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 & 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. doi: 10.1215/S0012-7094-52-01935-2. Google Scholar

[2]

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

[3]

D. P. Bertsekas, "Nonlinear Programming,", Athena Scientific, (2003). Google Scholar

[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. doi: 10.1109/34.121791. Google Scholar

[5]

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

[6]

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

[7]

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

[8]

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

[9]

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

[10]

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

[11]

I. L. Dryden and K. V. Mardia, "Statistical Shape Analysis,", Wiley Series in Probability and Statistics: Probability and Statistics, (1998). Google Scholar

[12]

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

[13]

Lawrence C. Evans, Partial differential equations and Monge-Kantorovich mass transfer,, Current developments in mathematics, (1999), 65. Google Scholar

[14]

_____, "Partial Differential Equations,", Second edition, 19 (2010). Google Scholar

[15]

Lawrence C. Evans and Ronald F. Gariepy, "Measure Theory and Fine Properties of Functions,", Studies in Advanced Mathematics, (1992). Google Scholar

[16]

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

[17]

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

[18]

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

[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. Google Scholar

[20]

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

[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. Google Scholar

[22]

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

[23]

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

[24]

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

[25]

Jürgen Jost and Xianqing Li-Jost, "Calculus of Variations,", Cambridge Studies in Advanced Mathematics, 64 (1998). Google Scholar

[26]

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

[27]

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

[28]

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

[29]

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

[30]

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

[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. doi: 10.1007/978-3-540-73273-0_23. Google Scholar

[32]

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

[33]

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

[34]

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

[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. Google Scholar

[36]

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

[37]

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

[38]

Svetlozar T. Rachev, "Probability Metrics and the Stability of Stochastic Models,", Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics, (1991). Google Scholar

[39]

A. Rényi, On measures of entropy and infromation,, Proc. 4th Berkeley Symp. Math. Stat. and Prob., (1961), 547. Google Scholar

[40]

R. Tyrrell Rockafellar, "Convex Analysis,", Princeton Mathematical Series, (1970). Google Scholar

[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. doi: 10.1023/A:1026543900054. Google Scholar

[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. Google Scholar

[43]

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

[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. doi: 10.1109/TPAMI.1987.4767872. Google Scholar

[45]

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

[46]

D. W. Thompson, "On Growth and Form,", Cambridge University Press, (1917). Google Scholar

[47]

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

[48]

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

[49]

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

[50]

S. Ullman, "The Interpretation of Visual Motion,", MIT Press, (1979). Google Scholar

[51]

Grace Wahba, "Spline Models for Observational Data,", CBMS-NSF Regional Conference Series in Applied Mathematics, 59 (1990). Google Scholar

[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. Google Scholar

[53]

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

[54]

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

[55]

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

[56]

Laurent Younes, Computable elastic distances between shapes,, SIAM J. Appl. Math., 58 (1998), 565. Google Scholar

[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. Google Scholar

[58]

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

[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. Google Scholar

[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. doi: 10.1109/34.865184. Google Scholar

[61]

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

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. doi: 10.1215/S0012-7094-52-01935-2. Google Scholar

[2]

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

[3]

D. P. Bertsekas, "Nonlinear Programming,", Athena Scientific, (2003). Google Scholar

[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. doi: 10.1109/34.121791. Google Scholar

[5]

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

[6]

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

[7]

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

[8]

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

[9]

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

[10]

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

[11]

I. L. Dryden and K. V. Mardia, "Statistical Shape Analysis,", Wiley Series in Probability and Statistics: Probability and Statistics, (1998). Google Scholar

[12]

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

[13]

Lawrence C. Evans, Partial differential equations and Monge-Kantorovich mass transfer,, Current developments in mathematics, (1999), 65. Google Scholar

[14]

_____, "Partial Differential Equations,", Second edition, 19 (2010). Google Scholar

[15]

Lawrence C. Evans and Ronald F. Gariepy, "Measure Theory and Fine Properties of Functions,", Studies in Advanced Mathematics, (1992). Google Scholar

[16]

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

[17]

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

[18]

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

[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. Google Scholar

[20]

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

[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. Google Scholar

[22]

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

[23]

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

[24]

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

[25]

Jürgen Jost and Xianqing Li-Jost, "Calculus of Variations,", Cambridge Studies in Advanced Mathematics, 64 (1998). Google Scholar

[26]

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

[27]

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

[28]

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

[29]

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

[30]

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

[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. doi: 10.1007/978-3-540-73273-0_23. Google Scholar

[32]

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

[33]

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

[34]

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

[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. Google Scholar

[36]

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

[37]

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

[38]

Svetlozar T. Rachev, "Probability Metrics and the Stability of Stochastic Models,", Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics, (1991). Google Scholar

[39]

A. Rényi, On measures of entropy and infromation,, Proc. 4th Berkeley Symp. Math. Stat. and Prob., (1961), 547. Google Scholar

[40]

R. Tyrrell Rockafellar, "Convex Analysis,", Princeton Mathematical Series, (1970). Google Scholar

[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. doi: 10.1023/A:1026543900054. Google Scholar

[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. Google Scholar

[43]

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

[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. doi: 10.1109/TPAMI.1987.4767872. Google Scholar

[45]

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

[46]

D. W. Thompson, "On Growth and Form,", Cambridge University Press, (1917). Google Scholar

[47]

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

[48]

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

[49]

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

[50]

S. Ullman, "The Interpretation of Visual Motion,", MIT Press, (1979). Google Scholar

[51]

Grace Wahba, "Spline Models for Observational Data,", CBMS-NSF Regional Conference Series in Applied Mathematics, 59 (1990). Google Scholar

[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. Google Scholar

[53]

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

[54]

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

[55]

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

[56]

Laurent Younes, Computable elastic distances between shapes,, SIAM J. Appl. Math., 58 (1998), 565. Google Scholar

[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. Google Scholar

[58]

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

[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. Google Scholar

[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. doi: 10.1109/34.865184. Google Scholar

[61]

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

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