A Harten's Multiresolution Framework for Subdivision Schemes

TL;DR

This paper explores a reversed approach in Harten's multiresolution framework, allowing flexible selection of prediction and decimation operators in subdivision schemes, broadening their applicability in data manipulation.

Contribution

It introduces a method to invert the traditional order of choosing decimation and prediction, enabling more versatile subdivision schemes within multiresolution analysis.

Findings

Any decimation can be derived from a chosen prediction.

The approach broadens the types of data and schemes that can be used.

The framework enhances flexibility in multilevel data manipulation.

Abstract

Harten's Multiresolution framework has been applied in different contexts, such as in the numerical simulation of PDE with conservation laws or in image compression, showing its flexibility to describe and manipulate the data in a multilevel fashion. Two basic operators form the basis of this theory: the decimation and the prediction. The decimation is chosen first and determines the type of data that is being manipulated. For instance, the data could be the point evaluations or the cell-averages of a function, which are the two classical environments. Next, the prediction is chosen, and it must be compatible with the decimation. Subdivision schemes can be used as prediction operators, but sometimes they not fit into one of the two typical environments. In this paper we show how to invert this order so we can choose a prediction first and then define a compatible decimation from that prediction. Moreover, we also prove that any possible decimation can be obtained in this way.