## 4 The Proposed Method
### Joint Descriptor Computing  
The robustness of descriptor is very important to achieve robust feature tracking, which has been analyzed deeply in [43]. According to the last recent evaluation work made by [PERSON] et al. [71], the SURF feature has desirable performance on aspect of matching speed and precision. However, the SURF feature easily suffers from affine transform, this may break the compactness of point-cloud model when it is used in 3D reconstruction system. To improve the quality of 3D reconstruction system, we propose a joint computing procedure that include SURF and binary test [36], the former is use to describe the keypoints located in the texture areas, then the latter is used in the textureless areas. For convenience, we called the proposed feature descriptor as joint feature descriptor (JFD), the pipeline for computing a JFD feature descriptor is depicted in Figure 4, in which it is run on GPU device for accelerating. In addition to the matching precision and fast speed, the proposed JFD feature is also robust to various perturbations such as noise, illumination or contrast change.  
For the \(k_{j}\) located in the texture areas, we first use SURF feature to compute a vector of 64 dimensional which is an normalized gradient statistics extracted from a spatial grid \(R\) divided into \(4\times 4\) regions. These subregions are referred to as \(\mathrm{R}=\left\{R_{i,j}\middle|1\leq i,j\leq 4\right\}\). According to [20], the weighted gradient at point \((u,v)\) is defined as,  
\[\left(\begin{array}{c}d_{x}(u,v)\\ d_{y}(u,v)\end{array}\right)=R_{-0k}\left(\begin{array}{c}D_{x}^{L_{k}}\\ D_{y}^{L_{k}}\end{array}\right)\varphi(x,y)\times G_{1}(u,v) \tag{10}\]  
where \(D_{x}^{L_{k}}\) and \(D_{y}^{L_{k}}\) denote first order box filters, which are used to compute the gradient components.  
To this end, the SURF uses first order statistical results on vertical and horizontal gradient responses to produce the good description that achieves the best performance between accuracy and efficiency, then the resulting statistical vector with respect to \(R_{i,j}\) can be calculated by the following formula,  
\[\mu_{k}(i,j)=\left(\begin{array}{c}\sum\limits_{u,v}^{R_{i,j}}d_{x}(u,v)\\ \sum\limits_{u,v}^{R_{i,j}}d_{y}(u,v)\\ \sum\limits_{u,v}^{R_{i,j}}\left|d_{x}(u,v)\right|\\ \sum\limits_{u,v}^{R_{i,j}}\left|d_{y}(u,v)\right|\end{array}\right)i,j\in[1,4] \tag{11}\]  
The SURF descriptor of \(k_{i}\) can be directly computed by concatenating the \(\mu_{k}(i,j)\), which is defined as  
\[\mu_{k}=v\textit{stack}(\mu_{k}(i,j)) \tag{12}\]  
where \(\textit{stack}(\cdot)\) is function that represents stacking the matrix in vertical direction.  
To improve the invariance to linear transform, the SURF descriptor should be normalized to a unit vector by L2 normal, the enhanced SURF descriptor can be calculated by the following formula  
\[\mathrm{SURF}(k_{i})=\mu_{k}/\|\mu_{k}\|_{2} \tag{13}\]However, for the keypoints distributed in textureless regions, we use binary test to produce robust descriptors in the neighbor regions that labeled by the superpixel-based segmentation. The binary test \(\tau\) in [36] is defined as  
\[\tau(L,x,y)=\left\{\begin{array}{l}0,p(x)\geq p(y)\\ 1,p(x)<p(y)\end{array}\right. \tag{14}\]  
where \(p(x)\) represents the intensity of \(p\) at a point \((x,y)\). Thus, the resulting feature vector is defined as  
\[v_{n}(k_{i})=v_{n}(p(x,y))=\sum_{1\leq i\leq n}2^{i-1}(\tau(L_{i},x_{i},y_{i})) \tag{15}\]  
Note that \(n\) is set to 32 for saving computation time in the whole experiment, thus the resulting feature vector has 32 binary elements.  
To this end, the JKD descriptor can be obtained by concatenating the SURF(\(k_{i}\)) and \(v_{n}(k_{i})\), then resulting a 96 dimensional of feature descriptor.  
\[\text{JKD}(k_{i})=concat(\text{SURF}(k_{i}),v_{n}(k_{i})) \tag{16}\]  
Owing to the JKD is hybrid type, namely it not only includes float type elements, but also contains binary type ones, thus, we need urgently a novel matching approach to match them.