Abstract
Some years ago, several authors tried to construct fractal surfaces which pass through a given set of data points. They used bivariable functions on rectangular grids, but the resulting surfaces failed to be continuous. A method based on their work for generating fractal interpolation surfaces is presented. Necessary conditions for the attractor of an iterated function system to be the graph of a continuous bivariable function which interpolates a given set of data are also presented here. Moreover, a comparative study for four of the most important constructions and attempts on rectangular grids is considered which points out some of their limitations and restrictions.
Highlights
Historical BackgroundAn iterated function system is a general method for constructing fractals; it makes the basis of most fractal-based image compression and pattern recognition methods
A fractal interpolation function can be considered as a continuous function whose graph is the attractor, a fractal set, of an appropriately chosen iterated function system; see [1] or [2]
In [18], a wide class of three-dimensional iterated function systems based on [14,16] was considered and it was shown that their attractors are a class of fractal interpolation surfaces
Summary
An iterated function system is a general method for constructing fractals; it makes the basis of most fractal-based image compression and pattern recognition methods. In [9], a different construction of an attractor that contains the interpolation points of a rectangular data set was introduced, but generally is not a graph of a continuous function. This methodology is repeated by the same authors in [10]. In [18], a wide class of three-dimensional iterated function systems based on [14,16] was considered and it was shown that their attractors are a class of fractal interpolation surfaces Another construction of bivariate fractal interpolation surfaces was presented in [19].
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