For instance, in the three-dimensional case, where two variables X and Y are given as coordinates, the elevation function between any two points (x1, y1) and (x2, y2) can be set to have a mean or expected value that increases as the vector distance between (x1, y1) and (x2, y2).[1] There are, however, many ways of defining the elevation function. For instance, the fractional Brownian motion variable may be used, or various rotation functions may be used to achieve more natural looking surfaces.[2]
Generation of fractional Brownian surfaces
Efficient generation of fractional Brownian surfaces poses significant challenges.[4] Since the Brownian surface represents a Gaussian process with a nonstationary covariance function, one can use the Cholesky decomposition method. A more efficient method is Stein's method,[5] which generates an auxiliary stationary Gaussian process using the circulant embedding approach and then adjusts this auxiliary process to obtain the desired nonstationary Gaussian process. The figure below shows three typical realizations of fractional Brownian surfaces for different values of the roughness or Hurst parameter. The Hurst parameter is always between zero and one, with values closer to one corresponding to smoother surfaces. These surfaces were generated using a Matlab implementation of Stein's method.
Fractional Brownian surfaces for different values of the Hurst parameter. The larger the parameter, the smoother the surface.
^ abRuss, John C. (1994). Fractal surfaces, Volume 1. p. 167. ISBN0-306-44702-9.
^ abXie, Heping (1993). Fractals in rock mechanics. p. 73. ISBN90-5410-133-4.
^Vicsek, Tamás (1992). Fractal growth phenomena. p. 40. ISBN981-02-0668-2.
^Kroese, D.P.; Botev, Z.I. (2015). "Spatial Process Generation". Lectures on Stochastic Geometry, Spatial Statistics and Random Fields, Volume II: Analysis, Modeling and Simulation of Complex Structures, Springer-Verlag, Berlin: 369–404. arXiv:1308.0399. Bibcode:2013arXiv1308.0399K. doi:10.1007/978-3-319-10064-7_12.
^Stein, M. L. (2002). "Fast and exact simulation of fractional Brownian motion". Journal of Computational and Graphical Statistics. 11 (3): 587–599. doi:10.1198/106186002466. S2CID 121718378.
January 07, 2023
brownian, surface, fractal, surface, generated, fractal, elevation, function, single, realisation, three, dimensional, with, brownian, motion, named, after, 19th, century, biologist, robert, brown, contents, example, generation, fractional, also, referencesexa. A Brownian surface is a fractal surface generated via a fractal elevation function 1 2 3 A single realisation of three dimensional Brownian surface As with Brownian motion Brownian surfaces are named after 19th century biologist Robert Brown Contents 1 Example 2 Generation of fractional Brownian surfaces 3 See also 4 ReferencesExample EditFor instance in the three dimensional case where two variables X and Y are given as coordinates the elevation function between any two points x1 y1 and x2 y2 can be set to have a mean or expected value that increases as the vector distance between x1 y1 and x2 y2 1 There are however many ways of defining the elevation function For instance the fractional Brownian motion variable may be used or various rotation functions may be used to achieve more natural looking surfaces 2 Generation of fractional Brownian surfaces EditEfficient generation of fractional Brownian surfaces poses significant challenges 4 Since the Brownian surface represents a Gaussian process with a nonstationary covariance function one can use the Cholesky decomposition method A more efficient method is Stein s method 5 which generates an auxiliary stationary Gaussian process using the circulant embedding approach and then adjusts this auxiliary process to obtain the desired nonstationary Gaussian process The figure below shows three typical realizations of fractional Brownian surfaces for different values of the roughness or Hurst parameter The Hurst parameter is always between zero and one with values closer to one corresponding to smoother surfaces These surfaces were generated using a Matlab implementation of Stein s method Fractional Brownian surfaces for different values of the Hurst parameter The larger the parameter the smoother the surface See also EditWiener process Fractional Brownian motionReferences Edit a b Russ John C 1994 Fractal surfaces Volume 1 p 167 ISBN 0 306 44702 9 a b Xie Heping 1993 Fractals in rock mechanics p 73 ISBN 90 5410 133 4 Vicsek Tamas 1992 Fractal growth phenomena p 40 ISBN 981 02 0668 2 Kroese D P Botev Z I 2015 Spatial Process Generation Lectures on Stochastic Geometry Spatial Statistics and Random Fields Volume II Analysis Modeling and Simulation of Complex Structures Springer Verlag Berlin 369 404 arXiv 1308 0399 Bibcode 2013arXiv1308 0399K doi 10 1007 978 3 319 10064 7 12 Stein M L 2002 Fast and exact simulation of fractional Brownian motion Journal of Computational and Graphical Statistics 11 3 587 599 doi 10 1198 106186002466 S2CID 121718378 Retrieved from https en wikipedia org w index php title Brownian surface amp oldid 1057532344, wikipedia, wiki, book, books, library,
