3D Faces --> The Fishersurface Method

In this section we provide details of the fishersurface method of face recognition. We apply PCA and LDA (linear discriminant analysis) to surface representations of 3D face models, producing a subspace projection matrix, as with Belhumier et al’s fisherface approach [12], taking advantage of ‘within-class’ information, minimising variation between multiple face models of the same person, yet maximising class separation. To accomplish this we use a training set containing several examples of each subject, describing facial structure variance (due to influences such as facial expression), from one model to another. From the training set we compute three scatter matrices, representing the within-class (SW), between-class (SB) and total (ST) distribution from the average surface _ and classes averages _n, as shown in equation 1.

The training set is partitioned into c classes, such that all surface vectors ni in a single class Xn are of the same person and no person is present in multiple classes. Calculating eigenvectors of the matrix ST, and taking the top 250 (number of surfaces minus number of classes) principal components, we produce a projection matrix Upca. This is then used to reduce dimensionality of the within-class and between-class scatter matrices (ensuring they are non-singular) before computing the top c-1 eigenvectors of the reduced scatter matrix ratio, Ufld, as shown in equation 2.

Finally, the matrix Uff is calculated, such that it projects a face surface vector into a reduced space of c-1 dimensions, in which between-class scatter is maximised for all c classes, while within-class scatter is minimised for each class Xn. Like the fisherface system [12], components of the projection matrix Uff can be viewed as images, as shown in Figure 1 for the depth map surface space.



Figure 1: The average surface (left) and first five fishersurfaces (right)
Once surface space has been defined, we project a facial surface into reduced surface
space by a simple matrix multiplication, as shown in equation 3.


^=(p−!)TU (3)

The vector _T___1__2____c-1] is taken as a ‘face-key’ representing the facial structure in reduced dimensionality space. Face-keys are compared using either euclidean or cosine distance measures as shown in equation 4.



An acceptance (facial surfaces match) or rejection (surfaces do not match) is determined by applying a threshold to the distance calculated. Any comparison producing a distance value below the threshold is considered an acceptance.