Figure 11.12. The gure illustrates the drift of facial features observed on average faces at

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different ages, for men and women. The origin of reference that was identi ed for the two classes is illustrated as well.

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Learning Facial Aging Models: A Face Recognition Perspective

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Computing Facial Growth Parameters

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Upon computing the origin of reference for the craniofacial growth model, the facial landmarks for different ages are represented in polar coordinates ((xi , yi ) (ri , i ) where i corresponds to the feature index and j corresponds to the age in years. Let the growth parameters corresponding to facial landmarks designated by [tr, n, sn, ls, sto, li, sl, gn, en, ex, ps, pi, zy, al, ch, go] be k = [k1 , k2 , . . . , k16 ], respectively. Assuming bilateral symmetry of faces, symmetric facial features share the same growth parameters and hence the 24 facial features result in 16 unique growth parameters. The 52 proportion indices that were discussed in the previous section play a fundamental role in computing the facial growth parameters. By studying the transformation in proportion indices from ages u years to v years, we can compute the facial growth parameters corresponding to the speci c age transformation. The agebased proportion indices translate into linear and nonlinear equations in facial growth parameters. Proportion indices derived from facial measurements that were extracted across facial features that lie on the same horizontal or vertical axis result in linear equations in the respective growth parameters, and those extracted across features that do not lie on the same horizontal or vertical axis result in nonlinear equations in growth parameters. n gn For example, the age-based transformation observed in the proportion index zy zy on features n, gn, and zy, for an age transformation from u years to v years, results in a linear equation in the relevant growth parameters. The following equations illustrate u u u the same. (Ru , n , Ru , gn , Ru , zy , and cv were derived from the projective facial n gn zy measurements provided in reference 33). Rv Rv (n gn)v gn n = cv = cv (zy zy)v 2 Rv cos( zy ) zy Ru (1 + kgn (1 cos( gn ))) Ru (1 + kn (1 cos( n ))) gn n = 2 cv cos( zy ) Ru (1 + kzy (1 cos( zy ))) zy 1 kgn + 2 kn + 3 kzy = 1 . (11.5) (11.4)

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Similarly, the age-based transformation observed in the proportion index sto gn gn zy on features sto, gn, and zy, for an age transformation from u years to v years, results in a nonlinear equation in the relevant growth parameters, as illustrated below. (Again, u u u Ru , gn , Ru , sto , Ru , zy , and dv were derived from the projective facial measuregn sto zy ments provided in reference 33). (sto gn)v = dv (gn zy)v Rv Rv sto gn (Rv Rv sin( zy ))2 + (Rv cos( zy ))2 gn zy zy = dv

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11.2 Age Progression During Formative Years

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Ru (1 + ksto (1 cos( sto ))) Ru (1 + kgn (1 cos( gn ))) = sto gn {[Ru (1 + kgn (1 cos( gn ))) Ru (1 + kzy (1 cos( zy ))) gn zy sin( zy )]2 + [Ru (1 + kzy (1 cos( zy ))) cos( zy )]2 )} 2 dv zy

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2 2 1 ksto + 2 kgn + 3 kzy + 4 ksto + 5 kgn 2 + 6 kzy + 7 ksto kgn + 8 kgn kzy = 2 .

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Thus, the set of 52 proportion indices that were identi ed for our study result in a set of linear and nonlinear equations on growth parameters solving whereby one can identify the growth parameters for speci c age transformations. Let the constraints derived using proportion indices be denoted as r1 (k) = 1 , r2 (k) = 2 , . . . , r52 (k) = 52 . The objective function f (k) that needs to be minimized with respect to k is de ned as f (k) = 1 2

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(ri (k) i )2 .

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(11.6)

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The following equations illustrate the constraints that were derived using different facial proportion indices. n gn (1) (1) (1) r1 : = c1 1 k1 + 2 k7 + 3 k12 = 1 zy zy r2 : r3 : r4 : al al (2) (2) = c2 1 k13 + 2 k14 = 2 ch ch li sl (3) (3) (3) = c3 1 k4 + 2 k5 + 3 k6 = 3 sto sl sto gn (4) (4) (4) 2 (4) 2 = c4 1 k4 + 2 k7 + 3 k4 + 4 k7 gn zy

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2 + 5 k12 + 6 k12 + 7 k4 k7 + 8 k7 k12 = 4 (4) (4) (4) (4)

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( i and i are constants. ci is the proportion index value computed from the ratios j of mean values of facial measurements corresponding to the target age, which were obtained from reference 33.) To compute the growth parameters k, we minimize the objective function in an iterative fashion using the Levenberg Marquardt nonlinear optimization algorithm [37]. We use the craniofacial growth model de ned in Eq. (11.2) to compute the initial estimate of the facial growth parameters. The initial estimates are obtained using the age-based facial measurements provided for each facial landmark, individually. The iterative step involved in the optimization process is de ned as ki+1 = ki (H + diag[H]) 1 f (ki ),

f (ki ) =

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ri (k) ri (k),

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