Fast over-complete wavelet implementation for spline family

We have studied the computational complexity of the over-complete wavelet representation for the commonly used spline wavelet family with an arbitrary order. By deriving a general expression for the complexity, it is shown that the inverse transform is nearly three times more costly in computation than the forward transform. In order to reduce the computational complexity, a new spatial implementation is proposed. This new implementation is based on the exploitation of redundancy between the lowpass and the bandpass outputs that is inherent to the over-complete wavelet scheme. It is shown that the new implementation can greatly simplify computation and give an efficient inverse structure.