Linear Interpolation of Opacity

Although interpolating the attenuation parameter ($\tau$) linearly is common, it can lead to problems. As the attenuation changes linearly, the opacity ($\alpha$), the fraction of incoming light occluded by the volume, changes exponentially. Because the observable effect is not changing in proportion to the modified parameter, building a transfer function is difficult.

Instead, a preferable option is to parameterize the opacity rather than the attenuation of the volume--a subtle but important difference. Wilhelms and van Gelder [102] give a simple relationship between the two.¹⁰

$$\alpha = 1 - e^{-\tau }$$ (62)

$$\tau = -\ln\left(1-\alpha \right)$$ (63)