Enhanced Multi-Resolution Analysis for Multi-Dimensional Data Utilizing Line Filtering Techniques

TL;DR

This paper introduces LSIAC-MRA, a novel multi-resolution analysis method that uses convolution kernel post-processors to improve accuracy and error reduction in multi-dimensional data analysis, especially during scale transitions.

Contribution

The paper presents a new filtering-projection technique that enhances multi-resolution analysis by analytically evaluating convolutions and defining detail coefficients during data refinement.

Findings

Effective error reduction in 2D and 3D data

Improved accuracy in multi-resolution transitions

Analytical evaluation of convolutions for basis transformation

Abstract

In this article we introduce Line Smoothness-Increasing Accuracy-Conserving Multi-Resolution Analysis\linebreak (LSIAC-MRA). This is a procedure for exploiting convolution kernel post-processors for obtaining more accurate multi-dimensional multi-resolution analysis (MRA) in terms of error reduction. This filtering-projection tool allows for the transition of data between different resolutions while simultaneously decreasing errors in the fine grid approximation. It specifically allows for defining detail multi-wavelet coefficients when translating coarse data onto finer meshes. These coefficients are usually not defined in such cases. We show how to analytically evaluate the resulting convolutions and express the filtered approximation in a new basis. This is done by combining the filtering procedure with projection operators that allow for computational implementation of this scale transition procedure. Further, this procedure can be applied to piecewise constant approximations to functions, as it provides error reduction. We demonstrate the effectiveness of this technique in two and three dimensions.