Can One Break the “Collage Barrier” in Fractal Image Coding?

Most fractal image coding methods rely upon collage coding, that is, finding a fractal transform operator T c that sends a target image u as close as possible to itself. The fixed point attractor ū c of T c is generally a good approximation to u. However, it is well known that collage coding does not necessarily yield an optimal attractor, i.e., one for which the approximation error u − ū c is minimized with respect to variations in the fractal transform parameters. A number of studies have employed the “collage attractor” ū c as a starting point from which to obtain better approximations to u. In this paper, we show that attractors ū are differentiate functions of the (affine) fractal parameters. This allows us to use gradient descent methods that search for optimal attractors in fractal parameter space, i.e., local minima of the approximation error u ℒ ū. We report on results of corresponding computer experiments and compare them with those obtained by related (nondifferentiable) methods based on the simplex hill climbing and annealing approaches.