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On the Computation of Optimal Transport Maps Using Gradient Flows and Multiresolution Analysis

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Recent Advances in Learning and Control

Part of the book series: Lecture Notes in Control and Information Sciences ((LNCIS,volume 371))

Abstract

The optimal mass transport methodology has numerous applications in econometrics, fluid dynamics, automatic control, statistical physics, shape optimization, expert systems, and meteorology. Further, it leads to some beautiful mathematical problems. Motivated by certain issues in image registration, visual tracking and medical image visualization, we outline in this note a straightforward gradient descent approach for computing the optimal L 2 optimal transport mapping which may be easily implemented using a multiresolution scheme. We discuss the well-posedness of our scheme, and indicate how the optimal transport map may be computed on the sphere.

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References

  1. Angenent, S., Haker, S., Tannenbaum, A., Kikinis, R.: On area preserving maps of minimal distortion. In: Djaferis, T., Schick, I. (eds.) System Theory: Modeling, Analysis, and Control, pp. 275–287. Kluwer, Holland (1999)

    Google Scholar 

  2. Angenent, S., Haker, S., Tannenbaum, A., Kikinis, R.: Laplace-Beltrami operator and brain surface flattening. IEEE Trans. on Medical Imaging 18, 700–711 (1999)

    Article  Google Scholar 

  3. Angenent, S., Haker, S., Tannenbaum, A.: Minimizing flows for the Monge-Kantorovich problem. SIAM J. Math. Analysis 35, 61–97 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  4. Benamou, J.-D., Brenier, Y.: A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem. Numerische Mathematik 84, 375–393 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  5. Brenier, Y.: Polar factorization and monotone rearrangement of vector-valued functions. Comm. Pure Appl. Math. 64, 375–417 (1991)

    Article  MathSciNet  Google Scholar 

  6. Dacorogna, B., Moser, J.: On a partial differential equation involving the Jacobian determinant. Ann. Inst. H. Poincaré Anal. Non Linéaire 7, 1–26 (1990)

    MATH  MathSciNet  Google Scholar 

  7. Do Carmo, M.P.: Differential Geometry of Curves and Surfaces. Prentice-Hall, Inc., Englewood Cliffs (1976)

    MATH  Google Scholar 

  8. Feldman, M., McCann, R.J.: Monge’s transport problem on a Riemannian manifold. Trans. Amer. Math. Soc. 354, 1667–1697 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  9. Haker, S., Zhu, L., Tannenbaum, A., Angenent, S.: Optimal mass transport for registration and warping. Int. Journal Computer Vision 60, 225–240 (2004)

    Article  Google Scholar 

  10. Kantorovich, L.V.: On a problem of Monge. Uspekhi Mat. Nauk. 3, 225–226 (1948)

    Google Scholar 

  11. McCann, R.: Polar factorization of maps on Riemannian manifolds. Geom. Funct. Anal. 11, 589–608 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  12. Moser, J.: On the volume elements on a manifold. Trans. Amer. Math. Soc. 120, 286–294 (1965)

    Article  MATH  MathSciNet  Google Scholar 

  13. Niethammer, M., Vela, P., Tannenbaum, A.: Geometric observers for dynamically evolving curves. IEEE PAMI (submitted, 2007)

    Google Scholar 

  14. Rachev, S., Rüschendorf, L.: Mass Transportation Problems and Probability and Its Applications, vol. I, II. Springer, New York (1998)

    Google Scholar 

  15. Schröder, P., Sweldens, W.: Spherical wavelets: Efficiently representing functions on the sphere. In: Proceedings SIGGRAPH 1995 Computer Graphics, ACM Siggraph, pp. 161–172 (1995)

    Google Scholar 

  16. Schröder, P., Sweldens, W.: Spherical wavelets: Texture processing. In: Hanrahan, P., Purgathofer, W. (eds.) Rendering Techniques 1995, pp. 252–263. Springer, New York (1995)

    Google Scholar 

  17. Sweldens, W.: The Lifting Scheme: A New Philosophy in Biorthogonal Wavelet Constructions. In: Laine, A.F., Unser, M. (eds.) Proc. SPIE, Wavelet Applications in Signal and Image Processing III, pp. 68–79 (1995)

    Google Scholar 

  18. Sweldens, W.: The lifting scheme: A construction of second generation wavelets. SIAM J. Math. Anal. 29, 511–546 (1997)

    Article  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Electrical Engineering, Technion-Israel Institute of Technology, Haifa, Israel

    Ayelet Dominitz & Allen Tannenbaum

  2. Department of Mathematics, University of Wisconsin, Madison, WI, 53706, USA

    Sigurd Angenent

  3. Schools of Electrical & Computer and Biomedical Engineering, Georgia Institute of Technolgy, Atlanta, GA, 30332-0250, USA

    Allen Tannenbaum

Authors
  1. Ayelet Dominitz
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  2. Sigurd Angenent
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  3. Allen Tannenbaum
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Editor information

Editors and Affiliations

  1. Département d’Ingénierie Mathématique, Université catholique de Louvain, avenue Georges Lemaître, 4, B-1348, Louvain-la-Neuve, Belgium

    Vincent D. Blondel PhD

  2. Information Systems Laboratory Department of Electrical Engineering, Stanford University, Stanford, CA, 94305-9510, USA

    Stephen P. Boyd PhD

  3. Bio-Mimetic Control Research Center, RIKEN (The Institute of Physical and Chemical Research), Anagahora, Shimoshidami, Moriyama-ku, Nagoya, 463-0003, Japan

    Hidenori Kimura PhD

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© 2008 Springer-Verlag Berlin Heidelberg

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Dominitz, A., Angenent, S., Tannenbaum, A. (2008). On the Computation of Optimal Transport Maps Using Gradient Flows and Multiresolution Analysis. In: Blondel, V.D., Boyd, S.P., Kimura, H. (eds) Recent Advances in Learning and Control. Lecture Notes in Control and Information Sciences, vol 371. Springer, London. https://doi.org/10.1007/978-1-84800-155-8_5

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  • DOI: https://doi.org/10.1007/978-1-84800-155-8_5

  • Publisher Name: Springer, London

  • Print ISBN: 978-1-84800-154-1

  • Online ISBN: 978-1-84800-155-8

  • eBook Packages: EngineeringEngineering (R0)

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