# Optimal View Estimation Algorithm and Evaluation with Deviation Angle Analysis
## 3 Methods
### Image Generated According to Viewpoints  
After selecting viewpoints using Algorithm 1, the dataset for image viewpoint estimation can be constructed. We utilize 3D models to perform parallel projection based on the specified viewpoint directions, thereby generating 2D projection images. This process is illustrated in Figure 6. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline
**Method** & **minAveD** & **maxAveD** & **Range** & **stdAveD** & **UniHks** & **UniP** & **NrmHks** & **NrmP** & **Time** \\ \hline CFLS & 0.3297 & 0.3585 & 0.0287 & **0.0029** & 1 & 0.0000 & 1 & 0.0000 & 0.0010 \\ CSDyn & 0.2480 & 0.4877 & 0.2397 & 0.0464 & 1 & 0.0000 & 0 & 0.2444 & 0.8326 \\ InveT & 0.1933 & 0.5313 & 0.3380 & 0.0562 & 1 & 0.0000 & 0 & 0.2507 & 0.0008 \\ Spiral & 0.2039 & 0.4757 & 0.2719 & 0.0522 & 1 & 0.0000 & 0 & 0.9607 & 0.0044 \\ Kspace & 0.2323 & 0.4702 & 0.2379 & 0.0419 & 1 & 0.0000 & 0 & 0.6081 & 0.0003 \\ SCS-AP & 0.0000 & 0.4435 & 0.4435 & 0.1392 & 1 & 0.0000 & 1 & 0.0000 & 0.0009 \\ Ours & 0.3207 & **0.3381** & **0.0174** & 0.0033 & 1 & 0.0000 & **0** & 0.9953 & 2.1910 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Calculation results of the sampling points (N = 250). Bold numbers indicate the best results; The underlined number is the second-best result. Figure 6: A schematic diagram of generating a projection image through parallel projection according to the specified viewpoint position. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline
**Method** & **minAveD** & **maxAveD** & **Range** & **stdAveD** & **UniHks** & **UniP** & **NrmHks** & **NrmP** & **Time** \\ \hline CFLS & 0.1738 & 0.1891 & **0.0154** & **0.0011** & 1 & 0.0000 & 1 & 0.0000 & 0.0014 \\ CSDyn & 0.1220 & 0.2449 & 0.1229 & 0.0210 & 1 & 0.0000 & 0 & 0.8633 & 10.3144 \\ InveT & 0.0799 & 0.2852 & 0.2053 & 0.0301 & 1 & 0.0000 & 0 & 0.2412 & 0.0010 \\ Spiral & 0.0984 & 0.2966 & 0.1983 & 0.0296 & 1 & 0.0000 & 1 & 0.0027 & 0.0015 \\ Kspace & 0.1024 & 0.2589 & 0.1565 & 0.0272 & 1 & 0.0000 & 0 & 0.8009 & 0.0013 \\ SCS-AP & 0.0000 & 0.2329 & 0.2329 & 0.0643 & 1 & 0.0000 & 1 & 0.0000 & 0.0011 \\ Ours & 0.1680 & **0.1840** & 0.0160 & 0.0027 & 1 & 0.0000 & **0** & 0.3879 & 3.7868 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Calculation results of the sampling points (N = 900). Bold numbers indicate the best results; The underlined number is the second-best result. Due to the different sizes of the models, it is necessary to standardize them into a fixed size. Let \(N_{v}\) be the number of vertices of the 3D object model, and \(V_{i}(x_{i},y_{i},z_{i})\left(i=1,2,\ldots,N_{v}\right)\) the \(i\)th vertex of the model. Then, this model can be standardized using the following formula:  
\[Q_{i}^{new}=\frac{\gamma_{o}}{R_{o}}(V_{i}-O_{cnt}),\;i=1,2,\ldots,N_{v}. \tag{5}\]  
where \(\gamma_{o}\) is a scale factor, and we use \(\gamma_{o}=0.45\) as the default value; \(R_{o}\) is the diameter of the object; and \(O_{cnt}\) is the geometric center of the object. That is,  
\[O_{cnt}=\frac{1}{N_{v}}\sum_{i=1}^{N_{v}}V_{i} \tag{6}\]  
After standardizing the model using Equation (5), it can be completely enclosed within the unit sphere. With the standardized model and the viewpoints obtained according to Section 3.1, projection images with viewpoint labels can be obtained through parallel projection based on the viewpoint direction. The mathematical formulas for parallel projection based on the direction of the viewpoint \(p_{view}\) (\(\|p_{view}\|=1\)) can be formulated as  
\[Q_{n}=Q_{o}-\left(p_{view}\overrightarrow{OQ_{o}}\right)p_{view}. \tag{7}\]  
where \(Q_{o}\) is any vertex of the 3D object of which the center has been translated to the origin, and \(Q_{n}\) is the projection point of \(Q_{o}\) on the projected image. The algorithm for generating images according to specified viewpoints (Algorithm 2) is described as follows. In the process of generating this image with a viewpoint label, Algorithm 1 is the first step in Algorithm 2.