An alternate model for the face is an ellipsoid or other simple geometric structure such as a cylinder as in Figure . Unlike the ``thin sheet'' model which cannot account for yaw or pitch, the ellipsoid has the ability to roughly mimic the out-of-plane rotations the face can undergo. This is due to the curvature of the ellipsoid which exhibits non-homogenous warping in a 2D sense. Unfortunately, a simple ellipsoid cannot encompass all the nuances of the face and fully normalize its 2D projection. For example, the nose can cause occlusion by rotating in front of the cheek. In addition, the human head is not quite ellipsoidal and is difficult to approximate with standard 3D geometric models.
Clearly, the most accurate 3D model of a face would be the true 3D range data of the individual obtained from laser range-finder scanning. This cumbersome process is not only time-consuming and non-automated, it requires the use of sophisticated equipment which is not readily available4.1. Some sample data obtained from such devices is shown in Figure as radial range and radial intensity images. The images are in a cylindrical coordinate system and the axes are appropriately labelled.
From the radial range data, we compute a polygonal mesh by converting the cylindrical coordinates into Cartesian form. The Cartesian 3D data can then be rendered and displayed as shown in Figure (a). Subsequently, we can ``colorize'' the 3D model with the radial intensity data and obtain a texture-mapped 3D model of the individual as shown in Figure (b). This 3D model can then be used to synthesize any view of the individual by treating the head as a rigid object and rotating and translating it with 6 degrees of freedom (see Figure and ).
Unfortunately, we do not and cannot have a 3D model for each individual that we will photograph for our recognition system. Thus, we shall attempt to use another individual's 3D model to normalize the photograph under the assumption that the 3D structure of most faces is somewhat constant. Therefore, we can use one 3D model of a face and texture-map new photographed faces onto it. Unfortunately, some individuals will have thinner or wider faces and the model will not fit them as well as it did with the original texture. We suggest deforming the model along its vertical axis to stretch or squash it to fit it to the new face, as shown in Figure . Ideally, we would like to deform the model arbitrarily with various small stretchings and warpings so that it can be locally adapted to each new individual. However, such a process is quite computationally expensive. Nevertheless, the single vertical stretch of the model and its six degrees of freedom gives us quite a good approximation of the faces we will encounter and is, by far, more accurate than the planar or ellipsoidal models used in previous experiments.