关于PIFS多分辨率特性的一个补充定理

虽然多分辨率分形解码策略是根据ＰＩＦＳ（ＰａｒｔｉｔｉｏｎｅｄＩｔｅｒａｔｅｄＦｕｎｃｔｉｏｎＳｙｓｔｅｍ）不动点在一定条件下的多分辨率特性提出的，但其优越性仍缺乏理论保证．本文从理论上证明了，在进行分形图像解码时，采用多分辨率策略先进行ｋ次低分辨率迭代再进行分辨率提升所得结果与直接进行ｋ＋１次高分辨率迭代所得结果是相同的，从而为这种策略提供了完备的理论依据，从理论上保证了，与直接迭代解码方法相比，采用多分辨率解码方法能在不损失重建图像质量的前提下大大减少计算开销，因为绝大多数迭代是在低分辨率图像上进行的．

A THEOREM ON PIFS'S MULTIRESOLUTION PROPERTIES

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.