Browse > Article
http://dx.doi.org/10.4134/BKMS.b190467

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

Kim, Jeong-Gyoo (School of Games Hongik University)
Publication Information
Bulletin of the Korean Mathematical Society / v.57, no.3, 2020 , pp. 751-762 More about this Journal
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.
Keywords
Average of the set of two-variable functions; two-parameter Wiener space; two-parameter Wiener process; average of surfaces;
Citations & Related Records
연도 인용수 순위
  • Reference
1 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   DOI
2 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   DOI
3 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   DOI
4 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   DOI
5 R. G. Bartle, The Elements of Real Analysis, second edition, John Wiley & Sons, New York, 1976.
6 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   DOI
7 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   DOI
8 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   DOI
9 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   DOI
10 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   DOI
11 H. L. Royden, Real Analysis, third edition, Macmillan Publishing Company, New York, 1988.
12 W. Rudin, Principles of Mathematical Analysis, third edition, McGraw-Hill Book Co., New York, 1976.
13 W. Rudin, Real and Complex Analysis, third edition, McGraw-Hill Book Co., New York, 1987.
14 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   DOI
15 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   DOI
16 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   DOI
17 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.
18 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.
19 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.