Legacy DLT

On prod today, extrinsics are often still produced with calibrate.py: click image pixels, enter matching LiDAR $(x,y,z)$, solve a free 3×4 projection matrix by Direct Linear Transform (DLT), write projection_matrix_{l|r}.

For each correspondence $(X,Y,Z) \leftrightarrow (u,v)$, the homogeneous model $p \sim P X$ rearranges into two linear rows in the 12 unknowns of $P$:

$$\begin{aligned} &-X\,p_{11} - Y\,p_{12} - Z\,p_{13} - p_{14} + u\,X\,p_{31} + u\,Y\,p_{32} + u\,Z\,p_{33} + u\,p_{34} = 0 \\ &-X\,p_{21} - Y\,p_{22} - Z\,p_{23} - p_{24} + v\,X\,p_{31} + v\,Y\,p_{32} + v\,Z\,p_{33} + v\,p_{34} = 0 \end{aligned}$$

Stack ≥6 point pairs (the script asks for more; default minimum is 10), take the SVD of the design matrix, and reshape the singular vector for the smallest singular value into $P \in \mathbb{R}^{3\times4}$. Hartley normalization helpers exist in the file but are commented out in the current solve path.

Unlike the feature-branch path, this $P$ is not forced to equal $K[R \mid t]$ for a particular $K$. The free $P$ can absorb extrinsic error, residual distortion, and click noise in one matrix.

Post-solve adjuster

dlt_proj_adjuster.py fine-tunes an existing $P$:

• Rotation: right-multiply the left 3×3 block: $P_{:,1:3} \leftarrow P_{:,1:3}\,R_\Delta$, so the delta acts on incoming LiDAR points.
• Pixel shift: add $\Delta u\,P_{3,:}$ to row 0 and $\Delta v\,P_{3,:}$ to row 1. Because $u = (P_{1,:}X)/(P_{3,:}X)$, that yields a depth-independent pixel offset $u \mapsto u + \Delta u$ (same for $v$).

Runs after calibrate.py for small rotation and pixel-offset corrections.

Comparison

	Click DLT (calibrate.py)	Multi-pose board (calibrate_extrinsics.py)
Status	Common on prod	feature/perc/physical-calibration
Inputs	Manual 3D–2D pairs	Synced images + dense /lidar_points
Model	Free $P$	$P = K_{\text{runtime}}[R \mid t]$
Failure mode	Bad clicks / sparse geometry	Bad planes, labeling consensus, observability

Feature-branch math: extrinsics, projection.