# Estimation of Antenna Pose in the Earth Frame Using Camera and IMU Data from Mobile Phones
## 3 Problem Formulation and Method Overview
### Method Overview  
A direct optimization-based solution to Equation (2a) is impractical because of its high dimensionality; therefore, we will break the problem down into smaller parts to solve it.  
Referring to the original graph model presented in Figure 2, we find that the first item in Equation (2a), which corresponds to the red-outlined region in the upper right of the figure, describes \(P\) model-based visual pose estimation problems, and similarly, the second item, corresponding to the blue-outlined region in the lower left of Figure 2, describes the IMU orientation estimation problem seeking the rotation of \(IF\) in \(EF\) denoted by \(\prescript{E}{I}{R_{i}}\), for which effective solutions (e.g., [52]) are available under the assumption that we have already aligned \(A^{i}\) and \(M^{i}\) via \(\prescript{A}{M}{R}\).  
With these insights, given that \(\prescript{C}{R}\), \(\prescript{C}{R_{i}}\) and \(\prescript{E}{I}{R_{i}}\) are available, using the graph model to determine \(\prescript{C}{R}\) becomes a simple process of passing messages through a chain, as follows:  
\[\prescript{CR}{R}=avg\left(\prescript{C}{R_{i}}\right)=avg\left(\prescript{C}{ I}{R_{i}}\prescript{CR}{G}{R_{i}}\right),\]  
where \(avg\) is used to fuse estimations from multiple viewpoints. Using \(\prescript{C}{R}\), we can calculate the downtilt and azimuth angles of the antenna.  
In addition, we note that the priors \(\prescript{A}{CR}\) and \(\prescript{A}{M}{R}\) (i.e., the relative poses between the camera, accelerometer and magnetometer) are inherent to each specific mobile phone; thus, they need to be calculated only once and can then be stored for later use. To acquire the exact values of these rotations, we employ a dedicated offline camera-IMU calibration process, which will be described in Section 4.  
To summarize, our solution for antenna pose estimation in the Earth frame consists of the following four main steps:  
1. For a given phone, we compute the relative poses between its camera and IMU sensors using an offline camera-IMU calibration procedure. Once calculated, the relative poses of the camera and IMU sensors will not change during the antenna pose estimation process.
2. Using the antenna model and images obtained from calibrated viewpoints, we estimate the relative pose between the antenna frame and the camera frame for each viewpoint.
3. We correct the IMU data using the relative rotation between the accelerometer and magnetometer from (1), and we calculate the rotation of the IMU in the Earth frame for each viewpoint using existing IMU orientation estimation techniques.
4. We concatenate the antenna pose in the camera frame and the IMU orientation in each viewpoint with the relative camera-IMU rotation from (1) to obtain the antenna rotation in the Earth frame; antenna poses in all viewpoints are averaged to calculate the resulting downtilt and azimuth angle.  
Figure 3 provides an overview of our method. The two key elements of our method are the determination of \(\prescript{A}{R}\) and \(\prescript{A}{M}\) in Step (1) and the estimation of \(\prescript{V}{R}\) in Step (2). We describe the corresponding procedures in detail in Sections 4 and 5.