transformations on face pro les of children than were other transformation models. Upon studying different transformation functions that were proposed to characterize facial growth, Mark et al. [15] identi ed certain geometric properties of objects that remained invariant across growth related transformations. The geometric invariants that were identi ed are described below: r Preserving the angular coordinates of object features across transformations. r Preserving the continuity of all contours and their directions of curvature across transformations. r Preserving the bilateral symmetry of the object across transformations. Furthermore, they suggested that only those transformation functions that preserve the aforementioned geometric properties of objects undergoing the transformation could result in transformations associated with growth. Table 11.1 illustrates some of transformation functions that were proposed to model craniofacial growth [15, 16]. Figure 11.4 illustrates the outcomes of employing the transformation functions on pro le faces. The approaches discussed above limited their analysis to growth deformations induced on pro le faces (silhouettes) which invariably are devoid of intricacies associated with facial structures. Hence, the effectiveness of such transformation models in inducing growth-like transformations on real 3D faces were unclear. Mark and Todd [17] extended the revised cardioidal-strain transformation model into three dimensions and demonstrated the effectiveness of the model in simulating facial growth on 3D head structures of children. For a concise account on the above discussed approaches toward developing a craniofacial growth model, the readers are referred to references 18 and 19. On another note, O Toole et al. [20] studied the effects of inducing wrinkles and facial creases on 3D caricatures of human faces and observed that such variations had a direct impact on the perceived age the caricatured faces. Figure 11.5 illustrates the effects of inducing exaggerations on 3D caricatures. From a computer vision perspective, studies pertaining to facial aging largely address the following tasks: (i) age estimation, (ii) appearance prediction, and (iii) face recognition/veri cation across ages.

Table 11.1. Some Geometric Transformations that Were Proposed to Model Craniofacial Growth Applied Transformation Cardioidal strain (polar coordinates) Spiral strain (polar coordinates) Af ne shear (carthesian coordinates) Revised cardioidal strain (polar coordinates) Model t+1 = t Rt+1 = Rt (1 k cos( t )) t+1 = t Rt+1 = Rt (1 + k| t |) Yt+1 = Yt Xt+1 = Xt + kYt t+1 = t Rt+1 = Rt (1 + k(1 cos( t )))

11

Figure 11.4. Strain transformations versus shear transformations : An illustration of the effects of

applying varying combinations of the transformations on pro le faces. (This illustration was derived from that which originally appeared in reference 13.)

Age Estimation. Kwon and Vitoria da Lobo [21] observed that the facial aspect ratios underwent notable changes from infancy to adulthood and performed age-based classi cation of faces using facial aspect ratios and the density of wrinkles on predesignated facial regions. They classi ed faces as that of infants or young adults or senior adults. While facial aspect ratios were predominantly used to classify faces into that of infants or young adults, facial wrinkle density was used to classify faces into that of young adults or senior adults. Lanitis et al. [22, 23] performed a comparative study on different age-based classi ers. They represent faces by means of their shape ( ducial features) and texture (shape warped facial texture) and create an eigenspace to perform dimensionality reduction. Assuming that such a representation inherently captures the age information, they built regression functions and trained hierarchical neural-network-based classi ers to estimate the age from face images. Gandhi [24] trains a support vector regression machine to derive an age prediction function from face images. Geng et al. [25] learn the aging pattern subspace (addressed as AGES ) from a sequence of age progressed images of many individuals and use the same in estimating the age from face images. Appearance Prediction. Burt and Perrett [26] created prototype faces for different age groups using face samples from the respective age groups. They studied the variations in shape and texture between face prototypes from different age groups and observed that by incorporating such variations on real face images, the perceived age of the face images could be altered. But the average textures derived by averaging across warped face images belonging to the same age group were often devoid of wrinkles and other textural variations that are commonly associated with images from different age groups. Tiddeman et al. [27] extended the above approach by compensating for the loss of textural information in the facial prototypes that occurred during the blending process. Using wavelet-based methods, they created texture enhanced prototypes by adjusting the amplitude of edges in the composite edges. Following the shape and texture transformations described in reference 26, they transformed the texture of face images using locally weighted wavelet functions at different scales and orientations and thereby increased the perceived age of a face image. Both the aforementioned studies were performed on adult face images. Lanitis et al. [22] proposed age transformation functions for individuals in the age group 1 30 years. Representing faces by means of their shape and texture, as described earlier in this section, they proposed methods to transform the eigencoef cients in such