In order to obtain smoothly progressing renderings, we would like to compensate for the exponential dependence of rendered color on opacity as we blend and . This can be done by devising an appropriate .
In principle, there cannot exist an ideal compensating . The exact relationship between rendered color and opacity depends on the distance the ray travels through voxels with this opacity. Hence a globally applied cannot compensate at once for all mismatches since they have different thickness. Even a locally chosen cannot work, as different viewpoints cast different rays through the morph.
In practice, the mismatches between and are small in number and extent. Hence, the above theoretical objections do not prevent us from empirically deriving a successful . Our design goal is to compensate for the exponential relation of rendered color to opacity by interpolating opacities at the rate of an inverse exponential. The sigmoid curve given by
satisfies this requirement. It suppresses the contribution of 's opacity in the early part of the morph, the degree of suppression controlled by the blending parameter s. Similarly, the contribution of 's opacity is enhanced in the latter part of the morph. Figure 5h, illustrates the application of compensated interpolation to the morph of figure 5: in contrast to figure 5g, figure 5h looks very much like the human head, as an early frame in the morph sequence should.