# Estimation of Antenna Pose in the Earth Frame Using Camera and IMU Data from Mobile Phones
## 5 Antenna Poses Estimated from Captured Images
### Pose Refinement  
#### 5.2.1 Bounds on Pose Parameters  
The pose obtained above is inaccurate as a result of three factors: (1) displacements between the detected 2D/3D principal axes and their true positions; (2) potential errors in the triangulation of the 3D principal axis from the images; and (3) imprecision in the determination of the roll angle. Consequently, we wish to further refine this pose. A simple approach is to treat the approximate pose as an initialization and then iterate until convergence is achieved, as done in PWP3D [35] and D2 CO [32]. Although our approximated poses function well as initializations in most cases, they still cannot guarantee the avoidance of local minima. For higher accuracy, we attempt to find bounds on the pose search space that will allow us to use global optimization techniques to solve Equation (6). We first concentrate on the rotational component of the pose. The first two sources of imprecision are predominantly related to the yaw and pitch of the antenna. We observe that the projection of the 3D principal axis based on the approximate pose never falls out of the area enclosed by the two long side edges of the antenna silhouette for each viewpoint, which means that the recovered 3D principal axis always lies within the double cone enclosing the visual hull of the antenna suggested by contours from multi-view images, as indicated in Figure 10a. Let the viewing angle between the two most distant viewpoints be denoted by \(\vartheta\), and let the radius of the bounding cylinder of the antenna be denoted by \(r\); then, the diameter \(D\) of the cone is \(2r/\sin\left(\frac{\vartheta}{2}\right)\) (see Figure 10b), and the opening angle \(\varepsilon\) of the cone is \(2\arctan(\frac{D}{H})\), where \(H\) is the height of the antenna (see Figure 10a). In this way, we can obtain bounding constraints on the refinements to the yaw and pitch. In practice, we have found that the approximate rotation is usually much closer to the true value than these bounds would suggest, so we scale the bounds by an empirical factor \(w_{1}\) to further shrink the search space. Regarding the last source of inaccuracy, i.e., that affecting the roll angle, we already have a natural bound, namely the granularity used when enumerating the roll angle in Section 5.1.2, which we let be denoted by \(\varrho\). To summarize, the bounded search space for the refined rotation is:  
\[R\left(\alpha,\beta,\gamma\right)=R\left(\alpha^{\prime}+ \Delta\alpha,\beta^{\prime}+\Delta\beta,\gamma^{\prime}+\Delta\gamma\right)=R ^{\prime}_{\parallel}\left(\gamma^{\prime}+\Delta\gamma\right)R^{\prime}_{ \perp}\left(\alpha^{\prime}+\Delta\alpha,\beta^{\prime}+\Delta\beta\right)\,,\] \[\text{s.t.}\quad|\Delta\alpha,\Delta\beta,\Delta\gamma|<\left(w _{1}\varepsilon,w_{1}\varepsilon,\varrho\right)^{T}\,,\]  
where \(\Delta\alpha\), \(\Delta\beta\) and \(\Delta\gamma\) describe the difference between the approximate rotation and the real value. Figure 10: Constraints on the refinements of rotation and translation. (**a**) The double cone defines the rotation constraints, and the cylinder defines the translation constraints; (**b**) the radius of the cone is related to the bottom size of the antenna and the angle between two most distant viewpoints. Regarding the translation of the model, it can be similarly observed that the projection of the center of the model always falls within the antenna foreground in the image and is usually not far from its true position. This means that in \(OF\), the true translation is confined to the cylinder formed by the top and bottom of the double cone found above (see Figure 10a), yielding bounds of \((D,H,D)\) on translations in \(OF\) based on \(T^{\prime}\). Moreover, for reasons similar to those motivating the introduction of the scale factor \(w_{1}\), we also introduce an empirical factor \(w_{2}\) for the translation along the principal axis.