An often used approximation for the volume rendering integral is to average the luminance over the ray segment [64,86,102]. Although the computation is fast enough to compute in real time, the errors introduced by the approximation are noticeable. A closed form for the volume rendering integral with linearly interpolated luminance and attenuation is known [104,105], but it takes more than an order of magnitude longer to compute than the approximation, making it suitable only for off-line rendering.
In Section 1.2, I introduced partial pre-integration. Partial pre-integration performs the same linear interpolation on luminance and attenuation as [105], but in a fraction of the time. Partial pre-integration is accurate yet fast enough to use in real time applications.
Rather than linearly interpolate attenuation, linearly interpolating opacity can facilitate the building of transfer functions. There is no closed form of the volume rendering integral for linearly interpolated luminance and opacity, but [102] provides a rough approximation. In Section 1.3, I provide improvements to the approximation that eliminate visual artifacts.