Mathematical Modeling with Multidisciplinary Applications

CHAPTER 13

FRACTALS, WITH APPLICATIONS TO SIGNAL AND IMAGE MODELING

H. KUNZE¹ AND D. LA TORRE²

¹Department of Mathematics and Statistics, University of Guelph, Canada

²Department of Economics, Business and Statistics, University of Milan, Italy

Looking for the definition of “fractals” on the internet, you find fractals are sets whose Hausdorff-Besicovitch dimension exceeds their topological dimension.

That description seems very heavy, so we will write instead that fractals are complicated, often irregular, sets generally produced by iteration of some kind of operation.

In the next section, we’ll see how to construct fractal sets through iterated function systems and we will develop an understanding of some odd facts about their dimension or size. In the subsequent section, we will see some modeling uses of fractals.

13.1 ITERATED FUNCTION SYSTEMS

We develop the ideas through two examples, in parallel. In each case, we consider the line segment of length one lying on the interval [0, 1] along the x-axis. We think of two operations:

1. Delete the middle third of the line segment. To be careful, we mean delete the open middle third of the line segment, , leaving behind the closed set on the x-axis.

2. Replace the middle third of the line segment by the other two sides of the corresponding ...