Jacobian for Bundle Adjustment

Implement a Jacobian matrix computation for bundle adjustment optimization, a crucial step in image alignment and stitching. This process involves minimizing the reprojection error between observed and predicted image points.

Bundle adjustment is a non-linear least squares problem, where the goal is to find the optimal camera parameters and 3D point coordinates that minimize the reprojection error. The error is typically measured as the sum of squared differences between observed image points $x_{ij}$ and predicted points $\pi(C_j, P_i)$, where $\pi$ is the projection function, $C_j$ represents the camera parameters, and $P_i$ represents the 3D point coordinates.

To compute the Jacobian matrix, we need to calculate the partial derivatives of the projection function with respect to the camera parameters and 3D point coordinates.

1. Compute the partial derivatives of the projection function $\pi$ with respect to the camera parameters $C_j$.
2. Compute the partial derivatives of the projection function $\pi$ with respect to the 3D point coordinates $P_i$.

This technique is widely used in structure from motion and stereo vision applications.