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k = max i , j .di-elect cons. C k η ij min i C k , j .di-elect cons. C k η ij , ( 3 )

[0097]in which Ck denotes indexes of the k-th subject's faces, C=kCk and ηij denotes dissimilarities between faces i and j) for facial and canonical surface matching. This criterion is convenient being scale-invariant; it measures the tightness of each cluster of faces that belong to the same subject and its distance from other clusters. Ideally, k should tend to zero. The approach of the present embodiments, 9th column, outperforms rigid facial surface matching, the 10th column, by up to 790:3% in the sense of k.

We would like to find the "most isometric" embedding, the one that deforms the manifold distances the least. In practice, we have a finite discrete set of N manifold samples {xi}i=1N (represented as a 3 X N matrix X=(xn; . . . ; xN)) and a set of N2 mutual geodesic distances between these samples. We consider a mapping of the form φ: (M,d)→(m,d'), which maps the manifold samples xi into points x'i in an m-dimensional Euclidean space, such that the geodesic distances dij are replaced by Euclidean ones d'ij=∥x'i-x'j∥2




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