## 3 Proposed Methods
### Translation and Focal Length Estimation  
After retrieving the matching CAD model, it is now possible to estimate the scale factor (\(f\), \(t_{z}\)) by projecting the CAD model to the RGB image space. In the pin-hole camera model, the projection size of the CAD model is affected by both \(f\) and \(t_{z}\). However, solvingboth parameters together is an ambiguous problem. Thus, one parameter must be set as an arbitrary value while the remaining parameter is estimated. In this study, we define the limited range of focal length and \(z\)-axis translation because there is no GT focal length parameter in the RGB images from the Internet. Using the limited ranges and an iterative algorithm, we estimate both parameters.  
In this iterative algorithm, \(t_{x}\) and \(t_{y}\) are set to 0 because we assume the center of the bounding box corresponds to the center of the object. By the pin-hole camera model, a 3D vertex point \(\mathbf{P}(X_{v},\ Y_{v},Z_{v})\) on the 3D CAD model is projected to a 2D image point \(\mathbf{p}(x_{v},\ y_{v})\), where \(v\in\{1,2,\dots,n\}\) and \(n\) is the number of vertices in the 3D CAD model. The scale parameters are ranged as follows: \(f\in\{50,70,90,\dots,1460,1480\}\) and \(t_{z}\in\{0.05,0.1,0.15,\dots,1.9,1.95\}\). Principal points \(c_{x}\) and \(c_{y}\) are decided to be at the center of the cropped image. By applying these values from Equations (1)-(3), the 3D vertices are projected into the image plane. A new bounding box is created using the 2D projection points such as \(x\in\{\underset{x}{\text{min}}\mathbf{p}(x_{v},y_{v}),\underset{x}{\text{max}}\mathbf{ p}(x_{v},y_{v})\}\) and \(y\in\{\underset{y}{\text{min}}\mathbf{p}(x_{v},y_{v}),\underset{y}{\text{max}}\mathbf{ p}(x_{v},y_{v})\}\). Then, the Intersection Over Union (IoU) between this newly created bounding box and the existing RGB object bounding box is calculated. The values of \(f\) and \(t_{z}\) that maximize the IoU are selected as the final estimation. An example of CAD vertex projection and bounding boxes is shown in Figure 8.  
After estimating the scale factor, \(t_{x}\) and \(t_{y}\) must be recalculated from the original RGB image using Equations (1)-(3). Principal points \(c_{x}\) and \(c_{y}\) represent the center points of the original image. Scale factors are estimated from the cropped image without changing the pixel size; thus, these values remain valid in the original image. Since the center of the 3D CAD model is at the origin of the reference coordinate system, \(t_{x}\) and \(t_{y}\) can be determined by Equations (1)-(3). By setting \(\mathbf{P}(X_{v},Y_{v},Z_{v})=(0,0,0)\) and projecting it to the image space, we have the center of the 2D bounding box \(\mathbf{p}(x_{v},y_{v})\) in the cropped RGB images. In the original image space, the center of the projected bounding box becomes \(\mathbf{p}\Big{(}x_{v}+x_{off},y_{v}+y_{off}\Big{)}\), where \(\Big{(}x_{off},y_{off}\Big{)}\) is the upper-left corner of the bounding box in the original image. Then external translation \(t_{x}\) and \(t_{y}\) can be calculated as:  
\[t_{x}=t_{z}\frac{x_{v}+x_{off}}{f}\text{ and }t_{y}=t_{z}\frac{y_{v}+y_{ off}}{f}. \tag{25}\]  
Figure 8: Projection result of a car CAD model.