# Tightly-Coupled Data Compression for Efficient Face Alignment
## 3 Tightly-Coupled Data Compression Algorithm  
As shown in Section 2, the key components of SDM are \(\mathbf{r}_{d}\) for all \(D\) steps. Thus, the trained model of SDM stores all the information of \(\mathbf{r}_{d}\) for all \(D\) steps. Since all the features calculated from every landmark are concatenated together and form a large feature vector, the dimension of the feature vector can easily achieve 27,200 if HoG features are used. So the dimension of \(\mathbf{A}_{d}\) is 136 \(\times\) 27,200 and \(\mathbf{b}_{d}\) is a 136-dimensional vector. Typically, each component in \(\mathbf{A}_{d}\) and \(\mathbf{b}_{d}\) is represented by a single-precision floating point number. Table 1 shows the data ranges of \(\mathbf{A}_{d}\) and \(\mathbf{b}_{d}\) for a typical HoG-based 6-step regressor.  
From the table it can be concluded that the data ranges vary in different regression steps. So in this paper, the method of compressing the data separately for each step is proposed. Furthermore, the data range of \(\mathbf{A}_{d}\) is very different from that of \(\mathbf{b}_{d}\), and \(\mathbf{b}_{d}\) only contains 136 floating point numbers in each step, which is much smaller (only about 3 KB for all 6 steps) than the size of \(\mathbf{A}_{d}\). Therefore, in this paper, only the data in \(\mathbf{A}_{d}\) are compressed.  
Since the data distribution of \(\mathbf{A}_{d}\) can not be described by a parametric model, a non-parametric method for estimating the data distribution of \(\mathbf{A}_{d}\) is applied in this paper. The number of elements in \(\mathbf{A}_{d}\) is large enough, so for the sake of computational efficiency, the Parzen window method [39] is used here. Assuming that the number of elements in \(\mathbf{A}_{d}\) is \(T\), and window size is \(h\), the probability density function (PDF) of the element \(x\) in \(\mathbf{A}_{d}\) can be estimated through the following equation:  
\[p(x)=\frac{1}{T}\sum_{i=1}^{T}\frac{1}{h}\phi\bigg{(}\frac{x-x_{i}}{h}\bigg{)} \tag{4}\]  
where \(\phi(x)\) is the square window function  
\[\phi(x)=\left\{\begin{array}{rl}&1\quad|x|\leq 0.5\\ &0\quad otherwise\end{array}\right. \tag{5}\]  
Figure 4 shows the estimated PDF of the elements in \(\mathbf{A}_{1}\). The shapes of the PDFs in other steps are similar. From the figure it can be concluded that most of the values in \(\mathbf{A}_{d}\) concentrate around 0, and the distribution also shows some long tail effects. As a result, it is inappropriate to compress the data according to uniform quantization. In this paper, a K-means based data quantization algorithm with a probability density-aware initialization technique is proposed in order to cope with the above mentioned difficulties.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Step Index**} & \multicolumn{2}{c}{\(\mathbf{A}_{d}\)} & \multicolumn{2}{c}{\(\mathbf{b}_{d}\)} \\ \cline{2-5}  & **Minimum Value** & **Maximum Value** & **Minimum Value** & **Maximum Value** \\ \hline
1 & \(-\)0.0072 & 0.0075 & \(-\)0.2597 & 0.1381 \\
2 & \(-\)0.0081 & 0.0075 & \(-\)0.1774 & 0.1308 \\
3 & \(-\)0.0050 & 0.0060 & \(-\)0.1093 & 0.0644 \\
4 & \(-\)0.0040 & 0.0042 & \(-\)0.0680 & 0.0400 \\
5 & \(-\)0.0032 & 0.0037 & \(-\)0.0295 & 0.0283 \\
6 & \(-\)0.0024 & 0.0031 & \(-\)0.0180 & 0.0157 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Data ranges for a typical HoG-based 6-step regressor.