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 & 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.  doi: 10.1137/0915014.  Google Scholar

[2]

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

[3]

F. L. Bookstein and W. D. K. Green, Edge information at landmarks in medical images,, in, (1992), 242.   Google Scholar

[4]

M. Bro-Nielsen and C. Gramkow, Fast fluid registration of medical images,, in, (1996), 267.   Google Scholar

[5]

C. Broit, "Optimal Registration of Deformed Images,", Ph.D. thesis, (1981).   Google Scholar

[6]

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

[7]

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

[8]

G. E. Christensen, R. D. Rabbitt and M. I. Miller, A deformable neuroanatomy textbook based on viscous fluid mechanics,, in, (1993), 211.   Google Scholar

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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.  doi: 10.1088/0031-9155/39/3/022.  Google Scholar

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

[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.  doi: 10.1016/S1361-8415(03)00039-2.  Google Scholar

[12]

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

[13]

J. C. Gee, D. R. Haynor, M. Reivich and R. Bajcsy, Finite element approach to warping of brain images,, in, (1994), 327.   Google Scholar

[14]

R. Gonzalez and R. Woods, "Digital Image Processing,", Addison-Wesley, (1992).   Google Scholar

[15]

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

[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.  doi: 10.1023/B:VISI.0000036836.66311.97.  Google Scholar

[17]

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

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S. Kabus, A. Franz and B. Fischer, Variational image registration allowing for discontinuities in the displacement field,, in, (2007).  doi: 10.1007/978-3-540-33267-1_20.  Google Scholar

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L. D. Landau and E. M. Lifshitz, "Fluid Mechanics,", Pergamon, (1987).   Google Scholar

[20]

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

[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.  doi: 10.1109/TMI.2004.843256.  Google Scholar

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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.  doi: 10.1109/42.563664.  Google Scholar

[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.   Google Scholar

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A. Mills, S. Marsland and T. Shardlow, Computing the geodesic interpolating spline,, in, (2006), 169.   Google Scholar

[25]

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

[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.  doi: 10.1097/00004728-198901000-00004.  Google Scholar

[27]

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

[28]

R. Peyret and T. Taylor, "Computational Methods for Fluid Flow,", Springer, (1983).   Google Scholar

[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.  doi: 10.1109/TMI.2003.815867.  Google Scholar

[30]

T. Rehman and A. Tannenbaum, Multigrid optimal mass transport for image registration and morphing,, in, (2007).   Google Scholar

[31]

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

[32]

R. Temam and A. Miranville, "Mathematical Modeling in Continuum Mechanics,", Cambridge, (2000).   Google Scholar

[33]

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

[34]

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

[35]

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

[36]

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

show all references

References:
[1]

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

[2]

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

[3]

F. L. Bookstein and W. D. K. Green, Edge information at landmarks in medical images,, in, (1992), 242.   Google Scholar

[4]

M. Bro-Nielsen and C. Gramkow, Fast fluid registration of medical images,, in, (1996), 267.   Google Scholar

[5]

C. Broit, "Optimal Registration of Deformed Images,", Ph.D. thesis, (1981).   Google Scholar

[6]

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

[7]

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

[8]

G. E. Christensen, R. D. Rabbitt and M. I. Miller, A deformable neuroanatomy textbook based on viscous fluid mechanics,, in, (1993), 211.   Google Scholar

[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.  doi: 10.1088/0031-9155/39/3/022.  Google Scholar

[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.  doi: 10.1109/83.536892.  Google Scholar

[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.  doi: 10.1016/S1361-8415(03)00039-2.  Google Scholar

[12]

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

[13]

J. C. Gee, D. R. Haynor, M. Reivich and R. Bajcsy, Finite element approach to warping of brain images,, in, (1994), 327.   Google Scholar

[14]

R. Gonzalez and R. Woods, "Digital Image Processing,", Addison-Wesley, (1992).   Google Scholar

[15]

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

[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.  doi: 10.1023/B:VISI.0000036836.66311.97.  Google Scholar

[17]

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

[18]

S. Kabus, A. Franz and B. Fischer, Variational image registration allowing for discontinuities in the displacement field,, in, (2007).  doi: 10.1007/978-3-540-33267-1_20.  Google Scholar

[19]

L. D. Landau and E. M. Lifshitz, "Fluid Mechanics,", Pergamon, (1987).   Google Scholar

[20]

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

[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.  doi: 10.1109/TMI.2004.843256.  Google Scholar

[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.  doi: 10.1109/42.563664.  Google Scholar

[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.   Google Scholar

[24]

A. Mills, S. Marsland and T. Shardlow, Computing the geodesic interpolating spline,, in, (2006), 169.   Google Scholar

[25]

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

[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.  doi: 10.1097/00004728-198901000-00004.  Google Scholar

[27]

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

[28]

R. Peyret and T. Taylor, "Computational Methods for Fluid Flow,", Springer, (1983).   Google Scholar

[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.  doi: 10.1109/TMI.2003.815867.  Google Scholar

[30]

T. Rehman and A. Tannenbaum, Multigrid optimal mass transport for image registration and morphing,, in, (2007).   Google Scholar

[31]

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

[32]

R. Temam and A. Miranville, "Mathematical Modeling in Continuum Mechanics,", Cambridge, (2000).   Google Scholar

[33]

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

[34]

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

[35]

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

[36]

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

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