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coefficients in equation 6.8 and obtain the approximated radiance environment

map.

Note that it is an under-constrained problem to determine all the 9 coefficients

from a single frontal image of a face according to [Ramamoorthi‚ 2002]. To

produce plausible illumination approximation results without conflicting with

the information provided by the frontal image‚ we make assumptions about

lighting in the back to constrain the problem. One of the assumptions that

we make is to assume a symmetric lighting environment‚ that is‚ the back has

the same lighting distribution as the front. This assumption is equivalent to

assuming for in equation (6.6). The

rest of the coefficients can then be solved uniquely according to [Ramamoorthi‚

2002]. One nice property about this assumption is that it generates the correct

lighting results for the front‚ and it generates plausible results for the back

if faces rotate in the lighting environment. For applications which deal with

only frontal lighting‚ the symmetric assumption produces correct results. If

we want to synthesize face appearance after the face is rotated in the lighting

environment‚ we need to make the assumption based on the scene. For example‚

in the cases where the lights mainly come from the two sides of the face‚ we

use symmetric assumptions. In the cases where the lights mainly come from

the front‚ we assume the back is dark.

2.2 Reduce person dependency based on ratio-image

technique

2.2.1 Ratio image

In Section 2.1.1‚ we have derived the formula (equation 6.4) for the intensity

of the neutral face point

at

After the face surface is deformed‚ the

intensity of

is

We denote

It can be observed that

called the ratio image‚ is independent of surface

reflectance property

[Liu et al.‚ 2001a]. Therefore‚

can be used

to as a facial motion representation independent of face albedos.

2.2.2 Transfer motion details using ratio image

The albedo-independency of ratio image give a novel representation of facial

motion field which is less person dependent than the original image. Liu et

al. [Liu et al.‚ 2001a] use this property to map facial expressions from one

person to another and achieve photo-realistic results.

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