# American Institute of Mathematical Sciences

March  2013, 5(1): 131-150. doi: 10.3934/jgm.2013.5.131

## Computing metamorphoses between discrete measures

 1 Center for Imaging Science, Johns Hopkins University, 3400 North Charles Street, Baltimore, MD 21218-2686, United States 2 Center for Imaging Science and Department of Applied Mathematics and Statistics, Johns Hopkins University, 3400 North Charles Street, Baltimore, MD 21218-2686, United States

Received  August 2012 Revised  January 2013 Published  April 2013

Metamorphosis is a mathematical framework for diffeomorphic pattern matching in which one defines a distance on a space of images or shapes. In the case of image matching, this distance involves computing the energetically optimal way in which one image can be morphed into the other, combining both smooth deformations and changes in the image intensity. In [12], Holm, Trouvé and Younes studied the metamorphosis of more singular deformable objects, in particular measures. In this paper, we present results on the analysis and computation of discrete measure metamorphosis, building upon the work in [12]. We show that, when matching sums of Dirac measures, minimizing evolutions can include other singular distributions, which complicates the numerical approximation of such solutions. We then present an Eulerian numerical scheme that accounts for these distributions, as well as some numerical experiments using this scheme.
Citation: Casey L. Richardson, Laurent Younes. Computing metamorphoses between discrete measures. Journal of Geometric Mechanics, 2013, 5 (1) : 131-150. doi: 10.3934/jgm.2013.5.131
##### References:
 [1] Vladimir I. Arnold, "Mathematical Methods of Classical Mechanics," Springer, 1989. [2] M. Faisal Beg, Michael I. Miller, Alain Trouvé and Laurent Younes, Computing large deformation metric mappings via geodesic flows of diffeomorphisms, Int. J. Comp. Vis., 61 (2005), 139-157. doi: 10.1023/B:VISI.0000043755.93987.aa. [3] Martins Bruveris, Francois Gay-Balmaz and Darryl D. Holm, The momentum map representation of images, J. Nonlinear Sci., 21 (2011), 115-150. doi: 10.1007/s00332-010-9079-5. [4] Alberto P. Calderón and Antoni Zygmund, On the existence of certain singular integrals, Acta Math., 88 (1952), 85-139. [5] Paul Dupuis, Ulf Grenander and Michael Miller, Variational problems on flows of diffeomorphisms for image matching, Quarterly of Applied Math, 56 (1998), 587-600. [6] Lawrence C. Evans, "Partial Differential Equations," American Mathematical Society, 1998. [7] Laurent Garcin and Laurent Younes, Geodesic image matching: A wavelet based energy minimization scheme, EMM-CVPR'05, (2005), pages 349-364. doi: 10.1007/11585978_23. [8] Laurent Garcin and Laurent Younes, Geodesic matching with free extremities, J. Math. Imag. Vis., 25 (2006), 329-340. doi: 10.1007/s10851-006-6729-1. [9] Joan Glaunès, Alain Trouvé and Laurent Younes, Diffeomorphic matching of distributions: A new approach for unlabelled point-sets and sub-manifolds matching, Proceedings of CVPR '04, (2004). [10] A. Henderson, "ParaView Guide, A Parallel Visualization Application," Kitware, Inc., 2007. [11] Darryl D. Holm, Tanya Schmah and Cristina Stoica, "Geometric Mechanics and Symmetry: From Finite to Infinite Dimensions," Oxford University Press, 2009. [12] Darryl D. Holm, Alain Trouvé and Laurent Younes, The Euler-Poincare theory of metamorphosis, Quart. Appl. Math., 67 (2009), 661-685. [13] Lars Hörmander, "The Analysis of Linear Partial Differential Operators I-IV," Classics in Mathematics. Springer-Verlag, New York, 1984. doi: 10.1007/978-3-642-96750-4. [14] Eric Jones, Travis Oliphant, Pearu Peterson, et al., "SciPy: Open Source Scientific Tools for Python," 2001. [15] Sarang Joshi and Michael I. Miller, Landmark matching via large deformation diffeomorphisms, IEEE Transactions in Image Processing, 9 (2000), 1357-1370. doi: 10.1109/83.855431. [16] Tsoy-Wo Ma, Higher chain formula proved by combinatorics, The Electronic Journal of Combinatorics, 16 (2009). [17] Richard Melrose, "Introduction to Microlocal Analysis," Unpublished book, December 2007. [18] Yves Meyer, "Wavelets and Operators," Cambridge University Press, 1992. [19] Yves Meyer and Ronald Coifman, "Wavelets: Calderón-Zygmund and Multilinear Operators," Cambridge University Press, 1997. [20] Michael I. Miller and Laurent Younes, Group action, diffeomorphism and matching: A general framework, Int. J. Comp. Vis., 41 (2001), 61-84. [21] John G. Proakis, "Digital Communications," McGraw-Hill, Inc., 1995. [22] Yu. Safarov, Distributions, Fourier transforms and microlocal analysis, Working Paper, (1996). [23] Elias M. Stein, "Singular Integrals and Differentiability Properties of Functions," Princeton University Press, Princeton, 1970. [24] Elias M. Stein, "Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals," Princeton University Press, Princeton, 1993. [25] Michael E. Taylor, "Pseudodifferential Operators and Nonlinear PDE," Birkhauser, 1991. doi: 10.1007/978-1-4612-0431-2. [26] Alain Trouvé and Laurent Younes, Local geometry of deformable templates, SIAM Journal on Mathematical Analysis, 37 (2005), 17-59. doi: 10.1137/S0036141002404838. [27] Alain Trouvé and Laurent Younes, Metamorphoses through lie group action, Found. Comp. Math., (2005), 173-198. doi: 10.1007/s10208-004-0128-z. [28] Marc Vaillant and Joan Glaunès, Surface matching via currents, Proceedings of Information Processing in Medical Imaging (IPMI 2005), 3565 in Lecture Notes in Computer Science, (2005). doi: 10.1007/11505730_32. [29] Marc Vaillant, Michael I. Miller, Alain Trouv'e and Laurent Younes, Statistics on diffeomorphisms via tangent space representations, Neuroimage, 23 (2004), S161-S169. doi: 10.1016/j.neuroimage.2004.07.023. [30] Lei Wang, Faisal Beg, Tilak Ratnanather, Can Ceritoglu, Laurent Younes, John C. Morris, John G. Csernansky and Michael I. Miller, Large deformation diffeomorphism and momentum based hippocampal shape discrimination in dementia of the alzheimer type, IEEE Transactions on Medical Imaging, 26 (2007), 462-470. doi: 10.1109/TMI.2006.887380. [31] Lei Wang, Jeffrey S. Swank, Irena E. Glick, Mokhtar H. Gado, Michael I. Miller, John C. Morris and John G. Csernansky, Large deformation diffeomorphism and momentum based hippocampal shape discrimination in dementia of the alzheimer type, NeuroImage, 20 (2003), 667-682. doi: 10.1109/TMI.2006.887380. [32] Laurent Younes, Computable elastic distances between shapes, SIAM J. Appl. Math, 58 (1998), 565-586. doi: 10.1137/S0036139995287685. [33] Laurent Younes, "Shapes and Diffeomorphisms," 171 of Applied Mathematical Sciences, Springer, Berlin, 2010. doi: 10.1007/978-3-642-12055-8. [34] Eberhard Zeidler, "Applied Functional Analysis: Applications to Mathematical Physics," Applied Mathematical Sciences. Springer, 1995.

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##### References:
 [1] Vladimir I. Arnold, "Mathematical Methods of Classical Mechanics," Springer, 1989. [2] M. Faisal Beg, Michael I. Miller, Alain Trouvé and Laurent Younes, Computing large deformation metric mappings via geodesic flows of diffeomorphisms, Int. J. Comp. Vis., 61 (2005), 139-157. doi: 10.1023/B:VISI.0000043755.93987.aa. [3] Martins Bruveris, Francois Gay-Balmaz and Darryl D. Holm, The momentum map representation of images, J. Nonlinear Sci., 21 (2011), 115-150. doi: 10.1007/s00332-010-9079-5. [4] Alberto P. Calderón and Antoni Zygmund, On the existence of certain singular integrals, Acta Math., 88 (1952), 85-139. [5] Paul Dupuis, Ulf Grenander and Michael Miller, Variational problems on flows of diffeomorphisms for image matching, Quarterly of Applied Math, 56 (1998), 587-600. [6] Lawrence C. Evans, "Partial Differential Equations," American Mathematical Society, 1998. [7] Laurent Garcin and Laurent Younes, Geodesic image matching: A wavelet based energy minimization scheme, EMM-CVPR'05, (2005), pages 349-364. doi: 10.1007/11585978_23. [8] Laurent Garcin and Laurent Younes, Geodesic matching with free extremities, J. Math. Imag. Vis., 25 (2006), 329-340. doi: 10.1007/s10851-006-6729-1. [9] Joan Glaunès, Alain Trouvé and Laurent Younes, Diffeomorphic matching of distributions: A new approach for unlabelled point-sets and sub-manifolds matching, Proceedings of CVPR '04, (2004). [10] A. Henderson, "ParaView Guide, A Parallel Visualization Application," Kitware, Inc., 2007. [11] Darryl D. Holm, Tanya Schmah and Cristina Stoica, "Geometric Mechanics and Symmetry: From Finite to Infinite Dimensions," Oxford University Press, 2009. [12] Darryl D. Holm, Alain Trouvé and Laurent Younes, The Euler-Poincare theory of metamorphosis, Quart. Appl. Math., 67 (2009), 661-685. [13] Lars Hörmander, "The Analysis of Linear Partial Differential Operators I-IV," Classics in Mathematics. Springer-Verlag, New York, 1984. doi: 10.1007/978-3-642-96750-4. [14] Eric Jones, Travis Oliphant, Pearu Peterson, et al., "SciPy: Open Source Scientific Tools for Python," 2001. [15] Sarang Joshi and Michael I. Miller, Landmark matching via large deformation diffeomorphisms, IEEE Transactions in Image Processing, 9 (2000), 1357-1370. doi: 10.1109/83.855431. [16] Tsoy-Wo Ma, Higher chain formula proved by combinatorics, The Electronic Journal of Combinatorics, 16 (2009). [17] Richard Melrose, "Introduction to Microlocal Analysis," Unpublished book, December 2007. [18] Yves Meyer, "Wavelets and Operators," Cambridge University Press, 1992. [19] Yves Meyer and Ronald Coifman, "Wavelets: Calderón-Zygmund and Multilinear Operators," Cambridge University Press, 1997. [20] Michael I. Miller and Laurent Younes, Group action, diffeomorphism and matching: A general framework, Int. J. Comp. Vis., 41 (2001), 61-84. [21] John G. Proakis, "Digital Communications," McGraw-Hill, Inc., 1995. [22] Yu. Safarov, Distributions, Fourier transforms and microlocal analysis, Working Paper, (1996). [23] Elias M. Stein, "Singular Integrals and Differentiability Properties of Functions," Princeton University Press, Princeton, 1970. [24] Elias M. Stein, "Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals," Princeton University Press, Princeton, 1993. [25] Michael E. Taylor, "Pseudodifferential Operators and Nonlinear PDE," Birkhauser, 1991. doi: 10.1007/978-1-4612-0431-2. [26] Alain Trouvé and Laurent Younes, Local geometry of deformable templates, SIAM Journal on Mathematical Analysis, 37 (2005), 17-59. doi: 10.1137/S0036141002404838. [27] Alain Trouvé and Laurent Younes, Metamorphoses through lie group action, Found. Comp. Math., (2005), 173-198. doi: 10.1007/s10208-004-0128-z. [28] Marc Vaillant and Joan Glaunès, Surface matching via currents, Proceedings of Information Processing in Medical Imaging (IPMI 2005), 3565 in Lecture Notes in Computer Science, (2005). doi: 10.1007/11505730_32. [29] Marc Vaillant, Michael I. Miller, Alain Trouv'e and Laurent Younes, Statistics on diffeomorphisms via tangent space representations, Neuroimage, 23 (2004), S161-S169. doi: 10.1016/j.neuroimage.2004.07.023. [30] Lei Wang, Faisal Beg, Tilak Ratnanather, Can Ceritoglu, Laurent Younes, John C. Morris, John G. Csernansky and Michael I. Miller, Large deformation diffeomorphism and momentum based hippocampal shape discrimination in dementia of the alzheimer type, IEEE Transactions on Medical Imaging, 26 (2007), 462-470. doi: 10.1109/TMI.2006.887380. [31] Lei Wang, Jeffrey S. Swank, Irena E. Glick, Mokhtar H. Gado, Michael I. Miller, John C. Morris and John G. Csernansky, Large deformation diffeomorphism and momentum based hippocampal shape discrimination in dementia of the alzheimer type, NeuroImage, 20 (2003), 667-682. doi: 10.1109/TMI.2006.887380. [32] Laurent Younes, Computable elastic distances between shapes, SIAM J. Appl. Math, 58 (1998), 565-586. doi: 10.1137/S0036139995287685. [33] Laurent Younes, "Shapes and Diffeomorphisms," 171 of Applied Mathematical Sciences, Springer, Berlin, 2010. doi: 10.1007/978-3-642-12055-8. [34] Eberhard Zeidler, "Applied Functional Analysis: Applications to Mathematical Physics," Applied Mathematical Sciences. Springer, 1995.
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