Human Face Recognition 
3D Face Matching  

3D Face matching As discussed before, statistical models of 3D faces have shown promising results in face recognition and also outside face recognition. The basic premise of statistical face models is that given the structural regularity of the faces, one can exploit the redundancy in order to describe a face with fewer parameters. To exploit this redundancy, dimensionality reduction techniques such as PCA can be used. For 2D face images the dimensionality of the face space depends on the number of pixels in the input images. For 3D face images it depends on the number of points on the surface or on the resolution of the range images. Let us assume a set of 3D faces Γ1,Γ2,Γ3…..ΓM can be described as surfaces with n surface points each. The average 3D face surface is then calculated by:
and using the vector difference
the covariance matrix C is computed by:
An eigenanalysis of C yields the eigenvectors ui and their associated eigenvalues λi sorted by decreasing eigenvalue. All surfaces are then projected on the facespace by:
where k = 1, ...,m. In analogy to active shape models in 2D (Cootes et al., 1995), every 3D surface can then be described by a vector of weights β^{T}= [β_{1}, β_{2}, ..., β_{m}], which dictates how much each of the principal eigenfaces contributes to describing the input surface. The value of m is application and dataspecific, but in general a value is used such that 98% of the population variation can be described. More formally :
The similarity between two faces A and B can be assessed by comparing the weights β_{A} and β_{B} which are required to parameterize the faces. We will use two measurements for measuring the distance between the shape parameters of the two faces. The first one is the Euclidean distance which is defined as:
In addition it is also possible calculated the distance of a face from the featurespace . This effectively calculates how “face”like the face is. Based on this, there are four distinct possibilities: (1) the face is near the featurespace and near a face class (the face is known), (2) the face is near the featurespace but not near a face class (face is unknown), (3) the face is distant from the featurespace and face class (image not a face) and finally (4) the face distant is from featurespace and near a face class (image not a face). This way images that are not faces can be detected. Typically case (3) leads to false positives in most recognition systems. By computing the sample variance along each dimension one can use the Mahalanobis distance to calculate the similarity between faces . In the Mahalanobis space, the variance along each dimension is normalized to one. In order to compare the shape parameters of two facial surfaces, the difference in shape parameters is divided by the corresponding standard deviation σ:
In principle it should also be possible to construct 3D face models which are optimal in some sense, e.g. with regard to a certain performance metric in face recognition. This would entail an optimization of the correspondences across all faces (e.g. by using a groupwise in such a way that the resulting model produces the best possible face recognition performance. Finally, the 3D statistical face models discussed so far include only shape information. Of course texture is also a very important aspect of the face and should be included into the 3D statistical face model (similarily to the 3D morphable face model .
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