A Theoretical Analysis of the Projection Error onto Discrete Wavelet Subspaces

A filterbank decomposition can be seen as a series of projections onto several discrete wavelet subspaces. In this presentation, we analyze the projection onto one of them—the low-pass one, since many signals tend to be low-pass. We prove a general but simple formula that allows the computation of the $l _{ 2 }$ -error made by approximating the signal by its projection. This result provides a norm for evaluating the accuracy of a complete decimation/interpolation branch for arbitrary analysis and synthesis filters; such a norm could be useful for the joint design of an analysis and synthesis filter, especially in the non-orthonormal case. As an example, we use our framework to compare the efficiency of different wavelet filters, such as Daubechies' or splines. In particular, we prove that the error made by using a Daubechies' filter downsampled by 2 is of the same order as the error using an orthonormal spline filter downsampled by 6. This proof is valid asymptotically as the number of regularity factors tends to infinity, and for a signal that is essentially low-pass. This implies that splines bring an additional compression gain of at least 3 over Daubechies' filters, asymptotically.