Sub-band operators and saddle point wavelets

In this paper, a concept of sub-band operator is defined, a 2-circular matrix method is developed and the exact bounds of the sub-band operators are obtained. The method to calculate the bounds of the sub-band operators is described by computing the maximum of a function. In addition, it is found that the size of bound of a sub-band operator is very sensitive to the performance of wavelet transforms. So a model to minimize the bounds is built and then a class of wavelets, namely, saddle point wavelets, is designed. Experiments show that some saddle point wavelets with the filters of even lengths perform better than the well-known 9-7-tap wavelet in terms of image compression.