February  2011, 5(1): 263-284. doi: 10.3934/ipi.2011.5.263

An inviscid model for nonrigid image registration

1. 

Department of Computer Science, University of California, Los Angeles, CA 90095, United States

2. 

David R. Cheriton School of Computer Science, University of Waterloo, Waterloo, ON N2L 6N3, Canada

Received  January 2010 Revised  May 2010 Published  February 2011

We propose an inviscid model for nonrigid image registration in a particle framework, and derive the corresponding nonlinear partial differential equations for computing the spatial transformation. Our idea is to simulate the template image as a set of free particles moving toward the target positions under applied forces. Our model can accommodate both small and large deformations, with sharper edges and clear texture achieved at less computational cost. We demonstrate the performance of our model on a variety of images including 2D and 3D, mono-modal and multi-modal images.
Citation: Zhao Yi, Justin W. L. Wan. An inviscid model for nonrigid image registration. Inverse Problems and Imaging, 2011, 5 (1) : 263-284. doi: 10.3934/ipi.2011.5.263
References:
[1]

Y. Amit, A nonlinear variational problem for image matching, SIAM Scientific Computing, 15 (1994), 207-224. doi: 10.1137/0915014.

[2]

R. Bajcsy and S. Kovacic, Multiresolution elastic matching, Computer Vision, Graphics, and Image Processing, 46 (1989), 1-21. doi: 10.1016/S0734-189X(89)80014-3.

[3]

F. L. Bookstein and W. D. K. Green, Edge information at landmarks in medical images, in "Proceedings of Visualization in Biomedical Computing," (1992), 242-258.

[4]

M. Bro-Nielsen and C. Gramkow, Fast fluid registration of medical images, in "Proceedings of Visualization in Biomedical Computing," (1996), 267-276.

[5]

C. Broit, "Optimal Registration of Deformed Images," Ph.D. thesis, University of Pennsylvania, 1981.

[6]

T. F. Chan and L. A. Vese, Active contours without edges, IEEE Transactions on Image Processing, 10 (2001), 266-277. doi: 10.1109/83.902291.

[7]

G. E. Christensen, S. C. Joshi and M. I. Miller, Volumetric transformation of brain anatomy, IEEE Transactions on Medical Imaging, 16 (1997), 864-877. doi: 10.1109/42.650882.

[8]

G. E. Christensen, R. D. Rabbitt and M. I. Miller, A deformable neuroanatomy textbook based on viscous fluid mechanics, in "Proceeding of Information Science and Systems," (1993), 211-216.

[9]

G. E. Christensen, R. D. Rabbitt and M. I. Miller, 3D brain mapping using a deformable neuroanatomy, Physics in Medicine and Biology, 39 (1994), 609-618. doi: 10.1088/0031-9155/39/3/022.

[10]

G. E. Christensen, R. D. Rabbitt and M. I. Miller, Deformable templates using large deformation kinematics, IEEE Transactions on Image Processing, 5 (1996), 1435-1447. doi: 10.1109/83.536892.

[11]

E. D'Agostino, F. Maes, D. Vandermeulen and P. Suetens, A viscous fluid model for multimodal non-rigid image registration using mutual information, Medical Image Analysis, 7 (2003), 565-575. doi: 10.1016/S1361-8415(03)00039-2.

[12]

C. Davatzikos, Spatial transformation and registration of brain images using elastically deformable models, Computer Vision and Image Understanding, 66 (1997), 207-222. doi: 10.1006/cviu.1997.0605.

[13]

J. C. Gee, D. R. Haynor, M. Reivich and R. Bajcsy, Finite element approach to warping of brain images, in "SPIE Medical Imaging," (1994), 327-337.

[14]

R. Gonzalez and R. Woods, "Digital Image Processing," Addison-Wesley, 1992.

[15]

E. Haber and J. Modersitzki, A multilevel method for image registration, SIAM J. Sci. Comput., 27 (2006), 1594-1607. doi: 10.1137/040608106.

[16]

S. Haker, L. Zhu, A. Tannenbaum and S. Angenent, Optimal mass transport for registration and warping, International Journal of Computer Vision, 60 (2004), 225-240. doi: 10.1023/B:VISI.0000036836.66311.97.

[17]

D. D. Holm, J. T. Ratnanather, A. Trouve and L. Younes, Soliton dynamics in computational anatomy, NeuroImage, 23 (2004), 170-178. doi: 10.1016/j.neuroimage.2004.07.017.

[18]

S. Kabus, A. Franz and B. Fischer, Variational image registration allowing for discontinuities in the displacement field, in "Image Processing Based on Partial Differential Equations," (eds. X. Tai, K. Lie, T. F. Chan and S. Osher), Springer Berlin Heidelberg, 2007. doi: 10.1007/978-3-540-33267-1_20.

[19]

L. D. Landau and E. M. Lifshitz, "Fluid Mechanics," Pergamon, 1987.

[20]

R. J. LeVeque, "Finite Volume Methods for Hyperbolic Problems," Cambridge University Press, 2002. doi: 10.1017/CBO9780511791253.

[21]

A. Madabhushi and J. K. Udupa, Interplay between intensity standardization and inhomogeneity correction in MR image processing, IEEE Transactions on Medical Imaging, 24 (2005), 561-576. doi: 10.1109/TMI.2004.843256.

[22]

F. Maes, A. Collington, D. Vandermeulen, G. Marchal and P. Suetens, Multimodality image registration by maximization of mutual information, IEEE Transactions on Medical Imaging, 16 (1997), 187-198. doi: 10.1109/42.563664.

[23]

S. Marsland and C. J. Twining, Constructing diffeomorphic representations for the groupwise analysis of nonrigid registrations of medical images, IEEE Transactions on Medical Imaging, 23 (2004), 1006-1020.

[24]

A. Mills, S. Marsland and T. Shardlow, Computing the geodesic interpolating spline, in "Lecture Notes in Computer Science, volume 4057," (eds. J. Pluim, B. Likar and F. A. Gerritsen), Springer, (2006), 169-177.

[25]

J. Modersitzki, "Numerical Methods for Image Registration," Oxford University Press, Oxford, 2004.

[26]

C. A. Pelizzari, G. T. Y. Chen, D. R. Spelbring, R. R. Weichselbaum and C. T. Chen, Accurate three-dimensional registration of CT, PET, and/or MR images of the brain, Computer Assisted Tomography, 13 (1989), 20-26. doi: 10.1097/00004728-198901000-00004.

[27]

S. Periaswamy and H. Farid, Elastic registration in the presence of intensity variations, IEEE Transactions on Medical Imaging, 22 (2003), 865-874. doi: 10.1109/TMI.2003.815069.

[28]

R. Peyret and T. Taylor, "Computational Methods for Fluid Flow," Springer, 1983.

[29]

J. P. W. Pluim, J. B. A. Maintz and M. A. Viegever, Mutual information based registration of medical images: A survey, IEEE Transactions on Medical Imaging, 22 (2003), 986-1004. doi: 10.1109/TMI.2003.815867.

[30]

T. Rehman and A. Tannenbaum, Multigrid optimal mass transport for image registration and morphing, in "SPIE Conference on Computational Imaging V," 2007.

[31]

L. I. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms, Physica D, 60 (1992), 259-268. doi: 10.1016/0167-2789(92)90242-F.

[32]

R. Temam and A. Miranville, "Mathematical Modeling in Continuum Mechanics," Cambridge, 2000.

[33]

J. Thirion, Image matching as a diffusion process: An analogy with Maxwell's demons, Medical Image Analysis, 2 (1998), 243-260. doi: 10.1016/S1361-8415(98)80022-4.

[34]

A. Trouve, Diffeomorphisms groups and pattern matching in image analysis, International Journal of Computer Vision, 28 (1998), 213-221. doi: 10.1023/A:1008001603737.

[35]

P. Viola and W. M. Wells, Alignment by maximization of mutual information, International Journal of Computer Vision, 24 (1997), 137-154. doi: 10.1023/A:1007958904918.

[36]

G. Wollny and F. Kruggel, Computational cost of nonrigid registration algorithms based on fluid dynamics, IEEE Transactions on Medical Imaging, 21 (2002), 946-952. doi: 10.1109/TMI.2002.803113.

show all references

References:
[1]

Y. Amit, A nonlinear variational problem for image matching, SIAM Scientific Computing, 15 (1994), 207-224. doi: 10.1137/0915014.

[2]

R. Bajcsy and S. Kovacic, Multiresolution elastic matching, Computer Vision, Graphics, and Image Processing, 46 (1989), 1-21. doi: 10.1016/S0734-189X(89)80014-3.

[3]

F. L. Bookstein and W. D. K. Green, Edge information at landmarks in medical images, in "Proceedings of Visualization in Biomedical Computing," (1992), 242-258.

[4]

M. Bro-Nielsen and C. Gramkow, Fast fluid registration of medical images, in "Proceedings of Visualization in Biomedical Computing," (1996), 267-276.

[5]

C. Broit, "Optimal Registration of Deformed Images," Ph.D. thesis, University of Pennsylvania, 1981.

[6]

T. F. Chan and L. A. Vese, Active contours without edges, IEEE Transactions on Image Processing, 10 (2001), 266-277. doi: 10.1109/83.902291.

[7]

G. E. Christensen, S. C. Joshi and M. I. Miller, Volumetric transformation of brain anatomy, IEEE Transactions on Medical Imaging, 16 (1997), 864-877. doi: 10.1109/42.650882.

[8]

G. E. Christensen, R. D. Rabbitt and M. I. Miller, A deformable neuroanatomy textbook based on viscous fluid mechanics, in "Proceeding of Information Science and Systems," (1993), 211-216.

[9]

G. E. Christensen, R. D. Rabbitt and M. I. Miller, 3D brain mapping using a deformable neuroanatomy, Physics in Medicine and Biology, 39 (1994), 609-618. doi: 10.1088/0031-9155/39/3/022.

[10]

G. E. Christensen, R. D. Rabbitt and M. I. Miller, Deformable templates using large deformation kinematics, IEEE Transactions on Image Processing, 5 (1996), 1435-1447. doi: 10.1109/83.536892.

[11]

E. D'Agostino, F. Maes, D. Vandermeulen and P. Suetens, A viscous fluid model for multimodal non-rigid image registration using mutual information, Medical Image Analysis, 7 (2003), 565-575. doi: 10.1016/S1361-8415(03)00039-2.

[12]

C. Davatzikos, Spatial transformation and registration of brain images using elastically deformable models, Computer Vision and Image Understanding, 66 (1997), 207-222. doi: 10.1006/cviu.1997.0605.

[13]

J. C. Gee, D. R. Haynor, M. Reivich and R. Bajcsy, Finite element approach to warping of brain images, in "SPIE Medical Imaging," (1994), 327-337.

[14]

R. Gonzalez and R. Woods, "Digital Image Processing," Addison-Wesley, 1992.

[15]

E. Haber and J. Modersitzki, A multilevel method for image registration, SIAM J. Sci. Comput., 27 (2006), 1594-1607. doi: 10.1137/040608106.

[16]

S. Haker, L. Zhu, A. Tannenbaum and S. Angenent, Optimal mass transport for registration and warping, International Journal of Computer Vision, 60 (2004), 225-240. doi: 10.1023/B:VISI.0000036836.66311.97.

[17]

D. D. Holm, J. T. Ratnanather, A. Trouve and L. Younes, Soliton dynamics in computational anatomy, NeuroImage, 23 (2004), 170-178. doi: 10.1016/j.neuroimage.2004.07.017.

[18]

S. Kabus, A. Franz and B. Fischer, Variational image registration allowing for discontinuities in the displacement field, in "Image Processing Based on Partial Differential Equations," (eds. X. Tai, K. Lie, T. F. Chan and S. Osher), Springer Berlin Heidelberg, 2007. doi: 10.1007/978-3-540-33267-1_20.

[19]

L. D. Landau and E. M. Lifshitz, "Fluid Mechanics," Pergamon, 1987.

[20]

R. J. LeVeque, "Finite Volume Methods for Hyperbolic Problems," Cambridge University Press, 2002. doi: 10.1017/CBO9780511791253.

[21]

A. Madabhushi and J. K. Udupa, Interplay between intensity standardization and inhomogeneity correction in MR image processing, IEEE Transactions on Medical Imaging, 24 (2005), 561-576. doi: 10.1109/TMI.2004.843256.

[22]

F. Maes, A. Collington, D. Vandermeulen, G. Marchal and P. Suetens, Multimodality image registration by maximization of mutual information, IEEE Transactions on Medical Imaging, 16 (1997), 187-198. doi: 10.1109/42.563664.

[23]

S. Marsland and C. J. Twining, Constructing diffeomorphic representations for the groupwise analysis of nonrigid registrations of medical images, IEEE Transactions on Medical Imaging, 23 (2004), 1006-1020.

[24]

A. Mills, S. Marsland and T. Shardlow, Computing the geodesic interpolating spline, in "Lecture Notes in Computer Science, volume 4057," (eds. J. Pluim, B. Likar and F. A. Gerritsen), Springer, (2006), 169-177.

[25]

J. Modersitzki, "Numerical Methods for Image Registration," Oxford University Press, Oxford, 2004.

[26]

C. A. Pelizzari, G. T. Y. Chen, D. R. Spelbring, R. R. Weichselbaum and C. T. Chen, Accurate three-dimensional registration of CT, PET, and/or MR images of the brain, Computer Assisted Tomography, 13 (1989), 20-26. doi: 10.1097/00004728-198901000-00004.

[27]

S. Periaswamy and H. Farid, Elastic registration in the presence of intensity variations, IEEE Transactions on Medical Imaging, 22 (2003), 865-874. doi: 10.1109/TMI.2003.815069.

[28]

R. Peyret and T. Taylor, "Computational Methods for Fluid Flow," Springer, 1983.

[29]

J. P. W. Pluim, J. B. A. Maintz and M. A. Viegever, Mutual information based registration of medical images: A survey, IEEE Transactions on Medical Imaging, 22 (2003), 986-1004. doi: 10.1109/TMI.2003.815867.

[30]

T. Rehman and A. Tannenbaum, Multigrid optimal mass transport for image registration and morphing, in "SPIE Conference on Computational Imaging V," 2007.

[31]

L. I. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms, Physica D, 60 (1992), 259-268. doi: 10.1016/0167-2789(92)90242-F.

[32]

R. Temam and A. Miranville, "Mathematical Modeling in Continuum Mechanics," Cambridge, 2000.

[33]

J. Thirion, Image matching as a diffusion process: An analogy with Maxwell's demons, Medical Image Analysis, 2 (1998), 243-260. doi: 10.1016/S1361-8415(98)80022-4.

[34]

A. Trouve, Diffeomorphisms groups and pattern matching in image analysis, International Journal of Computer Vision, 28 (1998), 213-221. doi: 10.1023/A:1008001603737.

[35]

P. Viola and W. M. Wells, Alignment by maximization of mutual information, International Journal of Computer Vision, 24 (1997), 137-154. doi: 10.1023/A:1007958904918.

[36]

G. Wollny and F. Kruggel, Computational cost of nonrigid registration algorithms based on fluid dynamics, IEEE Transactions on Medical Imaging, 21 (2002), 946-952. doi: 10.1109/TMI.2002.803113.

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