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Random fractal surfaces (Mandeibrot surfaces) are finding more and more applications
in computer graphics, image analysis and the simulation of naturally occurring
topologies. A random fractal as a fractional geometry whose statistical properties
are scale invarient. In other words, the object looks similar (statistically) at all
magnifications. The generation of a random fractal surface involves the user having
to input two essential parameters: (i) the Fractal Dimension (a decimal number
D where 2 < D < 3) which controls the surface roughness and (ii) the seed of a
random number generator which determines the structure of the surface. By changing
the seed, the user can generate different surfaces and by increasing the fractal
dimension the surface roughness can be increased. In practice, algorithms of this
type do not allow the user to construct a random fractal with specific topological
features. Hence, in respect of the surface obtained, the user is ultimately at the
mercy of a random number generator.
In this paper, we address the problem of how to incorporate a priori information
into a Mandelbrot surface in such a way that the end product is still fractal. A
solution is provided to this problem which provides the user with control over the
general topology of the surface. We demonstrate its application for incorporating
low resolution data obtained from geographical/geological survey maps on the
topology of a given area. Also, we show how the method can be used to generate
synthetic terrain databases for the validation of certain surveying algorithms.
The technique employs the Fourier Synthesis Method for generating Mandelbrot
surfaces and is based on transmitting a predetermined proportion of the complex
Fourier coefficients used to describe a given topology. In addition to its use as a
complex terrain modeller, it is also shown how the same technique can be used for
data compression of general topologies. The idea here is to describe a surface in
terms of a few essential coordinate parameters (a prior information), a given seed
and a specific fractal dimension.
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