A THEOREM ON PIFS'S MULTIRESOLUTION PROPERTIES
Sun Huai · Chinese Journal of Computers · 1999
It is clarified that multiresolution properties of PIFS(Partitioned Iterated Function System)'s fixed point under some conditions, i.e., the fixed points at low resolutions can be obtained from those at high resolutions through spatial contraction operator, while the fixed points at high resolutions can be obtained from those at low resolutions through affine operator, are just justifications of the multiresolution fractal decoding strategy first obtaining approximate fixed point through iterating at low resolution then increasing resolution using affine operator, and strict theoretical basis is lacked because strictly speaking, the multiresolution properties are held only for accurate fixed points. Further, it is proved theoretically that the result of first iterating k times at low resolution and then increasing resolution using affine transforming is equivalent to that of direct iterating k+1 times at high resolution when the values of all pixels of initial images for the two fractal decoding methods are the same. This additional condition is not a severe restriction, because the initial images are taken in the same way for most fractal decoding methods in literatures. So it is theoretically quarantined that compared with direct iterated fractal decoding, using the multiresolution properties of PIFS's fixed point can dramatically reduce computations in the fractal decoding while keeping the quality of the reconstructed image is not sacrificed, because most iterations are carried out at low resolution images.