Discrete multi-resolution analysis and generalized wavelets

Abstract

In this paper we consider a situation where we are given a finite number of values which represent a sampling of weighted averages of a function ƒ(x) corresponding to a uniform grid. We show that is the weight function φ( x) satisfies a dilation equation, there is a discrete multi-resolution analysis of these values corresponding to a diadic coarsening of the grid. We introduce a reconstruction procedure R which predicts ƒx from its discrete weighted averages to any desired order of accuracy and is conservative in the sense that weighted averaging of R reproduces the given input data. Our formulation allows for adaptive data-dependent reconstruction techniques in which R is a nonlinear functional of the input data. At each level of resolution k we use the reconstruction R to predict ƒ(x) and its weighted averages at the ( k−1)th level, which is the next finer lever of resolution. We define Q k(x;ƒ) , the kth-scale component of ƒ(x), to be the difference between the reconstruction of ƒ(x) at level ( k−1) and that of level k, and { d k −1 j }, the kth-scale coefficients of ƒ(x), to be the weighted averages of Q k on the finer grid. We show that the given input data can be reconstructed from knowledge of the scale coefficients { d k j } for all k and the weighted averages of ƒ(x) at the coarsest grid. The observation leads to an efficient data compression technique. OnS the functional side, ƒ(x) can be reconstructed to the accuracy of the finest grid from knowledge of the scale components Q(x;ƒ) for all k and the reconstruction of ƒ(x) from the coarsest grid. When R is data-independent we show that each scale component Q k can be represented in a basis of linearly independent generalized wavelets. This leads to representation of ƒ(x) in a multi-resolution basis which is the union of these generalized wavelets for all levels of resolution. In this framework the original wavelets are obtained from a particular choices of reconstruction technique, namely taking R to be the projection of ƒ into the linear span of all dilates and translates of φ( x). This is a restrictive coupling between the approximation technique R and the sense of averaging φ, which is unnecessary from yhe point of view of numerical analysis.