A new method to approximate the volume-rendering equation using wavelet bases and piecewise polynomials

Abstract

In the following paper, we describe a new generic method to find an approximate solution for the volume-rendering equation using hierarchical, orthonormal wavelet basis functions. The approach is basel on the idea that an initial volume data set can be decomposed into a pyramidal representation by means of a 3D wavelet transform. Once the wavelet function is described analytically, it is possible to approximate the volume density function. Moreover, when employing piecewise polynomial spline functions, as in our method, the rendering intergral can also be approximated, and gradient functions or related features of the data can be computed immediately from the approximation. Due to the localization properties of the wavelet transform both in space and in frequency on the one side anddue to the pyramidal subband coding scheme on the other side, this technique allows additionally for the control of the local quality of the reconstruction and provides elegantly for level-of-detail operations. Aside from the solution of the rendering equation itself, isosurfaces of the data can also be computed with either standard techniques, like marching cubes, or by more sophisticated algorithms that render the basis functions. All these additional rendering techniques can be embedded in a hybrid surface/volume-rendering scheme. In our paper, we elucidate this new concept and show its capabilities by different examples.