Bundle Adjustment

Formula

$$\underset{X,P}{\min }\sum _{i,j}{w}_{i,j}{\Vert u_{i,j}-\pi (P_i,X_j)\Vert }_2^2$$

$w_{i,j}$: point $j$ visible in camera $i$. $u_{i,j}$: observed 2D point. $\pi (P_i,X_j)$: projection with the current estimates. The difference is the reprojection error.

Solved with Gauss-Newton or Levenberg-Marquardt, using the sparse structure (e.g. Google Ceres).

Why not solve SfM with bundle adjustment alone?

Exact methods are cubic in the number of cameras, and BA only finds local minima, so it needs a good initialization. It is used as a refinement step after epipolar geometry and triangulation, and after every added view in incremental SfM.

Appears in

• Lecture 3, Bundle Adjustment
• Lecture 3, Incremental SfM
• Lecture 3, Global SfM
• Lecture 9, Global alignment: 3D loss instead of reprojection error

Related concepts

• Structure from Motion
• Projection Matrix
• Triangulation