Choosing Number of Shape Parameters

Individual shape parameters contribute in different ways toward the overall shape variation. For a dense shape model which is used in this research, the contribution from individual shape parameters impacts on global as well as on local shape. The decision regarding the number of shape parameters depends on the minimum number of shape parameters needed for achieving a given accuracy and the role of the individual shape parameters in representing specific shape variation.

For example shape parameters 1, 2 & 3 contribute 34%, 24% and 11% of the overall shape vari- ation respectively; if these parameters are taken together then they contribute approximately 60% of the shape variation (see figure 5.1 and figure 5.2).

For 90-96% of shape variation 14-24 modes are needed. However to explain 99% of the shape variation 50 modes are needed (see figure 5.1). This clearly means that an additional 25 parameters are needed to explain an increase of only 3% of the shape variation. As shown in figure 5.2, parameters 16-25 and beyond individually represent less than 1% of shape variation.

5.3. Constraining Shape Space 111

This implies it is necessary to deal with a very high dimensional optimization problem for recovering very minute and person specific variation such as wrinkles or noise due to scanning artifacts. This kind of variation might be very difficult to capture using the feature space available and even if it is captured it is still very difficult to recover as the dimensionality of the parameter space shoots up thus making the optimization even more difficult.

The visual effect of varying individual shape modes between ±3 standard deviations is shown in figure 5.4. It is clear that the highest order modes are contributing toward global face size as well as the shape of the individual features such as the nose or lips. Thus, it is shown in figure 5.4 that using a small number of features it is still possible to capture local as well as global shape variation while using a shape model that encodes the overall variation in face shape.

The quantitative effect of varying individual shape modes while fixing their values between +3 standard deviations and -3 standard deviations is presented in table 5.1. It is clear that the highest order mode produces the most variation in terms of weighted eigen distance and geometric distance and the variation decreases as we move down the order.

In the table 5.1 along the x-axis and y-axis, mode 1 does give the highest shape variation in terms of geometric distance. However the decrease in shape variation is not uniform as we move down the order. When considering the z-axis mode 1 does not give the highest variation, instead the highest shape variation is given by mode 2.

What is perhaps surprising is the fact that geometric shape variation in quantitative terms is the same when the ith shape parameter is set to either +3 or -3 standard deviations despite giving totally different visual appearances as shown in figure 5.4. This indicates that shape varies by same amount when the shape parameter is moved by the same distance in either direction.

It has also been shown previously [Pap06] that most of the variation in the 3D face model comes from the eyes, nose and lips. At the same time most of the face area is taken up by the cheek and forehead thus the contribution of these areas might be marginalized in the statistical

shape model. This is achieved by placing a higher number of landmarks around the eyes, nose and lips.

It can be inferred that the decision regarding number of the parameters is also dependent on the characteristics of the feature space being used. Aspects of the feature space that are of interest to us are its discrimination power and spatial localization. Spatial localization describes how features are predominantly chosen from certain areas and not equally distributed over the whole.

Another challenging aspect is that there are a lot of scanner errors in the eyes, nose and lips. This implies that some of the shape variation in these areas might be noise. The feature space also has a limited ability to characterize minute shape variation because of scanning issues. The noisy features are often subject specific and are represented by parameters having less than 1% percentage of shape variation, these parameters are ignored.

This implies that 3D reconstruction can be achieved effectively using a global statistical shape model with a smaller number of parameters and a cost function that is able to represent global as well local shape variations.

Mode WtEigen Geometric Distance (mm) Distance (sd) xyz-axis x-axis y-axis z-axis 1 1.72 6.53 3.11 5.12 0.64 2 1.44 5.28 1.87 1.85 3.96 3 0.98 3.79 2.62 1.28 1.76 4 0.66 2.42 1.75 0.77 0.99 5 0.61 2.13 0.51 0.71 1.63 All 2.61 9.99 3.23 8.04 2.82

Table 5.1: Shape variation produced as each of the first 5 shape modes of mean shape is varied from +3 to −3 standard deviations.

5.3. Constraining Shape Space 113 0.00 10.00 20.00 30.00 40.00 50.00 60.00 70.00 80.00 90.00 100.00 1 5 9 _{13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93}

Total No. of Eigen Modes

% P e r S h a p e V a ri a ti o n

Figure 5.1: Total number of modes required for explaining given percentage of shape.

0.00 5.00 10.00 15.00 20.00 25.00 30.00 35.00 1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55 58 61 64 67 70 73 76 79 82 85 88 91 94 Ith Eigen Mode

% P e r S h a p e V a ri a ti o n

-3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 Mode1 M o d e 2

Figure 5.3: Distribution of first two modes (mode 1 & mode 2) showing these vary within ±3 standard deviation for the Notre Dame face database.

5.3. Constraining Shape Space 115 Modes -3 3 1st Mode 2nd Mode 3rd Mode 4th Mode 5th Mode All 5 Modes(±3)

Figure 5.4: Shape variation produced by varying each of the first five shape modes from -3 to +3. Also showing unlikely face shape produced as result of varying these shape modes.