l∞-Stability for linear multiresolution algorithms: A new explicit approach. Part I: The basic rules and the Daubechies case

This general study is motivated by recent experiments showing that a multiresolution scheme without control in the infinity norm can produce numerical artifacts. This class of stability for the Mallat's multiresolution transform associated to orthogonal wavelet filters that belong to the class of linear multiresolution algorithms is revisited. Explicit error bounds in the infinity norm are presented by using an appropriate reformulation of the successive convolutions of a vector and assuming a contraction property. In the case of the decomposition an alternative normalization is necessary. We apply our general stability framework to the specific case of Daubechies' filters. The knowledge of explicit error bounds has some advantages in real problems as: industry applications, medical pathologies or FBI fingerprint compression. Our workable bounds give the level of compression necessary to recover the signal with a reconstruction error smaller than a prefixed tolerance. Our study presents two important aspects, the first one is the fact that we give precise error bounds and the second one is that we only use basic rules that can be understood for a wide part of the scientific community. Moreover, the results should be useful in mathematical, medical, physical, biological and engineering applications.