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the object with normal

its old and new intensity values are respectively:

Together with equation 6.1 and 6.2, we have a formula of the ratio of the two

intensity values

Therefore the intensity at

under new lighting condition can be computed as

1.1.4 Interactive face relighting

In [Ramamoorthi and Hanrahan, 2001a] spherical harmonic basis functions

of irradiance environment map were visualized on sphere intuitively, which

makes it easy to modify lighting by manually changing the coefficients. Our

radiance environment map is the irradiance environment map scaled by constant

albedo. We can modify the coefficients in equation (6.6) to interactively create

novel radiance environment maps. Then these radiance environment maps can

be used to edit the lighting effects of the face appearance. Unlike [Ramamoorthi

and Hanrahan, 2001a], we do not need to know the face albedo.

1.2 Face relighting from a single image

Given a single photograph of a person's face, it is possible to approximate

the radiance environment map, using the technique described in Section 2.1.3

of Chapter 6,

Given a photograph and a generic 3D face geometry, we first align the face

image with the generic face model. Note that if the input image is a face texture

image, it is already aligned with geometry. From the aligned photograph and

the 3D geometry, we use the method described in Section 2.1.2 of Chapter 6

to approximate the radiance environment map. To relight the face image under

rotated lighting environment, we compute each face pixel's normal (with re-

spect to the lighting environment) before and after rotation. Then we compute

ratio where and are the new and old normal vectors of the

pixel respectively, and is the approximated radiance environment map.

Finally the new intensity of the pixel is given by equation (8.6).

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