Can One Break the \ Collage Barrier " in FractalImage Coding ?

Most fractal image coding methods rely upon collage coding , that is, nding a fractal transform operator Tc that sends a target image u as close as possible to itself. The xed point attractor uc of Tc 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 jju ? ucjj is minimized with respect to variations in the fractal transform parameters. A number of studies have employed the \collage attractor" uc as a starting point from which to obtain better approximations to u. In this paper, we show that at-tractors u are diierentiable functions of the (aane) 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 jju ? ujj. We report on results of corresponding computer experiments and compare them with those obtained by related (nondif-ferentiable) methods based on the simplex hill climbing and annealing approaches.