Exact solution of an inverse problem for affine images of IFS attractors

Abstract

A contractive iterated function system (IFS) defined on a complete metric space X possesses a unique compact attractor F⊂X$F\subset X$. Obtaining the transformations of F is generally nontrivial, due to the nature of construction of F. In this work, we review a method based on barycentric coordinates for constructing affine transformations of F (translation, scaling, reflection, rotation, shear, and their compositions) and develop its stronger theoretical framework. This approach yields an exact solution to a specific inverse problem: given the affine image w(F)$w(F)$ as a target set, we derive an explicit formula for the IFS whose fixed point is exactly w(F)$w(F)$. Taking into account the definition of the Hutchinson operator, we also obtain the corresponding Hutchinson operator, whose fixed point is also w(F)$w(F)$. Pseudocode and examples are included to facilitate implementation and verification.