An experiment was conducted to evaluate the effectiveness of tridimensional regression and its improvement over bidimensional regression. Three-dimensional landmark data obtained from human faces were used for this purpose. The landmarks were obtained by placing reflective markers on the faces of subjects and tracking the coordinates as the subjects moved through a series of poses using automated software. The landmarks were adapted from [14
]. They are shown in and described in .
Landmarks used for evaluating tridimensional regression.
Table 1 Description of landmarks used for evaluation (adapted from ).
The landmarks were obtained for three subjects at two different sittings and five poses per sitting. The objective was to compare R2 values within a subject to the R2 values between subjects using both tridimensional regression and bidimensional regression. One would expect the degree of similarity to be higher, thus a higher R2 value, for two samples from the same person than for samples from two different people. All pairwise R2 values were calculated for bidimensional and tridimensional regressions. Poses of the same individual within a sitting were not compared since the markers were not removed between poses and using these poses would result in inflated R2 values.
For each transformation, both in two and three dimensions, the distributions of R2 values for within and between subjects were obtained by fitting a theoretical distribution over the histograms of observed values. Overlaying these theoretical distributions allowed for the estimation of a threshold value (τ) as a cutoff for determining if two images were from the same subject. R2 values greater than τ lead to the decision that the two images are of the same subject (match) while R2 values less than τ indicate that the images are of two different subjects (nonmatch). The threshold value was determined to be where the two distributions cross, as to simultaneously minimize the false-positive and false-negative error rates. A false positive is when images of two different subjects are incorrectly determined to be from the same subject (an R2 value greater than τ for different subjects); a false-negative occurs when two images from the same subject are incorrectly determined to be from different subjects (an R2 value less than τ for the same subject). In addition to calculating the observed error rates, the expected error rates were found by evaluating the cumulative distribution functions of the R2 values at τ. summarizes the observed and expected error rates, and Figures , , and show the within-subject (dotted line) and between subject (solid line) distributions for each transformation.
Error rates for each transformation.
Within and between person R2 for bidimensional (l) and tridimensional (r) Euclidean transformation.
Within and between person R2 for bidimensional (l) and tridimensional (r) affine transformation.
Within and between person R2 for bidimensional (l) and tridimensional (r) projective transformation.
shows that both the observed and expected error rates for tridimensional regression are much smaller than those for bidimensional regression using any of the three transformations. Bidimensional regression resulted in both error rates being very high, false-positives often over fifty percent. Tridimensional regression shows a substantial decrease in both false-positive and false-negative error rates which indicates that the three-dimensional method is better at correctly matching a subject to him or herself.
In this application, the Euclidean and affine transformations were comparable to one another with the affine performing slightly better. The projective transformation had the largest observed false-positive rate. This result is not surprising as the flexibility of the projective transformation allows it to map objects into many other shapes. This flexibility results in the ability to match even two very dissimilar objects quite well with certain transformation parameters. Consequently, the R2 values are very high for all matches. This shifts the between-person distribution closer to the within person-distribution which results in a larger false-positive error rate.
Additionally, a sixth pose was taken on each of the subjects in each setting. This pose was not used to build the within- and between-subject distributions, or to determine the threshold. These six sets of points (two for each subject) were compared to all other poses not taken in the same setting of the same subject (30 comparisons per pose, 6 possible correct matches). The highest R2 for all six was a correct match. In addition, a minimum of the top 4 matches were correct matches, illustrating that tridimensional regression can be very good at identifying correct matches and discriminating between different objects.