Shape parameters have been shown to be able to successfully discriminate between 3D faces in recognition and verification applications [PR05]. Papatheodorou et al have also shown that shape parameters have much higher discrimination ability compared to texture parameters in an eigen space. Therefore, having already constructed the statistical shape model; its parameters can be used in a surface similarity measure for analyzing reconstruction accuracy.
5.9.1
Eigen Distance
The 3D shape error may consist of an error between shape parameters (α) of the reconstructed 3D model (IRecons,3D) and the original 3D model (laser scanned model,IOriginal,3D). The Eu-
clidean distance between shape parameters (Ω) can then be given as:
ΩN(IOriginal,3D, IRecons,3D) = v u u t N X k=1 (αk Original− αkRecons)2 (5.14)
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Equation 5.14 is expressed in units of standard deviations (SD), since the unit for normalized shape parameters is also the standard deviations. Let’s call this distance measure an eigen distance. It must be mentioned that the Euclidean distance in the space of shape parameters is same as the Mahalanobis distance, and is therefore scale invariant. A small value for the eigen distance therefore means that shapes are close together whereas a large eigen distance means that shapes are far apart in some geometric sense.
0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00 0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00 Eigen Distance G e o m e tr ic D is ta n c e (m m )
Figure 5.7: Relationship between eigen distance (24 modes) and geometric distance.
The eigen distance calculated using normalized shape parameters has been frequently used to measure similarity between shapes. However in practice, the eigen distance is difficult to interpret as it can mean a number of different things as pointed out by Turk in his PhD thesis (Turk 1991) , the possibilities include:
1. The object of interest is near the centre of the face space and near the target face which might imply that face images are identical.
2. The object of interest is near the centre of the face space but is not identical to the target face.
0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00 0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60 1.80
Weighted Eigen Distance
G e o m e tr ic D is ta n c e (m m )
Figure 5.8: Relationship between weighted eigen distance (24 modes) and geometric distance. 3. The object of interest is far from the centre of the face space and the target face in which
case it is not a valid face.
4. The object of interest is far from the centre of the face space but it is near the target face in which case it is also not a valid face.
The fourth possibility demonstrates that a low eigen distance is not guarantee of accurate re- construction. The relationship between geometric similarity and corresponding eigen distances is illustrated in figure 5.7, where eigen distance varies along x-axis and corresponding geometric distance vary along y-axis. From the figure 5.7 it is clear that the eigen distance and geometric distance are not correlated which implies that the eigen distance is not a good measure of shape similarity in geometric sense. In addition to the possibilities given above, the experimentation has pointed to some other possibilities as well:
1. The face is near the centre of the face space, and is near target face in terms of eigen distance. But it is still not identical according to geometric distance.
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eigen distance. However the geometric distance tells a different story i.e. the face is close to the target face.
These two possibilities are explained by the fact that lower order parameters tend to vary freely during an optimization process as these contribute only small shape variations. As a result of large variation in the lower order parameters, the eigen distance varies unevenly when related with geometric similarity. Because of all these reasons, a weighted Eigen distance is explored in the next section.
5.9.2
Weighted Eigen Distance
The problem with the above similarity metric is that it is not very intuitive in a geometric sense. The lower order shape parameters contribute very little shape variation thus it is difficult to relate the cumulative changes in distance between shape parameters to actual anatomical shape differences.
To address this problem, the distances between individual parameters are weighted against shape variation of individual parameters. The weighted Eigen distance (Ωwt) is then given as:
ΩwtN(IOriginal,3D, IRecons,3D) = v u u t N X k=1 (ωk(αk Original− αkRecons)2) (5.15) where, ωi = λ i PN i=1λi
Here λ is the eigen value and N is the number of eigen modes used for calculating the distance. The main advantage of the weighted eigen distance is that it has a linear relationship with geometric similarity as shown in figure 5.8, where the weighted eigen distance is represented along the x-axis and geometric distance is represented along the y-axis.
a threshold of 1 standard deviation which is equivalent to approximately 4 mm. The threshold on the weighted eigen distance may vary with the number of modes. Both the thresholds can vary with changes in the population used for building the statistical shape model.
Figure 5.9: Histograms for average eigen distances (using first 24 modes) representing inter- subject shape variation for Notre Dame database.
5.9.3
Distance from the Mean Shape
Assuming a Gaussian shape distribution, the shape reconstruction process starts with the mean shape and aims for the ground truth. However, in some cases it will not move sufficiently in the direction of ground truth. In such a case, the distance of the reconstructed model from the mean shape will be much smaller than the distance from the ground truth. From figure 5.5, it is clear that a minimum Euclidean distance of approximately 2SD from the mean model can be used to differentiate the reconstructed model from the mean shape. Therefore, the threshold for the Euclidean distance from mean shape using shape parameters can be set as ω2 = 2SD.
In other cases, it might not move in the right direction resulting in the larger distance from both the ground truth and mean shape. Therefore the distance of the reconstructed model from the mean shape is another important metric to be considered during the shape reconstruction process.