Learning Facial Aging Models: A Face Recognition Perspective in .NET

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Learning Facial Aging Models: A Face Recognition Perspective
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where H corresponds to the Hessian matrix of f evaluated at ki . At the end of each iteration, is updated as illustrated in reference 37. Since the computation of k discussed above is based on the average facial measurements tabulated in reference 33 and does not involve facial measurements from test face images, such computations can be performed of ine. 11.2.2.3 Applying Aging Model on Faces
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On each of the test face images, we locate the 24 facial features illustrated in Figure 11.9 in a semiautomatic manner. We adopt the face detection and feature localization approach proposed by Moon et al. [38] to detect facial features such as eyes, mouth, and the outer contour of the face. This operation enables the location of the following facial landmarks (tr, gn: forehead and chin), (en, ex, ps, pi: eyes) and (ch, sto, ls, li: mouth). Other features designated as n, zy, go, and so on, do not correspond to corners or edges on faces and hence were located manually. We enforce bilateral symmetry while locating facial features. In our observation, minor errors in feature localization do not affect the proposed method to compute facial growth parameters. Next, using the growth parameters computed over selected facial landmarks k, we compute the growth parameters over the entire facial region. This is formulated as a scattered data interpolation problem [39]. On a cartesian coordinate system de ned over the face region, the growth parameters k = [k1 , k2 , . . . , kn ] correspond to parameters obtained at facial landmarks located in (x1 , y1 ), (x2 , y2 ), . . . , (xn , yn ). Our objective is to nd an interpolating function f : R2 R such that g(xi ) = ki , i = 1, . . . , n, (11.7)
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where xi = (xi , yi ) and the thin-plate energy functional E, a measure of the amount of bending in the surface, is minimized. The thin-plate energy functional is de ned as E=
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2 2 2 gxx (x) + 2gxy (x) + gxx (x) dx,
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(11.8)
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where is the region of interest (face region, in our case). Using the method of radial basis function, the interpolating function that minimizes the energy functional can be shown to take the form
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g(x) = c0 + c1 x + c2 y +
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i (|x xi |),
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where i s are real numbers, |.| is the Euclidean norm in R2 , and the linear polynomial c0 + c1 x + c2 y accounts for af ne deformations in the system. We adopt the thin-plate splines functions de ned as (x) = |x|2 log(|x|) as the basis functions. As illustrated in reference 39, to remove af ne contributions from the basis functions, we introduce additional constraints n i = n i xi = n i yi = 0. Equations (11.7) and i=1 i=1 i=1 (11.9) coupled with the constraints above, results in the following linear system of
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11.3 Discussions and Conclusions Table 11.2. Age Transformation Models Age Transformation From t0 years to t1 years (t1 > t0 )
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Growth Model Rti1 = Ri0 [1 + ki0 1 (1 cos( i0 ))] t it1 = i0 Rti1 =
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equations, the solution of which yields the interpolating function g. The linear system of equations is An n Pn 3 PT 3 n 03 3
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where A is a matrix with entries Ai,j = (|xi xj |) i, j = 1, . . . , n, P is a matrix with rows (1, xi , yi ), = ( 1 , . . . , n )T , and c = (c0 , c1 , c2 )T . Thus, the growth parameters computed at selected facial features using age-based anthropometric data is used to compute the growth parameters over the entire facial region. Upon computing the facial growth parameters over the facial region, the craniofacial growth model can be applied in a pixel-wise manner, to transform the age of faces. The transformation models for different age transformations are illustrated in Table 11.2. The proposed craniofacial growth model nds direct applications in predicting the appearances of children across different ages. Figure 11.13 illustrates the original age-separated image pairs of different subjects and the age-transformed face image that was obtained using the personalized growth model. Furthermore, the facial growth parameters corresponding to the speci c age transformation on each subject are illustrated, in the form of range maps. The varying intensities observed in the range maps re ect the different growth rates observed across different facial features across ages. One can identify certain gender-based facial growth patterns that are similar across subjects undergoing a similar age transformation. We evaluate the performance of the proposed facial aging model by performing face recognition across age progression on the FG-NET aging database [40]. For a detailed account on the experiments, please refer to reference 32.