8 3D Face Recognition |
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leading diagonal allowing a suitable number of eigenvectors to be selected for the subspace, as in step 5 of the previous subsection.
8.7.3 PCA Testing
Once the above PCA training phase is completed, it is straightforward to implement a simple nearest neighbor face identification scheme, within the reduced k-dimensional space. We can also threshold a suitable distance metric to implement a face verification scheme.
Each test or probe face, xp , must undergo the same transformations as the training faces, namely subtraction of the training data mean and projection into the subspace:
x˜ pT = (xp − x¯ )T Vk . |
(8.21) |
Euclidean distance and cosine distance are common metrics used to find the nearest neighbor in the gallery. Given some probe face, x˜ p , and some gallery face, x˜ g , both of which have been projected into the PCA-derived subspace, the Euclidean distance between them is given as:
de (xp , xg ) |
xp |
xg |
= |
|
(xp |
xg )T (xp |
xg ) |
˜ ˜ |
= ˜ |
− ˜ |
|
˜ |
− ˜ ˜ |
− ˜ |
and the cosine distance is given as:
x˜ T x˜ g
dc (x˜ p , x˜ g ) = 1 − p .
x˜ p · x˜ g
(8.22)
(8.23)
In both cases, a small value of the metric (preferably close to zero) indicates a good match. In testing of a PCA-based 3D face recognition system Heseltine et al. [44] found that, usually the Euclidean distance outperformed the cosine distance, but the difference between the two metrics depended on the surface feature type (depth, curvature or gradient) and in a minority of cases, the cosine distance gave a marginally better performance.
The distance metrics described above don’t take any account of how the training data is spread along the different axes of the PCA-derived subspace. The Mahalanobis distance normalizes the spread along each axis, by dividing by its associated variance to give:
dm(xp , xg ) |
|
|
|
|
|
|
|
(8.24) |
= |
(xp |
xg )T D−1 |
(xp |
xg ). |
||||
˜ ˜ |
|
˜ |
− ˜ |
˜ |
− ˜ |
|
||
This expresses the distance in units of standard deviation. Note that the inverse of D is fast to compute due to its diagonal structure. Equivalently, we can whiten the training and test data, by premultiplying all feature vectors by D− 12 , which maps the covariance of the training data to the identity matrix, and then Eq. (8.22) for the Euclidean distance metric can be used in this new space. Similarly, we can use the cosine distance metric in the whitened feature space. Heseltine et al. [44] found that
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using information in D generally improved performance in their PCA-based 3D face recognition system. For many surface features, the cosine distance in the whitened space improved on the standard cosine distance so much that it became the best performing distance metric. This metric is also reported to be the preferred metric in the PCA-based work of Chang et al. [18].
Finally, 3D face recognition systems often display the match between the probe and the gallery. This can be done in terms of the original images, or alternatively the two m × 1 3D face vectors can be reconstructed from their k-dimensional subspace vectors as:
x = Vk x˜ + x¯ . |
(8.25) |
We summarize a PCA face recognition testing phase as follows:
1.Project the test (probe) face into the PCA derived subspace using Eq. (8.21).
2.For every face in the training data set (gallery), compute a distance metric between the probe and gallery. Select the distance metric with the smallest value as the rank-1 identification match.
3.Optionally display the probe and gallery as reconstructions from the PCA space using Eq. (8.25).
For a verification system, we replace step 2 with a check against the claimed identity gallery capture only, and if the distance metric is below some threshold, then the identity is verified. Performance metrics are then evaluated with reference to the true identities of probe and gallery, which are generally contained with the 3D face scan filenames. Obviously, for large scale performance evaluations, step 3 is omitted.
8.7.4 PCA Performance
PCA has been tested on 3D face datasets by many researchers. It is often used as a baseline to measure the performance of other systems (i.e. reported new systems are expected to be better than this.) As mentioned earlier, Chang et al. [19] report the PCA performance of rank-1 recognition on the FRGC dataset as 77.7 % and 61.3 % for neutral and non-neutral expressions respectively. Problems with the PCA include (i) a vulnerability to expressions due to the holistic nature of the approach and (ii) the difficulty to get good pose normalization, which is a requirement of the preprocessing stages of the method. The most time-consuming part of on-line face processing is usually the pose normalization stage, particularly if automatic feature localization and cropping is used as a precursor to this. Once we have sampled the face scan into a standard size feature vector, its projection into 3D face space is a fast operation (linear in the dimension of the feature vector) and, in a nearest neighbor matching scheme, matching time is linear in the size of the gallery.