# Low-Complexity Pupil Tracking for Sunglasses-Wearing Faces for Glasses-Free 3D HUDs
## 2 Methods
### Nose-Month-Face Boundary Tracking: Alignment and Tracker Checker  
After the detector identifies the eye-nose region, the tracking mode starts to extract the coordinates of the pupil centers based on eye-nose shape alignments using the SDM [30] and the scale-invariant feature transform (SIFT) [35]. The SDM mode trains a sequence of descent directions that minimize the mean of the nonlinear square functions from each landmark point [30]. SIFT extracts feature description from landmark points and is invariant to translations, rotations, and scaling transformations, and is widely used in face recognition technologies [35]. This non-CNN-based method has the advantages of low CPU consumption and fast speed compared to state-of-the-art CNN-based landmark point alignment methods. To prevent erroneous detection or tracking, we propose a novel tracker checker idea. The proposed tracker checker guarantees that the aligned results contain the eyes. For each frame, after the nose-mouth-face boundary points are aligned, the proposed nose-mouth-face boundary tracker checker performs a final examination of the tracking results, irrespective of whether it tracks eyes or not. Thus, more efficient, faster eye-tracking system computations can be achieved. A greedy search algorithm was performed to find the best nonoccluded area points for estimating the eye center positions. Among the 23 candidate points from nonoccluded areas, such as the nose-mouth-face boundary, we identified the 11 points that best suggested the  
Figure 2: Flowchart of the proposed eye-tracking method for bare and sunglasses-wearing faces. eye position via the greedy algorithm with end-to-end eye position calculation (further details are provided in Section 2.3). Owing to the computational complexity of the SIFT feature extraction increasing according to the number of points involved, we only utilized \(11\) points to ensure reliability and manageable complexity. Among these points, the nose-mouth and nose-mouth-face boundary points were identified as the best two positions (Figure 4). Figure 3: (**a**) Entire-face detector, which detects both bare faces (1 st row) and sunglasses-wearing faces (2 nd row). Each column shows the left and right image from the stereo camera. (**b**) Cropped eye regions for the classification of bare and sunglasses-wearing faces. Cropping was performed using the statistical mean shape determined according to training dataset labeling. Eye Center Position Estimation from Nose-Mouth-Face Boundary Points Using a Supervised Linear Regression Algorithm  
Using the 11 points representing nonoccluded face areas identified in Section 2.2, we estimated the eye center positions using a supervised linear regression method, meaning that we constructed the training image database with the ground truth of eye position on sunglasses according to other points (Figure 5). All the ground truths of eye positions were annotated as precisely as possible via comparison with the corresponding bare-face images. Ultimately, 32,400 sunglasses-wearing images with a head pose less than 20\({}^{\circ}\) were constructed with eye position labeling and used for eye position estimations using a supervised linear regression. Figure 4: Nonoccluded area shape point candidates identified by a greedy search algorithm: nose-mouth points (**1 st row**) and nose-mouth-face boundary points (**2 nd row**). Each column shows the left and right images from the stereo camera. Figure 5: Eye position estimation by a supervised linear regression algorithm. Nonoccluded area shape tracking points (**left**, green dots), with eye positions inferred using these points (**right**, red dots). In this paper, we set \(E_{left}\) as the left eye center position and let \(X\) represent the set of points corresponding to nonoccluded areas, i.e., the nose-mouth-face boundary points introduced in Section 2.1.