## 2 Bundle Adjustment for Motion Estimation Using HFR Cameras
### Bundle Adjustment Reconstructs Motion's Trajectory
The benefit of this design is that it can balance the magnitude gap between the position, orientation and intrinsic parameters. \(\big{[}\mathcal{J}^{T}\mathcal{J}+\lambda D\big{]}\) is also denoted as the Hessian matrix. To solve the bundle adjustment, the key is the Jacobian matrix. From Equation (3), we give the analytically partial derivatives of the reprojection error to each component of \(\varphi\) by using rotation lie algebra 50 (3) properties and the chain rule of derivatives. For \(\phi_{iw}\), the partial derivative can be expressed as:  
\[ \begin{array}{c}\frac{\partial F_{ij}}{\partial\phi_{iw}}=\frac{\partial F_{ij} }{\partial c_{ij}}\frac{\partial c_{ij}}{\partial c_{ij}}\frac{\partial c_{ij} }{\partial c_{ij}}\frac{\partial c_{ij}}{\partial c_{ij}}\frac{\partial c_{ij} }{\partial c_{iw}}\\ =\frac{\partial c_{ij}}{\partial c_{ij}}\frac{\partial c_{ij}}{\partial c_{ij} }\left(-\left(\overline{\exp(\widehat{\phi_{iw}})PW_{j}}\right)J_{l}(\phi_{iw })\right)\end{array}  \tag{6}\]  
where the equation in that means the skew matrix of the related vector; the other two partial derivatives can be easily calculated by using Equation (3). \(J_{l}(\phi)\) is the left Jacobian for the BCH formula, with  
\[\mathrm{s}J_{l}(\phi)=\frac{sin\|\phi\|}{\|\phi\|}I_{3}+\left(1-\frac{sin\| \phi\|}{\|\phi\|}\right)\frac{\phi\phi^{T}}{\|\phi\|}+\left(\frac{1-cos\| \phi\|}{\|\phi\|}\right)\frac{\phi}{\|\phi\|}\]  
For translation vector \(i_{i}^{iw}\), the partial derivate is expressed in Equation (7). Meanwhile, Equation (8) is for intrinsic \(y_{i}\), and Equation (9) is for position \(PW_{j}\) at time stamp \(j\),  
\[\frac{\partial F_{ij}}{\partial t_{i}^{iw}}=\frac{\partial F_{ij}}{\partial c _{ij}}\frac{\partial c_{ij}}{\partial c_{ij}^{\prime}}\frac{\partial c_{ij}^{ \prime}}{\partial c_{ij}}\frac{\partial PC_{ij}}{\partial c_{ij}}\frac{ \partial PC_{ij}}{\partial t_{i}^{iw}}=\frac{\partial c_{ij}}{\partial c_{ij}^ {1}}\frac{\partial c_{ij}^{\prime}}{\partial PC_{ij}} \tag{7}\]  
\[\frac{\partial F_{ij}}{\partial y_{i}}=\frac{\partial F_{ij}}{\partial c_{ij} }\frac{\partial c_{ij}}{\partial y_{i}}=\left[ \begin{array}{cc}dis\cdot c_{ ij,x}^{1}&f_{i}\|c_{ij}^{\prime}\|^{2}c_{ij,x}^{\prime}&f_{i}\|c_{ij}^{\prime}\|^{4}c_{ij,x} ^{\prime}\\ dis\cdot c_{ij,y}^{1}&f_{i}\|c_{ij}^{\prime}\|^{2}c_{ij,y}^{\prime}&f_{i}\|c_{ ij}^{\prime}\|^{4}c_{ij,y}^{\prime}\end{array} \right] \tag{8}\]  
\[\frac{\partial F_{ij}}{\partial PW_{j}}=\frac{\partial F_{ij}}{\partial c_{ij} }\frac{\partial c_{ij}}{\partial c_{ij}^{\prime}}\frac{\partial c_{ij}^{ \prime}}{\partial PC_{ij}}\frac{\partial PC_{ij}}{\partial PW_{j}}=\frac{ \partial c_{ij}}{\partial c_{ij}^{\prime}}\frac{\partial c_{ij}^{\prime}}{ \partial PC_{ij}}\exp\left(\hat{\phi}_{iw}\right) \tag{9}\]  
with \(dis=\left(1+k_{1i}\|c_{ij}^{1}\|^{2}+k_{2i}\|c_{ij}^{1}\|^{4}\right)\). Equations (6)-(9) form the Jacobian matrix. We denote \(\mathcal{J}_{c}^{ij}=\left[\frac{\partial F_{ij}}{\partial\phi_{iw}}\frac{ \partial F_{ij}}{\partial t_{i}^{iw}},\frac{\partial F_{ij}}{\partial y_{i}} \right]^{T}\) formed by Equations (6)-(8) as the camera's Jacobian for camera \(i\) at time stamp \(j\), with \(\mathcal{J}_{\mu\
u}^{i,j}\) formed by Equation (9) as the position's Jacobian observed by camera \(i\) at time stamp \(j\). The Jacobian matrix will be detailed by the subsequent section which combines the HFR camera's properties and deploying strategy. The optimization procedure of the bundle adjustment for motion estimation are summarized by pseudocode Algorithm 1.