Abstract. We present a general theory of fractal transformations and show how it leads to new type of method for filtering and transforming digital images. This work substantially generalizes earlier work on fractal tops. The approach involves fractal geometry, chaotic dynamics, and an interplay between discrete and continuous representations. The underlying mathematics is established and applications to digital imaging are described and exemplified.Key words. Iterated function systems, dynamical systems, fractal transformations.AMS subject classifications. 37B10, 54H20, 68U101. Introduction. Fractal transformations are mappings between pairs of attractors of iterated function systems. They are defined with the aid of code space structures, and can be quite simple to handle and compute. They can be applied to digital images when the attractors are rectangular subsets of R 2 . They are termed "fractal" because they can change the box-counting, Hausdorff, and other dimensions of sets and measures upon which they act. In this paper we substantially generalize and develop the theory and we illustrate how it may be applied to digital imaging. Previous work was restricted to fractal transformations defined using fractal tops.Fractal tops were introduced in [2] and further developed in [5,6,7,11]. The main idea is this: given an iterated function system with a coding map and an attractor, a section of the coding map, called a tops function, can be defined using the "top" addresses of points on the attractor. Given two iterated function systems each with an attractor, a coding map, and a common code space, a mapping from one attractor to the other can be constructed by composing the tops function, for the first iterated function system, with the coding map for the second system. Under various conditions the composed map, from one attractor to the other, is continuous or a homeomorphism. In the cases of affine and projective iterated function systems, practical methods based on the chaos game algorithm [8] are feasible for the approximate digital computation of such transformations. Fractal tops have applications to information theory and to computer graphics. They have been applied to the production of artwork, as discussed for example in [4], and to real-time image synthesis [18]. In the present paper we extend the theory and applications.Much of the material in this paper is new. The underlying new idea is that diverse sections of a coding map may be defined quite generally, but specifically enough to be useful, by associating certain dynamical systems with the iterated function system. These sections provide novel collections of fractal transformations; by their means we generalize the theory and applications of fractal tops. We establish properties of fractal transformations, including conditions under which they are continuous. The properties are illustrated by examples related to digital imaging.A notable result, Theorem 5.3, states the existence of nontrivial fractal homeomorphisms between attractors of some affi...