A fast and efficient computational structure for the 2D over-complete wavelet transform

We 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 significantly more costly in computation than the forward transform. In order to reduce the computational complexity, a new spatial implementation is proposed. This new implementation exploits the 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 the computations, give an efficient inverse structure and allow the use of an arbitrary boundary extension method without affecting the ease of the inverse transform.