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AN AVERAGE OF SURFACES AS FUNCTIONS IN THE TWO-PARAMETER WIENER SPACE FOR A PROBABILISTIC 3D SHAPE MODEL

  • Received : 2019.05.06
  • Accepted : 2019.06.26
  • Published : 2020.05.31

Abstract

We define the average of a set of continuous functions of two variables (surfaces) using the structure of the two-parameter Wiener space that constitutes a probability space. The average of a sample set in the two-parameter Wiener space is defined employing the two-parameter Wiener process, which provides the concept of distribution over the two-parameter Wiener space. The average defined in our work, called an average function, also turns out to be a continuous function which is very desirable. It is proved that the average function also lies within the range of the sample set. The average function can be applied to model 3D shapes, which are regarded as their boundaries (surfaces), and serve as the average shape of them.

References

  1. R. G. Bartle, The Elements of Real Analysis, second edition, John Wiley & Sons, New York, 1976.
  2. R. H. Cameron and D. A. Storvick, An operator valued Yeh-Wiener integral, and a Wiener integral equation, Indiana Univ. Math. J. 25 (1976), no. 3, 235-258. https://doi.org/10.1512/iumj.1976.25.25020 https://doi.org/10.1512/iumj.1976.25.25020
  3. A. H. C. Chan, Some lower bounds for the distribution of the supremum of the Yeh-Wiener process over a rectangular region, J. Appl. Probability 12 (1975), no. 4, 824-830. https://doi.org/10.1017/s0021900200048798 https://doi.org/10.2307/3212734
  4. K. S. Chang, Converse measurability theorems for Yeh-Wiener space, Pacific J. Math. 97 (1981), no. 1, 59-63. http://projecteuclid.org/euclid.pjm/1102734653 https://doi.org/10.2140/pjm.1981.97.59
  5. L. Devilliers, S. Allassonniere, A. Trouve, and X. Pennec. Template estimation in computational anatomy: Frechet means top and quotient spaces are not consistent, SIAM J. Imaging Sci. 10 (2017), no. 3, 1139-1169. https://doi.org/10.1137/16M1083931 https://doi.org/10.1137/16M1083931
  6. B. A. Gutman, P. T. Fletcher, G. Fleishman, and P. M. Thompson, Reconstructing Karcher means of shapes on a Riemannian manifold of metrics and curvatures, in X. Pennec, S. Joshi, M. Nielsen, T. Fletcher, S. Durrleman, and S. Sommer, editors, Proceedings of the fifth International workshop on Mathematical Foundations of Computational Anatomy, pages 25-34. MFCA, 2015.
  7. G. W. Johnson and M. L. Lapidus, The Feynman integral and Feynman's operational calculus, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 2000.
  8. J.-G. Kim and B. S. Kim, A 3D shape model on the Yeh-Wiener space. I, International Congress of Mathematicians 2014, page 572. IMU, 2014.
  9. J.-G. Kim, J. A. Noble, and J. M. Brady, Probabilistic models for shapes as continuous curves, J. Math. Imaging Vision 33 (2009), no. 1, 39-65. https://doi.org/10.1007/s10851-008-0104-3 https://doi.org/10.1007/s10851-008-0104-3
  10. S. R. Paranjape and C. Park, Distribution of the supremum of the two-parameter Yeh-Wiener process on the boundary, J. Appl. Probability 10 (1973), no. 4, 875-880. https://doi.org/10.2307/3212390 https://doi.org/10.2307/3212390
  11. C. Park and D. L. Skoug, Distribution estimates of barrier-crossing probabilities of the Yeh-Wiener process, Pacific J. Math. 78 (1978), no. 2, 455-466. http://projecteuclid.org/euclid.pjm/1102806143 https://doi.org/10.2140/pjm.1978.78.455
  12. C. Park and D. L. Skoug, Grid-valued conditional Yeh-Wiener integrals and a Kac-Feynman Wiener integral equation, J. Integral Equations Appl. 8 (1996), no. 2, 213-230. https://doi.org/10.1216/jiea/1181075936 https://doi.org/10.1216/jiea/1181075936
  13. I. Pierce and D. Skoug, Comparing the distribution of various suprema on two-parameter Wiener space, Proc. Amer. Math. Soc. 141 (2013), no. 6, 2149-2152. https://doi.org/10.1090/S0002-9939-2013-11497-8 https://doi.org/10.1090/S0002-9939-2013-11497-8
  14. H. L. Royden, Real Analysis, third edition, Macmillan Publishing Company, New York, 1988.
  15. W. Rudin, Principles of Mathematical Analysis, third edition, McGraw-Hill Book Co., New York, 1976.
  16. W. Rudin, Real and Complex Analysis, third edition, McGraw-Hill Book Co., New York, 1987.
  17. D. Skoug, Converses to measurability theorems for Yeh-Wiener space, Proc. Amer. Math. Soc. 57 (1976), no. 2, 304-310. https://doi.org/10.2307/2041211 https://doi.org/10.1090/S0002-9939-1976-0422563-9
  18. J. Yeh, Wiener measure in a space of functions of two variables, Trans. Amer. Math. Soc. 95 (1960), 433-450. https://doi.org/10.2307/1993566 https://doi.org/10.1090/S0002-9947-1960-0125433-1
  19. J. Yeh, Cameron-Martin translation theorems in the Wiener space of functions of two variables, Trans. Amer. Math. Soc. 107 (1963), 409-420. https://doi.org/10.2307/1993809 https://doi.org/10.1090/S0002-9947-1963-0189138-6