On the convergence of fractal transforms

This paper reports on investigations concerning the convergence of fractal transforms for signal modelling. Convergence is essential for the functionality of fractal based coding schemes. The coding process is described as non-linear transformation in the finite-dimensional vector space. Using spectral theory, a necessary and sufficient condition for the contractivity is derived from the eigenvalues of a special linear operator. In the same way some constraints for the choice of the encoding parameters are deduced which are less strict than those imposed so far. The proposed contractivity measure can be calculated directly from the transformation parameters during the encoding process. For complex encoding schemes the calculation of the eigenvalues may be infeasible. For those cases a contractivity criterion derived from the norm of the operator is suggested.