## 3 Proposed Method
### Pyramid Ternary Histogram of Oriented Gradients(PTHOG)  
Since, augmented reality based virtual objects can be poorly textured [7; 8], so it is not appropriate to utilize the texture based feature descriptor to obtain the features from the virtual objects. So, in order to obtain the prominent information from the poorly textured virtual objects, shape based feature descriptor can be utilized. Various shape based feature descriptor (e.g., SIFT [5]) are already proposed, however, those extraction approaches are not appropriate for real time augmented reality applications [6]. Since these approaches first detect keypoints and then calculate the descriptors, which increases the computational time. To address the above issues, in this paper, we present a novel frame based shape feature descriptor, namely Pyramid Ternary Histogram of Oriented Gradient (PTHOG), which is an extension of PHOG [26] and BHOG [27]. Shape information in PTHOG is described by the distribution of intensity gradients. Here at first, we divide each frame into the cell with the different level of the pyramid. Then, the square of the gradient magnitude and each pixel orientation with 3 \(\times\) 3 block is computed. After that, we create an orientation histogram \(HOG(p)\), \(p=0,1,...,8\); as like as HOG [28] feature extraction approach. Lastly, the feature vector is formed based on the histogram values by comparing with a threshold value. The following equation represents our proposed feature descriptor,  
\[THOG=\sum_{z=0}^{p-1}S(HOG(z)-\overline{a})*2^{z} \tag{1}\]  
\[S(i)=\left\{\begin{array}{cc}1,&\quad if\ i>threshold\\ -1,&\quad if\ i<threshold\\ 0,&\quad otherwise\end{array}\right. \tag{2}\]  
\[\overline{a}=\frac{1}{9}\sum_{z=0}^{p-1}(HOG(z)) \tag{3}\]  
where, \(z\) are the orientation histogram values of all the pixels within a block and i is the difference between \(HOG(z)-\overline{a}\). In the experiment, the threshold value is set to 10, which is chosen empirically. THOG is split into two binary codes (upper pattern code and lower pattern code) and represented as two separate histograms. After that, these histograms are concatenated to produce the intermediate THOG, which is represented as THOG = [UTHOG LTHOC]; where UTHOG is Upper Ternary HOG and LTHOG is Lower Ternary HOG. The final PTHOG descriptor for a frame or image is a concatenation of all the THOG vectors at each pyramid level, \(PTHOG=[THOG_{1}THOG_{2}...THO_{n}]\). The concatenation of all the THOG vectors introduces the spatial information of the frame or image. Figure 3 illustrates the proposed feature descriptor, Pyramid Ternary Histogram of Oriented Gradient (PTHOG). The main advantages of using PTHOG over PHOG [26] and HOG [28] are: PTHOG feature extraction method does not need to perform the square root action to get the gradient magnitude since it compares the value of histogram with the average of the orientation histogram of each pixel and a given threshold value. Moreover, it does not compute the normalization of the orientation histograms. PTHOG generates ternary codes which increase more discriminative power than BHOG [27] and reduces the issue of noise that BHOG may suffer.