August  2011, 5(3): 551-590. doi: 10.3934/ipi.2011.5.551

Alpha divergences based mass transport models for image matching problems

1. 

Department of Mathematics, National Taiwan University, Taiwan

2. 

School of Mathematics, Hunan University, Changsha 410082

Received  May 2009 Revised  June 2010 Published  August 2011

Registration methods could be roughly divided into two groups: area-based methods and feature-based methods. In the literature, the Monge-Kantorovich (MK) mass transport problem has been applied to image registration as an area-based method. In this paper, we propose to use Monge-Kantorovich (MK) mass transport model as a feature-based method. This novel image matching model is a coupling of the MK problem with the well-known alpha divergence from the probability theory. The optimal matching scheme is the one which minimizes the weighted alpha divergence between two images. A primal-dual approach is employed to analyze the existence and uniqueness/non-uniqueness of the optimal matching scheme. A block coordinate method, analogous to the Sinkhorn matrix balancing method, can be used to compute the optimal matching scheme. We also derive a distance function for image morphing. Similar to elastic distances proposed by Younes, the geodesic under this distance function has an explicit expression.
Citation: Pengwen Chen, Changfeng Gui. Alpha divergences based mass transport models for image matching problems. Inverse Problems and Imaging, 2011, 5 (3) : 551-590. doi: 10.3934/ipi.2011.5.551
References:
[1]

R. D. Anderson and V. L. Klee, Jr., Convex functions and upper semi-continuous collecti ons, Duke Math. J., 19 (1952), 349-357. doi: 10.1215/S0012-7094-52-01935-2.

[2]

Jean-David Benamou and Yann Brenier, A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem, Numer. Math., 84 (2000), 375-393.

[3]

D. P. Bertsekas, "Nonlinear Programming," Athena Scientific, 2003.

[4]

P. J. Besl and N. D. McKay, A method for registration of 3-d shapes, IEEE Trans. Pattern Anal. Mach. Intell., 14 (1992), 239-256. doi: 10.1109/34.121791.

[5]

F. L. Bookstein, Principal warps: Thin-plate splines and the decomposition of deformations, IEEE Trans. Pattern Anal. Mach. Intell., 11 (1989), 567-585. doi: 10.1109/34.24792.

[6]

Alberto Borobia and Rafael Cantó, Matrix scaling: A geometric proof of Sinkhorn's theorem, Linear Algebra Appl., 268 (1998), 1-8.

[7]

Yann Brenier, Polar factorization and monotone rearrangement of vector-valued functions, Comm. Pure Appl. Math., 44 (1991), 375-417.

[8]

P. Chen, A novel kernel correlation model with the corrrespondence estimation, JMIV, 39 (2011), 100-120.

[9]

H. Chui and A. Rangarajan, A feature registration framework using mixture models, IEEE workshop on MMBIA, (2000), 190-197.

[10]

_____, A new algorithm for non-rigid point matching, CVIU, 89 (2003), 114-141.

[11]

I. L. Dryden and K. V. Mardia, "Statistical Shape Analysis," Wiley Series in Probability and Statistics: Probability and Statistics, John Wiley & Sons, Ltd., Chichester, 1998.

[12]

L. C. Evans and W. Gangbo, Differential equations methods for the Monge-Kantorovich mass transfer problem, Mem. Amer. Math. Soc., 137 (1999), viii+66 pp.

[13]

Lawrence C. Evans, Partial differential equations and Monge-Kantorovich mass transfer, Current developments in mathematics, 1997, (Cambridge, MA), Int. Press, Boston, MA, (1999), 65-126.

[14]

_____, "Partial Differential Equations," Second edition, Graduate Studies in Mathematics, 19, American Mathematical Society, Providence, RI, 2010.

[15]

Lawrence C. Evans and Ronald F. Gariepy, "Measure Theory and Fine Properties of Functions," Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992.

[16]

Olivier Faugeras and Gerardo Hermosillo, Well-posedness of two nonrigid multimodal image registration methods, SIAM J. Appl. Math., 64 (2004), 1550-1587.

[17]

Wilfrid Gangbo, An elementary proof of the polar factorization of vector-valued functions, Arch. Rational Mech. Anal., 128 (1994), 381-399.

[18]

A. L. Gibbs and F. E. Su, On choosing and bounding probability metrics, Intl. Stat. Rev., 7 (2002), 419-435.

[19]

J. Glaunes, A. Trouve and L. Younes, Diffeomorphic matching of distributions: A new approach for unlabelled point-sets and sub-manifolds matching, CVPR, 2 (2004), 712-718.

[20]

S. Granger and X. Pennec, Multi-scale EM-ICP: A fast and robust approach for surface registration, ECCV, 4 (2002), 418-432.

[21]

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.

[22]

Roger A. Horn and Charles R. Johnson, "Matrix Analysis," Cambridge University Press, Cambridge, 1985.

[23]

B. Jian and B. C. Vemuri, A robust algorithm for point set registration using mixture of Gaussians, ICCV, 2 (2005), 1246-1251.

[24]

S. C. Joshi and M. I. Miller, Landmark matching via large deformation diffeomorphism, IEEE Image Proc., 9 (2000), 1357-1370.

[25]

Jürgen Jost and Xianqing Li-Jost, "Calculus of Variations," Cambridge Studies in Advanced Mathematics, 64, Cambridge University Press, Cambridge, 1998.

[26]

Thomas Kaijser, Computing the Kantorovich distance for images, J. Math. Imaging Vision, 9 (1998), 173-191.

[27]

L. V. Kantorovich, On the transfer of masses, Dokl. Akad. Nauk. SSSR, 37 (1942), 227-229.

[28]

M. Kass, A. Witkin and D. Terzopoulos, Snake: Active contour models, International Journal of Computer Vision, 1 (1988), 321-331. doi: 10.1007/BF00133570.

[29]

M. Leordeanu and M. Hebert, A spectral technique for correspondence problems using pairwise constraints, ICCV, (2005), 1482-1489.

[30]

B. Luo and E. R. Hancock, A unified framework for alignment and correspondence, Computer Vision and Image Understanding, 92 (2003), 26-55. doi: 10.1016/S1077-3142(03)00097-3.

[31]

B. Ma, R. Narayanan, H. Park, A. O. Hero, P. H. Bland and C. R. Meyer, Comparing pairwise and simultaneous joint registrations of decorrelating interval exams using entropic graphs, Information Processing in Medical Imaging, 4584 (2008), 270-282. doi: 10.1007/978-3-540-73273-0_23.

[32]

Robert J. McCann, Existence and uniqueness of monotone measure-preserving maps, Duke Math. J., 80 (1995), 309-323.

[33]

G. McNeil and S. Vijayakumar, A probabilistic approach to robust shape matching, IEEE ICIP, (2006), 937-940.

[34]

M. I. Miller and L. Younes, Group actions, homeomorphism, matching: A general framework, Int. J. Comput. Vis., 41 (2001), 61-84. doi: 10.1023/A:1011161132514.

[35]

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), 1068-1097.

[36]

Yurii A. Neretin, On Jordan angles and the triangle inequality in Grassmann manifolds, Geom. Dedicata, 86 (2001), 81-92.

[37]

J. Rabin, J. Delon and Y. Gousseau, A statistical approach to the matching of local features, SIAM J. Imaging Sci., 2 (2009), 931-958.

[38]

Svetlozar T. Rachev, "Probability Metrics and the Stability of Stochastic Models," Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics, John Wiley & Sons, Ltd., Chichester, 1991.

[39]

A. Rényi, On measures of entropy and infromation, Proc. 4th Berkeley Symp. Math. Stat. and Prob., Vol. 1, Univ. California Press, Berkeley, Calif., (1961), 547-561.

[40]

R. Tyrrell Rockafellar, "Convex Analysis," Princeton Mathematical Series, No. 28, Princeton University Press, Princeton, N.J., 1970.

[41]

Y. Rubner, C. Tomasi and L. J. Guibas, The earth mover's distance as a metric for image retrieval, Int. J. Comput. Vis., 40 (2000), 99-121. doi: 10.1023/A:1026543900054.

[42]

G. L. Scott and H. C. Longuet-Higgins, An algorithm for associating the features of two images, Proceedings of the Royal Society London: Biological Sciences, 244 (1991), 21-26.

[43]

T. Sebastian, P. Klein and B. Kimia, On aligning curves, IEEE Trans. Pattern Anal. Mach. Intell., 5 (2003), 116-125. doi: 10.1109/TPAMI.2003.1159951.

[44]

I. K. Sethi and R. Jain, Finding trajectories of feature points in a monocular image sequence, IEEE Trans. Pattern Anal. Mach. Intell., 9 (1987), 56-73. doi: 10.1109/TPAMI.1987.4767872.

[45]

Richard Sinkhorn, A relationship between arbitrary positive matrices and doubly stochastic matrices, Ann. Math. Statist., 35 (1964), 876-879. doi: 10.1214/aoms/1177703591.

[46]

D. W. Thompson, "On Growth and Form," Cambridge University Press, Cambridge, 1917.

[47]

A. Trouve, Diffeomorphism groups and pattern matching in image analysis, Int. J. Comput. Vis., 28 (1998), 213-221. doi: 10.1023/A:1008001603737.

[48]

Y. Tsin and T. Kanade, A correlation-based approach to robust point set registration, ECCV, 3 (2004), 558-569.

[49]

Zhuowen Tu, Songfeng Zheng and Alan Yuille, Shape matching and registration by data-driven EM, Computer Vision and Image Understanding, 109 (2008), 290-304.

[50]

S. Ullman, "The Interpretation of Visual Motion," MIT Press, Cambridge, MA, 1979.

[51]

Grace Wahba, "Spline Models for Observational Data," CBMS-NSF Regional Conference Series in Applied Mathematics, 59, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1990.

[52]

F. Wang, B. C. Vemuri, A. Rangarajan and S. J. Eisenschenk, Simultaneous nonrigid registration of multiple point sets and atlas construction, IEEE PAMI, 30 (2008), 2011-2022.

[53]

J. Warga, Minimizing certain convex functions, J. Soc. Indust. Appl. Math., 11 (1963), 588-593. doi: 10.1137/0111043.

[54]

W. M. Wells, Statistical approaches to feature-based object recognition, International Journal of Computer Vision, 22 (1997), 63-98. doi: 10.1023/A:1007923522710.

[55]

M. Werman, S. Peleg and A. Rosenfeld, A distance metric for multi-dimensional histograms, Comp. Vis. Graphics Image Proc., 32 (1985), 328-336. doi: 10.1016/0734-189X(85)90055-6.

[56]

Laurent Younes, Computable elastic distances between shapes, SIAM J. Appl. Math., 58 (1998), 565-586 (electronic).

[57]

Laurent Younes, Peter W. Michor, Jayant Shah and David Mumford, A metric on shape space with explicit geodesics, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 19 (2008), 25-57.

[58]

Z. Zhang, Iterative point matching for registration of free-form curves and surfaces, International Journal of Computer Vision, 13 (1994), 119-152.

[59]

L. Zhu, Y. Yang, S. Haker and A. Tannenbaum, An image morphing technique based on optimal mass preserving mapping, IEEE Trans. Image Processing, 16 (2007), 1481-1495.

[60]

C. L. Zitnick and T. Kanade, A cooperative algorithm for stereo matching and occlusion detection, IEEE Trans. Pattern Anal. Mach. Intell., 22 (2000), 675-684. doi: 10.1109/34.865184.

[61]

B. Zitova and J. Flusser, Image registration methods: A survey, Image and Vis. Compu., 21 (2003), 977-1000. doi: 10.1016/S0262-8856(03)00137-9.

show all references

References:
[1]

R. D. Anderson and V. L. Klee, Jr., Convex functions and upper semi-continuous collecti ons, Duke Math. J., 19 (1952), 349-357. doi: 10.1215/S0012-7094-52-01935-2.

[2]

Jean-David Benamou and Yann Brenier, A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem, Numer. Math., 84 (2000), 375-393.

[3]

D. P. Bertsekas, "Nonlinear Programming," Athena Scientific, 2003.

[4]

P. J. Besl and N. D. McKay, A method for registration of 3-d shapes, IEEE Trans. Pattern Anal. Mach. Intell., 14 (1992), 239-256. doi: 10.1109/34.121791.

[5]

F. L. Bookstein, Principal warps: Thin-plate splines and the decomposition of deformations, IEEE Trans. Pattern Anal. Mach. Intell., 11 (1989), 567-585. doi: 10.1109/34.24792.

[6]

Alberto Borobia and Rafael Cantó, Matrix scaling: A geometric proof of Sinkhorn's theorem, Linear Algebra Appl., 268 (1998), 1-8.

[7]

Yann Brenier, Polar factorization and monotone rearrangement of vector-valued functions, Comm. Pure Appl. Math., 44 (1991), 375-417.

[8]

P. Chen, A novel kernel correlation model with the corrrespondence estimation, JMIV, 39 (2011), 100-120.

[9]

H. Chui and A. Rangarajan, A feature registration framework using mixture models, IEEE workshop on MMBIA, (2000), 190-197.

[10]

_____, A new algorithm for non-rigid point matching, CVIU, 89 (2003), 114-141.

[11]

I. L. Dryden and K. V. Mardia, "Statistical Shape Analysis," Wiley Series in Probability and Statistics: Probability and Statistics, John Wiley & Sons, Ltd., Chichester, 1998.

[12]

L. C. Evans and W. Gangbo, Differential equations methods for the Monge-Kantorovich mass transfer problem, Mem. Amer. Math. Soc., 137 (1999), viii+66 pp.

[13]

Lawrence C. Evans, Partial differential equations and Monge-Kantorovich mass transfer, Current developments in mathematics, 1997, (Cambridge, MA), Int. Press, Boston, MA, (1999), 65-126.

[14]

_____, "Partial Differential Equations," Second edition, Graduate Studies in Mathematics, 19, American Mathematical Society, Providence, RI, 2010.

[15]

Lawrence C. Evans and Ronald F. Gariepy, "Measure Theory and Fine Properties of Functions," Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992.

[16]

Olivier Faugeras and Gerardo Hermosillo, Well-posedness of two nonrigid multimodal image registration methods, SIAM J. Appl. Math., 64 (2004), 1550-1587.

[17]

Wilfrid Gangbo, An elementary proof of the polar factorization of vector-valued functions, Arch. Rational Mech. Anal., 128 (1994), 381-399.

[18]

A. L. Gibbs and F. E. Su, On choosing and bounding probability metrics, Intl. Stat. Rev., 7 (2002), 419-435.

[19]

J. Glaunes, A. Trouve and L. Younes, Diffeomorphic matching of distributions: A new approach for unlabelled point-sets and sub-manifolds matching, CVPR, 2 (2004), 712-718.

[20]

S. Granger and X. Pennec, Multi-scale EM-ICP: A fast and robust approach for surface registration, ECCV, 4 (2002), 418-432.

[21]

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.

[22]

Roger A. Horn and Charles R. Johnson, "Matrix Analysis," Cambridge University Press, Cambridge, 1985.

[23]

B. Jian and B. C. Vemuri, A robust algorithm for point set registration using mixture of Gaussians, ICCV, 2 (2005), 1246-1251.

[24]

S. C. Joshi and M. I. Miller, Landmark matching via large deformation diffeomorphism, IEEE Image Proc., 9 (2000), 1357-1370.

[25]

Jürgen Jost and Xianqing Li-Jost, "Calculus of Variations," Cambridge Studies in Advanced Mathematics, 64, Cambridge University Press, Cambridge, 1998.

[26]

Thomas Kaijser, Computing the Kantorovich distance for images, J. Math. Imaging Vision, 9 (1998), 173-191.

[27]

L. V. Kantorovich, On the transfer of masses, Dokl. Akad. Nauk. SSSR, 37 (1942), 227-229.

[28]

M. Kass, A. Witkin and D. Terzopoulos, Snake: Active contour models, International Journal of Computer Vision, 1 (1988), 321-331. doi: 10.1007/BF00133570.

[29]

M. Leordeanu and M. Hebert, A spectral technique for correspondence problems using pairwise constraints, ICCV, (2005), 1482-1489.

[30]

B. Luo and E. R. Hancock, A unified framework for alignment and correspondence, Computer Vision and Image Understanding, 92 (2003), 26-55. doi: 10.1016/S1077-3142(03)00097-3.

[31]

B. Ma, R. Narayanan, H. Park, A. O. Hero, P. H. Bland and C. R. Meyer, Comparing pairwise and simultaneous joint registrations of decorrelating interval exams using entropic graphs, Information Processing in Medical Imaging, 4584 (2008), 270-282. doi: 10.1007/978-3-540-73273-0_23.

[32]

Robert J. McCann, Existence and uniqueness of monotone measure-preserving maps, Duke Math. J., 80 (1995), 309-323.

[33]

G. McNeil and S. Vijayakumar, A probabilistic approach to robust shape matching, IEEE ICIP, (2006), 937-940.

[34]

M. I. Miller and L. Younes, Group actions, homeomorphism, matching: A general framework, Int. J. Comput. Vis., 41 (2001), 61-84. doi: 10.1023/A:1011161132514.

[35]

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), 1068-1097.

[36]

Yurii A. Neretin, On Jordan angles and the triangle inequality in Grassmann manifolds, Geom. Dedicata, 86 (2001), 81-92.

[37]

J. Rabin, J. Delon and Y. Gousseau, A statistical approach to the matching of local features, SIAM J. Imaging Sci., 2 (2009), 931-958.

[38]

Svetlozar T. Rachev, "Probability Metrics and the Stability of Stochastic Models," Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics, John Wiley & Sons, Ltd., Chichester, 1991.

[39]

A. Rényi, On measures of entropy and infromation, Proc. 4th Berkeley Symp. Math. Stat. and Prob., Vol. 1, Univ. California Press, Berkeley, Calif., (1961), 547-561.

[40]

R. Tyrrell Rockafellar, "Convex Analysis," Princeton Mathematical Series, No. 28, Princeton University Press, Princeton, N.J., 1970.

[41]

Y. Rubner, C. Tomasi and L. J. Guibas, The earth mover's distance as a metric for image retrieval, Int. J. Comput. Vis., 40 (2000), 99-121. doi: 10.1023/A:1026543900054.

[42]

G. L. Scott and H. C. Longuet-Higgins, An algorithm for associating the features of two images, Proceedings of the Royal Society London: Biological Sciences, 244 (1991), 21-26.

[43]

T. Sebastian, P. Klein and B. Kimia, On aligning curves, IEEE Trans. Pattern Anal. Mach. Intell., 5 (2003), 116-125. doi: 10.1109/TPAMI.2003.1159951.

[44]

I. K. Sethi and R. Jain, Finding trajectories of feature points in a monocular image sequence, IEEE Trans. Pattern Anal. Mach. Intell., 9 (1987), 56-73. doi: 10.1109/TPAMI.1987.4767872.

[45]

Richard Sinkhorn, A relationship between arbitrary positive matrices and doubly stochastic matrices, Ann. Math. Statist., 35 (1964), 876-879. doi: 10.1214/aoms/1177703591.

[46]

D. W. Thompson, "On Growth and Form," Cambridge University Press, Cambridge, 1917.

[47]

A. Trouve, Diffeomorphism groups and pattern matching in image analysis, Int. J. Comput. Vis., 28 (1998), 213-221. doi: 10.1023/A:1008001603737.

[48]

Y. Tsin and T. Kanade, A correlation-based approach to robust point set registration, ECCV, 3 (2004), 558-569.

[49]

Zhuowen Tu, Songfeng Zheng and Alan Yuille, Shape matching and registration by data-driven EM, Computer Vision and Image Understanding, 109 (2008), 290-304.

[50]

S. Ullman, "The Interpretation of Visual Motion," MIT Press, Cambridge, MA, 1979.

[51]

Grace Wahba, "Spline Models for Observational Data," CBMS-NSF Regional Conference Series in Applied Mathematics, 59, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1990.

[52]

F. Wang, B. C. Vemuri, A. Rangarajan and S. J. Eisenschenk, Simultaneous nonrigid registration of multiple point sets and atlas construction, IEEE PAMI, 30 (2008), 2011-2022.

[53]

J. Warga, Minimizing certain convex functions, J. Soc. Indust. Appl. Math., 11 (1963), 588-593. doi: 10.1137/0111043.

[54]

W. M. Wells, Statistical approaches to feature-based object recognition, International Journal of Computer Vision, 22 (1997), 63-98. doi: 10.1023/A:1007923522710.

[55]

M. Werman, S. Peleg and A. Rosenfeld, A distance metric for multi-dimensional histograms, Comp. Vis. Graphics Image Proc., 32 (1985), 328-336. doi: 10.1016/0734-189X(85)90055-6.

[56]

Laurent Younes, Computable elastic distances between shapes, SIAM J. Appl. Math., 58 (1998), 565-586 (electronic).

[57]

Laurent Younes, Peter W. Michor, Jayant Shah and David Mumford, A metric on shape space with explicit geodesics, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 19 (2008), 25-57.

[58]

Z. Zhang, Iterative point matching for registration of free-form curves and surfaces, International Journal of Computer Vision, 13 (1994), 119-152.

[59]

L. Zhu, Y. Yang, S. Haker and A. Tannenbaum, An image morphing technique based on optimal mass preserving mapping, IEEE Trans. Image Processing, 16 (2007), 1481-1495.

[60]

C. L. Zitnick and T. Kanade, A cooperative algorithm for stereo matching and occlusion detection, IEEE Trans. Pattern Anal. Mach. Intell., 22 (2000), 675-684. doi: 10.1109/34.865184.

[61]

B. Zitova and J. Flusser, Image registration methods: A survey, Image and Vis. Compu., 21 (2003), 977-1000. doi: 10.1016/S0262-8856(03)00137-9.

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