A new approach to morphing 2D affine IFS fractals

摘要

The objective of this paper is metamorphosis of 2D affine IFS attractors, which is done in a correct manner. The criteria of correctness can be described in brief as follows. First of all, the topology of intermediate shapes generated via morphing should not be sensitive to affine deformations of the key shapes―the property we call affine stability of morphing. Secondly, intermediate shapes should not be less similar to the key ones as the latter are to each other. In a special case, when the key shapes are affine images of each other, the criterion above states that every intermediate shape has to be affine similar to the key ones. Although the postulates are obvious, no prior method of blending fractals satisfies them in general. In contrast, the approach presented in this paper obeys all the conditions. Another problem we focus on is blending fractals specified by IFSs of different numbers of maps, and we propose a number of solutions. To satisfy all the conditions we face in this paper among others moment matrices of IFS invariant measures, the SO(2) Procrustes problem, and numerical stability of the process of morphing. We also introduce a number of new notions, including IFS similarity equivalence classes, the IFS canonical form, the IFS morphing space, and the special SVD. Finally, we point out other possible applications of the presented ideas and indicate open problems.