variational methods for multimodal image matching,, Int. J. Computer Vision, 50 (2002), 329.  doi: 10.1023/A:1020830525823.  Google Scholar

[31]

M. Hintermüller and S. L. Keeling, Image registration and segmentation based on energy minimization,, in, (2009), 213.   Google Scholar

[32]

B. Jansen, "Interior Point Techniques in Optimization,", Kluwer, (1997).   Google Scholar

[33]

T. Kaijser, Computing the Kantorovich distance for images,, J. Math. Imaging Vision, 9 (1998), 173.  doi: 10.1023/A:1008389726910.  Google Scholar

[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.  doi: 10.1007/s10851-005-4967-2.  Google Scholar

[35]

C. Laird and A. Wächter, Introduction to IPOPT: A tutorial for downloading, installing, and using IPOPT,, Revision No. 1863, (1863).   Google Scholar

[36]

C. Le Guyader and L. Vese, A combined segmentation and registration framework with a nonlinear elasticity smoother,, in, (2009), 1.   Google Scholar

[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, (2003), 1092.   Google Scholar

[38]

G. Maess, "Vorlesungen über numerische Mathematik II,", Akademie-Verlag, (1988).   Google Scholar

[39]

J. Modersitzki, "Numerical Methods for Image Registration,", Oxford University Press, (2004).   Google Scholar

[40]

J. Modersitzki, "FAIR. Flexible Algorithms for Image Registration,", SIAM, (2009).   Google Scholar

[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.  doi: 10.1137/080721522.  Google Scholar

[42]

R. W. Ogden, Nonlinear elasticity, anisotropy, material stability and residual stresses in soft tissue,, in, (2003), 65.   Google Scholar

[43]

K. N. Plataniotis and A. N. Venetsanopoulos, "Color Image Processing and Applications,", Springer, (2000).   Google Scholar

[44]

C. Pöschl, J. Modersitzki and O. Scherzer, A variational setting for volume constrained image registration,, Inverse Probl. Imaging, 4 (2010), 505.  doi: 10.3934/ipi.2010.4.505.  Google Scholar

[45]

O. Scherzer, M. Grasmair, H. Grossauer, M. Haltmeier and F. Lenzen, "Variational Methods in Imaging,", Springer, (2009).   Google Scholar

[46]

B. C. Vemuri, J. Ye, Y. Chen and C. M. Leonard, A level-set based approach to image registration,, in, (2000), 86.   Google Scholar

[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.  doi: 10.1007/s10107-004-0559-y.  Google Scholar

[48]

M. Wagner, Pontryagin's maximum principle for multidimensional control problems in image processing,, J. Optim. Theory Appl., 140 (2009), 543.  doi: 10.1007/s10957-008-9460-9.  Google Scholar

[49]

M. Wagner, Elastic image registration in presence of polyconvex constraints,, submitted: Proceedings of the International Workshop on Optimal Control in Image Processing, (2010).   Google Scholar

[50]

M. Wagner, Quasiconvex relaxation of multidimensional control problems with integrands $ f(t,\xi,v)$,, ESAIM: Control, 17 (2011), 190.  doi: 10.1051/cocv/2010008.  Google Scholar

[51]

A. Yezzi, L. Zollei and T. Kapur, A variational framework for joint segmentation and registration,, in, (2001), 44.   Google Scholar

show all references

References:
[1]

A. Angelov, "Multimodale Bildregistrierung durch elastisches Matching von Kantenskizzen,", Diploma thesis, (2011).   Google Scholar

[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.  doi: 10.1023/A:1008170101536.  Google Scholar

[3]

J. M. Ball, Convexity conditions and existence theorems in nonlinear elasticity,, Arch. Rat. Mech. Anal., 63 (1977), 337.   Google Scholar

[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.  doi: 10.1016/j.ijsolstr.2005.07.048.  Google Scholar

[5]

S. Barbieri, M. Welk and J. Weickert, A variational approach to the registration of tensor-valued images,, in, (2009), 59.   Google Scholar

[6]

D. Breitenreicher and C. Schnörr, Robust 3D object registration without explicit correspondence using geometric integration,, Machine Vis. and Appl., (): 00138.   Google Scholar

[7]

C. Brune, "Berechnung des Optischen Flusses und Kantenerkennung mit Optimierungsmethoden,", Diploma thesis, (2007).   Google Scholar

[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.  doi: 10.1137/080725064.  Google Scholar

[9]

F. Chmelka and E. Melan, "Einführung in die Festigkeitslehre,", Springer, (1976).   Google Scholar

[10]

G. E. Christensen, R. D. Rabbitt and M. I. Miller, Deformable templates using large deformation kinematics,, IEEE Trans. Image Processing, 5 (1996), 1435.  doi: 10.1109/83.536892.  Google Scholar

[11]

B. Dacorogna, "Direct Methods in the Calculus of Variations,", Springer, (2008).   Google Scholar

[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.  doi: 10.1118/1.2748104.  Google Scholar

[13]

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

[14]

M. Droske and M. Rumpf, Multiscale joint segmentation and registration of image morphology,, IEEE Trans. Pattern Recognition Machine Intelligence, 29 (2007), 2181.  doi: 10.1109/TPAMI.2007.1120.  Google Scholar

[15]

,L. C. Evans and R. F. Gariepy, "Measure Theory and Fine Properties of Functions,", CRC Press, (1992).   Google Scholar

[16]

O. Faugeras and G. Hermosillo, Well-posedness of two nonrigid multimodal image registration methods,, SIAM J. Appl. Math., 64 (2004), 1550.  doi: 10.1137/S0036139903424904.  Google Scholar

[17]

B. Fischer and J. Modersitzki, Curvature based image registration,, J. Math. Imaging Vision, 18 (2003), 81.   Google Scholar

[18]

R. Fourer, D. M. Gay and B. W. Kernighan, "AMPL. A Modeling Language for Mathematical Programming,", Brooks/Cole - Thomson Learning, (2003).   Google Scholar

[19]

L. Franek, "Anwendung optimaler Steuerungsprobleme mit $L^\infty$-Steuerbeschrünkung auf ein Modell-problem der Bildverarbeitung,", Diploma thesis, (2007).   Google Scholar

[20]

M. Franek, "Bildentrauschung und Kantenerkennung mit $L^p$-Regularisierung und Gradienten-beschränkung bei Graustufenbildern,", Diploma thesis, (2007).   Google Scholar

[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.  doi: 10.1002/oca.996.  Google Scholar

[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).   Google Scholar

[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.  doi: 10.1007/s00466-002-0347-6.  Google Scholar

[24]

H. Goering, H.-G. Roos and L. Tobiska, "Finite-Element-Method,", Akademie-Verlag, (1993).   Google Scholar

[25]

E. Haber and J. Modersitzki, Numerical methods for volume preserving image registration,, Inverse Problems, 20 (2004), 1621.  doi: 10.1088/0266-5611/20/5/018.  Google Scholar

[26]

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

[27]

S. Haker, L. Zhu, A. Tannenbaum and S. Angenent, Optimal mass transport for registration and warping,, Int. J. Computer Vision, 60 (2004), 225.  doi: 10.1023/B:VISI.0000036836.66311.97.  Google Scholar

[28]

S. Henn and K. Witsch, A multigrid approach for minimizing a nonlinear functional for digital image matching,, Computing, 64 (2000), 339.  doi: 10.1007/s006070070029.  Google Scholar

[29]

S. Henn and K. Witsch, Iterative multigrid regularization techniques for image matching,, SIAM J. Sci. Comput., 23 (2001), 1077.  doi: 10.1137/S106482750037161X.  Google Scholar

[30]

G. Hermosillo, C. Chefd'hotel and O. Faugeras, Variational methods for multimodal image matching,, Int. J. Computer Vision, 50 (2002), 329.  doi: 10.1023/A:1020830525823.  Google Scholar

[31]

M. Hintermüller and S. L. Keeling, Image registration and segmentation based on energy minimization,, in, (2009), 213.   Google Scholar

[32]

B. Jansen, "Interior Point Techniques in Optimization,", Kluwer, (1997).   Google Scholar

[33]

T. Kaijser, Computing the Kantorovich distance for images,, J. Math. Imaging Vision, 9 (1998), 173.  doi: 10.1023/A:1008389726910.  Google Scholar

[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.  doi: 10.1007/s10851-005-4967-2.  Google Scholar

[35]

C. Laird and A. Wächter, Introduction to IPOPT: A tutorial for downloading, installing, and using IPOPT,, Revision No. 1863, (1863).   Google Scholar

[36]

C. Le Guyader and L. Vese, A combined segmentation and registration framework with a nonlinear elasticity smoother,, in, (2009), 1.   Google Scholar

[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, (2003), 1092.   Google Scholar

[38]

G. Maess, "Vorlesungen über numerische Mathematik II,", Akademie-Verlag, (1988).   Google Scholar

[39]

J. Modersitzki, "Numerical Methods for Image Registration,", Oxford University Press, (2004).   Google Scholar

[40]

J. Modersitzki, "FAIR. Flexible Algorithms for Image Registration,", SIAM, (2009).   Google Scholar

[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.  doi: 10.1137/080721522.  Google Scholar

[42]

R. W. Ogden, Nonlinear elasticity, anisotropy, material stability and residual stresses in soft tissue,, in, (2003), 65.   Google Scholar

[43]

K. N. Plataniotis and A. N. Venetsanopoulos, "Color Image Processing and Applications,", Springer, (2000).   Google Scholar

[44]

C. Pöschl, J. Modersitzki and O. Scherzer, A variational setting for volume constrained image registration,, Inverse Probl. Imaging, 4 (2010), 505.  doi: 10.3934/ipi.2010.4.505.  Google Scholar

[45]

O. Scherzer, M. Grasmair, H. Grossauer, M. Haltmeier and F. Lenzen, "Variational Methods in Imaging,", Springer, (2009).   Google Scholar

[46]

B. C. Vemuri, J. Ye, Y. Chen and C. M. Leonard, A level-set based approach to image registration,, in, (2000), 86.   Google Scholar

[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.  doi: 10.1007/s10107-004-0559-y.  Google Scholar

[48]

M. Wagner, Pontryagin's maximum principle for multidimensional control problems in image processing,, J. Optim. Theory Appl., 140 (2009), 543.  doi: 10.1007/s10957-008-9460-9.  Google Scholar

[49]

M. Wagner, Elastic image registration in presence of polyconvex constraints,, submitted: Proceedings of the International Workshop on Optimal Control in Image Processing, (2010).   Google Scholar

[50]

M. Wagner, Quasiconvex relaxation of multidimensional control problems with integrands $ f(t,\xi,v)$,, ESAIM: Control, 17 (2011), 190.  doi: 10.1051/cocv/2010008.  Google Scholar

[51]

A. Yezzi, L. Zollei and T. Kapur, A variational framework for joint segmentation and registration,, in, (2001), 44.   Google Scholar

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