6. Form the Weingarten matrix as:

7.Determine the principal curvatures, κ1, κ2 as the eigenvalues of the Weingarten matrix.

8.Form the Gaussian curvature, mean curvature and shape index, as required, using Eqs. (8.39) and (8.40).

Of course there are many variants of this approach. For example, in step 2, it is faster to compute normals using cross products rather than SVD, in which case we would use an arbitrary local basis to do the quadratic surface fit.

8.10 Recent Techniques in 3D Face Recognition

In this section we present a selection of promising recent techniques in the field of 3D face recognition including 3D face recognition using Annotated Face Models (AFMs) [51], local feature-based 3D face recognition [65], and expression-invariant 3D face recognition [15].

8.10.1 3D Face Recognition Using Annotated Face Models (AFM)

Kakadiaris et al. [50, 51] proposed a deformable Annotated Face Model (AFM), which is based on statistical data from 3D face meshes and anthropometric landmarks identified in the work of Farkas [32]. Farkas studied 132 measurements on the human face and head which are widely used in the literature. For example, DeCarlo and Metaxas [26] used 47 landmarks and their associated distance/angular measurements to describe a 3D deformable face model for tracking. Kakadiaris et al. [51] associated a subset of the facial anthropometric landmarks [32] with the vertices of the AFM and using knowledge of the facial physiology, segmented the AFM into different annotated areas. Figure 8.12 illustrates the AFM.

The input data is registered to the AFM using a multistage alignment procedure. Firstly, spin images [49] are used to coarsely align the data with the AFM. In the second step, the ICP algorithm [9] is used to refine the registration. In the final step, the alignment is further refined using Simulated Annealing to minimize the z-buffer difference. The z-buffers uniformly resample the data resulting in better registration. Uniform sampling was also used by Mian et al. [64] for better pose correction in an iterative approach. The AFM is fitted to the registered data using an elastically deformable model framework [62]. Additional constraints were imposed on the AFM similar to DeCarlo and Metaxas [26] to ensure that the fitting result is anthropometrically correct. Kakadiaris et al. [51] used a Finite Element Approximation to solve this equation and used a subdivision surface [60] as the model for greater flexibility

Fig. 8.12 Anthropometric landmarks on a 3D face mesh. Annotated Face Model (AFM) with different annotated areas shown different shaded. Demonstration of parameterization using a checkerboard texture mapped on the AFM. Figure courtesy of [50]

(compared to parametric surfaces). To speed up the fitting process, space partitioning techniques are used to reduce the cost of the nearest neighbor search per model vertex required at each iteration.

After the fitting, a continuous UV parameterization was applied to the AFM (polygonal mesh) transforming it to an equivalent geometry image. The geometry image has three channels each recording one of the three x, y or z position coordinates of the fitted AFM points. However, most of the information is contained in the z channel. Mapping between the AFM and the deformation image is performed using a spherical mapping function such that the ratio of area in 3D space to the area in UV space is approximately the same for every part of the surface.

A three channel (X, Y, Z) normal map is also extracted from the geometry image. The geometry and normal map images are treated as separate images and their individual wavelet coefficients are computed. Haar wavelets were used to extract features from the normal and geometry images. Additionally, the pyramid transform was also used to extract features from only the geometry images.

The Haar wavelet transform consisted of the tensor product of the 1D Walsh

tensor product resulted in the four channel filter bank gT g, gT h, hT g and hT h, which were applied to the images and the same subset of coefficients were used directly for comparison of two faces.

The pyramid transform [75] performs a linear multiscale, multiresolution image decomposition. The image is first divided into high and low-pass subbands and the latter is passed through a set of steerable bandpass filters resulting in a set of oriented subbands and a lower-pass subband. The latter is subsampled by a factor of two and recursively passed through the set of steerable bandpass filters resulting in a pyramid representation which is translation and rotation-invariant. These properties provide robustness to facial expressions. Kakadiaris et al. [51] applied 3 scale, 10 orientations decomposition and used the oriented subbands at the farthest scale only for comparison of two faces.

The AFM 3D face recognition system was evaluated on the FRGC v2 dataset and the Haar wavelet performed slightly better than the pyramid transform with approximately 97 % verification rate at 0.1 % FAR, as compared to 95 %. A fusion