D · Interactive tool · Module 06

Camera–LiDAR Projection Visualizer

A synthetic scene viewed by a camera and a LiDAR. Perturb the extrinsics and watch the projected points slide off the objects; read the reprojection error by distance.

Preliminary engineering guidance. Results depend on the assumptions listed below and on simplified models. Verify with manufacturer datasheets and a test on your own hardware before committing to a design.

Synthetic scene

Extrinsic error (applied to the camera←LiDAR transform, in the camera frame)

Roll about x 0 °

Pitch about y 0 °

Yaw about z 0 °

Shift x (right) 0 m

Shift y (down) 0 m

Shift z (forward) 0 m

Show where the correct extrinsics put the points

Three different operations

3D point transformation (Coordinate Transformation Calculator): P_C = T_CL·P_L. 3D to 3D, exact, reversible.
3D-to-2D projection (above): u = f_x·X/Z + c_x, v = f_y·Y/Z + c_y on the camera-frame point. Discards depth, not reversible.
Depth from a 2D detection (below): a pixel defines a ray, not a point. You need an extra assumption (a known plane, a known object size, or a range sensor) to place it in 3D.

Pixel column u

Pixel row v (image centre is 180)

Assumptions and limitations

The scene is synthetic: a box 7–9 m ahead, a pole at 13.5 m, a wall at 19 m and a flat floor 1.2 m below the LiDAR origin. The LiDAR points lie on those surfaces and on a ground grid.
Camera: ideal pinhole, f_x = f_y = 450 px, principal point at the image centre, 640×360 pixels, no lens distortion, optical frame x right, y down, z forward. Mount: 0.1 m ahead and 0.3 m above the LiDAR, looking forward. These are illustration values, not a product.
The “extrinsic error” is applied as an additional transform E in the camera frame: T_CL,est = E·T_CL,true. Real calibration errors also come from intrinsics, time offset and motion, which are not modelled here.
Reprojection error is measured in pixels between the projection with the true and the perturbed extrinsics for the same LiDAR point. It is a sensitivity demonstration, not a calibration quality metric for a real rig (that needs real detections or edges).
A rotation error of θ moves an on-axis point by about f·tan θ pixels at any distance; a translation error t moves it by about f·t/Z pixels, shrinking with distance Z.
The floor-depth example assumes the camera is level, 1.5 m above a flat floor, with the optical axis horizontal: Z = f_y·h / (v − c_y). A tilted camera, a slope or a non-flat floor breaks it.

References

Pinhole camera model: any computer-vision text (the formulas on this page are the complete model used).