FACE APPEARANCE SYNTHESIS USING FLEXIBLE APPEARANCE MODEL - 3D Face Processing: Modeling, Analysis and Synthesis - page 87

the old intensity (before rotation), we can obtain the new intensity
from the following equation:

Note that the REM could be approximated from a single input face

image.

1.1.2 Comparison with inverse rendering approach
It is interesting to compare our method with the inverse rendering approach.
Here we take the face rotating scenario as an example. To use inverse rendering
approach, we can capture the illumination environment map, and use spherical
harmonics technique [Ramamoorthi and Hanrahan, 2001a] to obtain
the diffuse components of the irradiance environment map. The reflectance
coefficient at point can be resolved from its intensity before rotation and
the irradiance, that is,

After rotation, its intensity is equal to From equation (8.7) and 8.4,

Thus, as expected, we obtain the same formula as equation (8.6).
The difference between our approach and the inverse rendering approach is
that our approach only requires a radiance environment map, while the inverse
rendering approach requires the illumination environment map (or irradiance
environment map). In some cases where only limited amount of data about the
lighting environment is available (such as a few photographs of some diffuse
objects), it would be difficult to separate illuminations from reflectance prop-
erties to obtain illumination environment map. Our technique allows us to do
image-based relighting of diffuse objects even from a single image.

1.1.3 Relighting in different lighting conditions
Let denote the old and new lighting distributions respectively. Suppose
we use the same material to capture the radiance environment maps for both
lighting conditions. Let and denote the radiance environment
maps of the old and new lighting conditions, respectively. For any point on