Integrated Volume Rendering and Data Analysis in Wavelet Spaces

Abstract

The following paper describes a framework for volume data analysis and visualization using wavelet transforms (WTs). It bases on the idea that on the one hand WTs have approved to provide powerful features for various applications in the field of data analysis. Due to the basic properties of this transform, such as local support and orientation selectivity, many researchers tried to exploit the WT for the extraction of local data features. In particular, in image texture analysis, wavelet based feature extractors accomplished highly accurate segmentation results that can be extended straightforwardly to volumetric data sets. On the other hand, due to the compact coding properties of orthonormal wavelets the WT allows to decompose any finite energy function and to approximate it from its bases. Therefore we can develop rendering methods, that provide an approximate solution of the low albedo volume rendering equation. Since generally, the wavelets are not given in a closed form, the approach reported in this paper bases on a piecewise polynomial representation. Isosurfaces can easily be obtained from the data either by ray tracing of the bases or by simple marching cubes techniques. Hence, the WT provides a uniform data representation that features both data analysis and data visualization. The paper introduces first to the mathematical foundations of the WT, it reviews briefly different types of basis functions and it stresses implementation details using iterated QMF–pair filters. Furthermore, separable extensions to multiple dimensions are explained and it is elucidated, how local data features can be derived from the wavelet pyramid. For data analysis purposes, a newly developed image texture analysis pipeline contains a WT, a principal component analysis, normalization procedures and a neural network. Volume rendering is accomplished by projecting the 3D wavelets onto the viewing ray and by piecewise analytic integration of the rendering equation. The methods reported here are illustrated by various examples.