Total Bundle Cost
You are given multiple observations (2D point measurements) of 3D points from different cameras, and need to compute the total bundle adjustment cost.
The total cost is the sum of squared reprojection errors across all observations:
C=βi,jββ₯Ο(RjβXiβ+tjβ)βxijββ₯2
Where:
- Xiβ is the i-th 3D point
- Rjβ,tjβ are the j-th camera's rotation and translation
- xijβ is the observation of point i in camera j
This is the objective function that bundle adjustment minimizes.
Example:
observations = [(0, 0, [0, 0])] points = [[0, 0, 10]] cameras = [[0, 0, 0]] focal = 1
0.0
Single observation: camera 0 sees point 0 at (0, 0)
- Get point: [0, 0, 10]
- Get camera translation: [0, 0, 0]
- Transform: [0+0, 0+0, 10+0] = [0, 0, 10]
- Project: u = 1Γ0/10 = 0, v = 1Γ0/10 = 0
- Error: (0-0)Β² + (0-0)Β² = 0
Total cost = 0.0
Constraints:
- observations: list of (camera_idx, point_idx, [u, v]) tuples
- points: list of [X, Y, Z] for each 3D point
- cameras: list of [tx, ty, tz] translation for each camera
- focal: focal length
- Assume identity rotation for all cameras
- Return total cost rounded to 4 decimal places
More from CV: Image Alignment and Stitching
Bundle Adjustment Cost Computation: Background and Strategy
Background Knowledge
Bundle Adjustment Fundamentals
Bundle adjustment is a nonlinear optimization technique that refines camera parameters and 3D point positions by minimizing reprojection errors. The core idea is that when you project a 3D point onto a camera's image plane using the camera's pose (rotation and translation), the projected location should match the actual observed 2D point. The difference between predicted and observed locations is the reprojection error. Bundle adjustment simultaneously optimizes all camera poses and 3D points to minimize the sum of these errors across all observations.
The Reprojection Error
The reprojection error for a single observation measures how well a 3D point, when transformed by a camera's pose and projected onto the image plane, matches the observed 2D measurement. Mathematically, this involves:
- Transforming the 3D point using the camera's rotation matrix Rjβ and translation vector tjβ: RjβXiβ+tjβ
- Projecting this transformed point onto the image plane using the projection function Ο (typically perspective projection)
- Computing the Euclidean distance between the projected point and the observed 2D point xijβ
The total cost function sums the squared reprojection errors across all point-camera pairs, which is standard in least-squares optimization.
Why Squared Errors?
Squaring the errors emphasizes larger deviations and makes the function differentiable everywhere, enabling efficient gradient-based optimization methods like Levenberg-Marquardt. This formulation is the foundation for modern SLAM and Structure from Motion systems.
Algorithm/Approach
The solution follows a straightforward summation pattern:
- Iterate over all observations (each pair of camera j and 3D point i)
- Transform the 3D point using the camera's pose
- Project the transformed point onto the image plane
- Compute the reprojection error (difference between projected and observed 2D points)
- Accumulate the squared error into the total cost
This is a direct computation problemβyou're evaluating the objective function at given parameter values, not optimizing it. The key is correctly implementing the geometric transformations.
Step-by-Step Strategy
Step 1: Understand the Input Structure
- Identify how 3D points, camera poses (rotation and translation), and 2D observations are represented
- Determine the format of rotation matrices (3Γ3 matrices or other parameterizations like quaternions)
- Understand the projection function Ο (typically perspective projection: Ο(p)=\frac{1}{p_z}[pxβ,pyβ]T for a point p=[pxβ,pyβ,pzβ]T)
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