On the Problem of Convergence in Fractal Coding Schemes

Most fractal coding schemes employ an iterative decoding algorithm in order to reconstruct the approximation of the original signal from the fractal code. A necessary condition for obtaining a unique solution is the convergence of the reconstruction process. This paper reports on investigations concerning a necessary and sufficient condition for convergence which is based upon the spectral radius of the transformation matrix. For a very general class of fractal transforms a simple calculation of the spectral radius can be performed in order to decide whether the reconstruction converges or not. This allows more freedom in the choice of the encoding parameters resulting in a better and faster reconstruction process. Also the proposed description leads to a more accurate theoretical foundation of fractal coding schemes.