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

In this article we introduce Line Smoothness-Increasing Accuracy-Conserving Multi-Resolution Analysis (LSIAC-MRA). This is a procedure for exploiting convolution kernel post-processors for obtaining more accurate multi-resolution analysis (MRA) in multiple dimensions. 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 allows for combining the filtering procedure with projection operators that allow for efficient computational implementation of this scale transition procedure. Further, this procedure can be applied to piecewise constant approximations to functions, contrary to the theory of SIAC filters. We demonstrate the effectiveness of this technique in two- and three-dimensions.