Arbitrary affine transformation and their composition effects for two-dimensional fractal sets

摘要

Fractal sets can be generated by the iterated function system (IFS) codes using the contractive affine transformation. This paper presents various geometric affine transformations and their composition effects for two-dimensional (2-D) fractal sets. Here, the geometric transformations include translation, rotation, shearing, dilation/contraction, and reflection. First, a hierarchical fixed point-searching algorithm is proposed to determine the original coordinates of a 2-D fractal set directly from its IFS code. Then, the IFS code is modified according to the desired transformation. Instead of post processing the generated result, arbitrary affine transformation on the original fractal set can be directly obtained. On the other hand, the composite geometric transformations for 2-D fractal sets are also available. Finally, a complicated image frame can be synthesized by multiple 2-D fractal sets.