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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

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