Construction of stationary self-similar generalized fields by random wavelet expansion |
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Authors: | Zhiyi Chi |
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Institution: | (1) Department of Statistics, The University of Chicago, Chicago, IL 60637, USA. e-mail: chi@galton.uchicago.edu, US |
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Abstract: | Random wavelet expansion is introduced in the study of stationary self-similar generalized random fields. It is motivated
by a model of natural images, in which 2D views of objects are randomly scaled and translated because the objects are randomly
distributed in the 3D space. It is demonstrated that any stationary self-similar random field defined on the dual space of
a Schwartz space of smooth rapidly decreasing functions has a random wavelet expansion representation. To explicitly construct
stationary self-similar random fields, random wavelet expansion representations incorporating random functionals of the following
three types are considered: (1) a multiple stochastic integral over the product domain of scale and translate, (2) an iterated
one, first integrating over the scale domain, and (3) an iterated one, first integrating over the translate domain. We show
that random wavelet expansion gives rise to a variety of stationary self-similar random fields, including such well-known
processes as the linear fractional stable motions.
Received: 11 December 1998 / Revised version: 31 January 2001 / Published online: 23 August 2001 |
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