Laserline Device Calibration

Laserline calibration jointly estimates camera parameters and a laser plane from observations of both a calibration board and laser line projections on the board. This is used in laser triangulation systems where a camera and laser are rigidly mounted together.

Problem Formulation

Parameters

• Camera intrinsics: $K=(f_x,f_y,c_x,c_y)$
• Distortion: $\mathbf{d}=(k_1,k_2,k_3,p_1,p_2)$
• Sensor tilt: $({\tau }_x,{\tau }_y)$ (optional, for Scheimpflug cameras)
• Per-view poses: $\{T_v\}$ (camera-to-target SE(3))
• Laser plane: normal $\hat{\mathbf{n}}\in S^2$, distance $d\in \mathbb{R}$

Observations

Each view provides two types of observations:

1. Calibration points: 2D-3D correspondences from the chessboard (same as planar intrinsics)
2. Laser pixels: 2D pixel positions where the laser line appears on the target

Objective

The cost function has two terms:

$$F=w_c\sum _{\text{calib}}{\rho }_c\left(\Vert \pi (K,\mathbf{d},S,T_v,{\mathbf{P}}_j)-{\mathbf{p}}_{vj}\Vert \right)+w_l\sum _{\text{laser}}{\rho }_l\left(r_{\text{laser}}(\cdot )\right)$$

where $w_c,w_l$ are weights, ${\rho }_c,{\rho }_l$ are (possibly different) robust loss functions, and $r_{\text{laser}}$ is the laser residual.

Laser Residual Types

Two approaches are implemented for the laser residual, selectable via configuration.

PointToPlane (3D Distance)

Algorithm:

1. Undistort laser pixel to normalized coordinates
2. Back-project to a ray in camera frame
3. Intersect the ray with the target plane (at $Z_{\text{target}}=0$, transformed by pose $T_v$) to get a 3D point ${\mathbf{P}}_c$
4. Compute signed distance from ${\mathbf{P}}_c$ to the laser plane:

$$r=\sqrt{w}\cdot ({\hat{\mathbf{n}}}^T{\mathbf{P}}_c-d)$$

Residual dimension: 1 (meters)

LineDistNormalized (2D Line Distance) — Default

Algorithm:

1. Compute the 3D intersection line of the laser plane and the target plane (in camera frame)
2. Project this 3D line onto the $Z=1$ normalized camera plane
3. Undistort the laser pixel to normalized coordinates (done once, not per-iteration)
4. Measure the perpendicular distance from the undistorted pixel to the projected line
5. Scale by $\sqrt{f_x\cdot f_y}$ for pixel-comparable units:

$$r=\sqrt{w}\cdot d_{\perp }\cdot \sqrt{f_x{f}_y}$$

Residual dimension: 1 (effective pixels)

Comparison

Both approaches yield similar accuracy in practice (<6% intrinsics error, <5° plane normal error in synthetic tests).

Derivation: Line-Distance Residual

Plane-Plane Intersection Line

The laser plane ${\hat{\mathbf{n}}}^T\mathbf{P}=d$ and the target plane (normal $\hat{\mathbf{m}}$, distance $e$ from camera, derived from pose $T_v$) intersect in a 3D line with:

$$\text{direction: }\hat{\mathbf{l}}=\hat{\mathbf{n}}\times \hat{\mathbf{m}}$$

$$\text{origin: solved from }\left[\begin{array}{c}{\hat{\mathbf{n}}}^T\\ {\hat{\mathbf{m}}}^T\end{array}\right]{\mathbf{P}}_0=\left[\begin{array}{c}d\\ e\end{array}\right]$$

Projection onto Normalized Plane

Project the 3D line to the $Z=1$ plane:

$${\mathbf{p}}_0=\text{point on line at }Z=1$$ $${\hat{\mathbf{d}}}_{\text{2D}}=\text{direction projected to }Z=1\text{ plane}$$

Perpendicular Distance

For an undistorted pixel $\mathbf{q}$ in normalized coordinates, the perpendicular distance to the 2D line is:

$$d_{\perp }=\frac{|(\mathbf{q}-{\mathbf{p}}_0)\times {\hat{\mathbf{d}}}_{\text{2D}}|}{|{\hat{\mathbf{d}}}_{\text{2D}}|}$$

Configuration

#![allow(unused)]
fn main() {
pub struct LaserlineDeviceConfig {
    // Initialization
    pub init_iterations: usize,        // Iterative intrinsics iterations (default: 2)
    pub fix_k3_in_init: bool,          // Fix k3 during init (default: true)
    pub fix_tangential_in_init: bool,  // Fix p1, p2 during init (default: false)
    pub zero_skew: bool,               // Enforce zero skew (default: true)
    pub sensor_init: ScheimpflugParams, // Initial sensor tilt (default: identity)

    // Optimization
    pub max_iters: usize,              // LM iterations (default: 50)
    pub verbosity: usize,
    pub calib_loss: RobustLoss,        // Default: Huber { scale: 1.0 }
    pub laser_loss: RobustLoss,        // Default: Huber { scale: 0.01 }
    pub calib_weight: f64,             // Weight for calibration residuals (default: 1.0)
    pub laser_weight: f64,             // Weight for laser residuals (default: 1.0)
    pub fix_intrinsics: bool,
    pub fix_distortion: bool,
    pub fix_k3: bool,                  // Default: true
    pub fix_sensor: bool,              // Default: true
    pub fix_poses: Vec<usize>,         // Default: vec![0]
    pub fix_plane: bool,
    pub laser_residual_type: LaserlineResidualType, // Default: LineDistNormalized
}
}

Weight Balancing

Since calibration residuals (in pixels) and laser residuals may have different scales, the weights allow balancing their relative influence. A common starting point is calib_weight = 1.0 and laser_weight = 1.0, adjusting if one term dominates.

Complete Example

#![allow(unused)]
fn main() {
use vision_calibration::prelude::*;
use vision_calibration::laserline_device::*;

let mut session = CalibrationSession::<LaserlineDeviceProblem>::new();
session.set_input(laserline_input)?;

run_calibration(&mut session, None)?;

let export = session.export()?;
println!("Plane normal: {:?}", export.estimate.params.plane.normal);
println!("Plane distance: {:.4}", export.estimate.params.plane.distance);
println!("Reprojection error: {:.4} px", export.mean_reproj_error);
println!("Laser error: {:.4}", export.stats.mean_laser_error);
}

Scheimpflug Support

For laser profilers with tilted sensors, the sensor model parameters $({\tau }_x,{\tau }_y)$ are jointly optimized. The ReprojPointPinhole4Dist5Scheimpflug2 factor handles the extended camera model.