4.3. The biased manifold embedding (BME)

The head pose estimation is subjected to the identity variation. The ideal case is to eliminate such negative effects, which means the face images with close pose angles should maintain nearer and the ones with quite different poses should stay farther in the low-dimensional manifold, even the poses are from the same identity. Based on this statement, the BME is proposed to modify the distance matrix according to the pose angles, which can be extended with almost all the classical algorithms [23].

The modified distance between a pair of data points x<sup>i</sup> and x<sup>j</sup> is given by:

$$d\_{ij} = \begin{cases} \frac{\beta \ast p(i, j)}{\max\_{m, n} \left( p(m, n) \right) - p(i, j)} \ast d\_{ij}, p(i, j) \ast 0 \\ 0, & p(i, j) = 0 \end{cases} \tag{16}$$

where p(i,j) = |pi - pj| is simply defined as the absolute difference of the angles of two poses. From the modified distance matrix, one can find that the distance between images with close poses is biased to be proportionally small. The images with the same poses are defined to be zero-distance.

In fact, the BME can be seen as a naÏve version of the supervised manifold learning. The head pose information is used as the supervision to enhance the construction of the graph. For the head pose estimation stage, the generalized regression neural network (GRNN) [24] is applied to learn the nonlinear mapping for the unseen data points, and linear multivariate regression is applied to estimate the head pose angle. This idea can be easily extended to the classical algorithms, e.g., Isomap, LLE, and LE, among which the biased LE achieves the lowest error rate on the data set of FacePix [25].

#### 4.4. Head pose estimation as frontal view search

The two remarkable head poses, i.e., yaw and pitch, cause the problem self-occlusion. Compared with pitch, the yaw makes the problem more serious. An extended manifold learning (EML) method is proposed to specify the head pose estimation only considering the variation in the yaw [15]. This work resorts to the frontal view search instead of directly estimating the head pose, which is more efficient and robust. The idea is based on the observation that the frontal face locates nearly at the vertex in the symmetrical shape of the embedding. However, if the pose distribution of the data is asymmetric, the location of the frontal face in the manifold will shift from the vertex. Therefore, the first trial of the EML method is data enhancement. All the images are horizontally flipped and both the original and flipped images are used for manifold learning. In order to make the method more robust to variations in environment, for example, illumination, the localized edge orientation histogram (LEOH) is presented to represent the original color mappings as more representative features. The idea is inspired by the classical HoG feature [14]. The first step of LEOH is to apply a Canny edge detector on the original images. Then, the whole image is divided into M +N cells. The gradient orientation is quantized into NB bins. Next, histograms of the gradient orientation of the cells locating in a block consisted of P +Q cells are accumulated and normalized. Finally, the LEOH feature is obtained by the block features concatenation. The proposed ideas can be easily incorporated in various manifold learning methods that improve the performance of the frontal view searching.
