Closed-form approximations to the volume rendering integral with Gaussian transfer functions

In direct volume rendering, transfer functions map data points to optical properties such as color and opacity. We have found transfer functions based on the Gaussian primitive to be particularly useful for multivariate volumes, because they are simple and rely on a limited number of free parameters. We show how this class of transfer function primitives can be analytically integrated over a line segment under the assumption that data values vary linearly between two sampled points. Analytically integrated segment can then be composited using standard techniques.