# 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 & 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
 [1] 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 [2] Xiangtuan Xiong, Jinmei Li, Jin Wen. Some novel linear regularization methods for a deblurring problem. Inverse Problems & Imaging, 2017, 11 (2) : 403-426. doi: 10.3934/ipi.2017019 [3] Yangang Chen, Justin W. L. Wan. Numerical method for image registration model based on optimal mass transport. Inverse Problems & Imaging, 2018, 12 (2) : 401-432. doi: 10.3934/ipi.2018018 [4] Huan Han. A variational model with fractional-order regularization term arising in registration of diffusion tensor image. Inverse Problems & Imaging, 2018, 12 (6) : 1263-1291. doi: 10.3934/ipi.2018053 [5] Dana Paquin, Doron Levy, Eduard Schreibmann, Lei Xing. Multiscale Image Registration. Mathematical Biosciences & Engineering, 2006, 3 (2) : 389-418. doi: 10.3934/mbe.2006.3.389 [6] Z. Foroozandeh, Maria do rosário de Pinho, M. Shamsi. On numerical methods for singular optimal control problems: An application to an AUV problem. Discrete & Continuous Dynamical Systems - B, 2019, 24 (5) : 2219-2235. doi: 10.3934/dcdsb.2019092 [7] Frank Pörner, Daniel Wachsmuth. Tikhonov regularization of optimal control problems governed by semi-linear partial differential equations. Mathematical Control & Related Fields, 2018, 8 (1) : 315-335. doi: 10.3934/mcrf.2018013 [8] Bartomeu Coll, Joan Duran, Catalina Sbert. Half-linear regularization for nonconvex image restoration models. Inverse Problems & Imaging, 2015, 9 (2) : 337-370. doi: 10.3934/ipi.2015.9.337 [9] Alina Toma, Bruno Sixou, Françoise Peyrin. Iterative choice of the optimal regularization parameter in TV image restoration. Inverse Problems & Imaging, 2015, 9 (4) : 1171-1191. doi: 10.3934/ipi.2015.9.1171 [10] Piermarco Cannarsa, Hélène Frankowska, Elsa M. Marchini. On Bolza optimal control problems with constraints. Discrete & Continuous Dynamical Systems - B, 2009, 11 (3) : 629-653. doi: 10.3934/dcdsb.2009.11.629 [11] Zhao Yi, Justin W. L. Wan. An inviscid model for nonrigid image registration. Inverse Problems & Imaging, 2011, 5 (1) : 263-284. doi: 10.3934/ipi.2011.5.263 [12] Jiongmin Yong. A deterministic linear quadratic time-inconsistent optimal control problem. Mathematical Control & Related Fields, 2011, 1 (1) : 83-118. doi: 10.3934/mcrf.2011.1.83 [13] Peter I. Kogut. On approximation of an optimal boundary control problem for linear elliptic equation with unbounded coefficients. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 2105-2133. doi: 10.3934/dcds.2014.34.2105 [14] James P. Nelson, Mark J. Balas. Direct model reference adaptive control of linear systems with input/output delays. Numerical Algebra, Control & Optimization, 2013, 3 (3) : 445-462. doi: 10.3934/naco.2013.3.445 [15] Maria do Rosário de Pinho, Ilya Shvartsman. Lipschitz continuity of optimal control and Lagrange multipliers in a problem with mixed and pure state constraints. Discrete & Continuous Dynamical Systems - A, 2011, 29 (2) : 505-522. doi: 10.3934/dcds.2011.29.505 [16] Georg Vossen, Torsten Hermanns. On an optimal control problem in laser cutting with mixed finite-/infinite-dimensional constraints. Journal of Industrial & Management Optimization, 2014, 10 (2) : 503-519. doi: 10.3934/jimo.2014.10.503 [17] Jan-Hendrik Webert, Philip E. Gill, Sven-Joachim Kimmerle, Matthias Gerdts. A study of structure-exploiting SQP algorithms for an optimal control problem with coupled hyperbolic and ordinary differential equation constraints. Discrete & Continuous Dynamical Systems - S, 2018, 11 (6) : 1259-1282. doi: 10.3934/dcdss.2018071 [18] Wenxiong Chen, Shijie Qi. Direct methods on fractional equations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (3) : 1269-1310. doi: 10.3934/dcds.2019055 [19] IvÁn Area, FaÏÇal NdaÏrou, Juan J. Nieto, Cristiana J. Silva, Delfim F. M. Torres. Ebola model and optimal control with vaccination constraints. Journal of Industrial & Management Optimization, 2018, 14 (2) : 427-446. doi: 10.3934/jimo.2017054 [20] Dana Paquin, Doron Levy, Lei Xing. Hybrid multiscale landmark and deformable image registration. Mathematical Biosciences & Engineering, 2007, 4 (4) : 711-737. doi: 10.3934/mbe.2007.4.711

Impact Factor: