Brown-Conrady Distortion

Real lenses introduce geometric distortion: straight lines in the world project as curves in the image. The Brown-Conrady model captures the two dominant distortion effects — radial and tangential — as polynomial functions of the distance from the optical axis.

Distortion Model

Given undistorted normalized coordinates ${\mathbf{n}}_u=[x,y{]}^T$, the distorted coordinates ${\mathbf{n}}_d=[x_d,y_d{]}^T$ are:

$$r^2=x^2+y^2$$

Radial distortion (barrel/pincushion):

$$\alpha =1+k_1{r}^2+k_2{r}^4+k_3{r}^6$$

Tangential distortion (decentering):

$${\delta }_x=2p_{1}xy+p_2(r^2+2x^2)$$ $${\delta }_y=p_1(r^2+2y^2)+2p_{2}xy$$

Combined:

$$x_d=x\cdot \alpha +{\delta }_x$$ $$y_d=y\cdot \alpha +{\delta }_y$$

The model has five parameters: radial coefficients $(k_1,k_2,k_3)$ and tangential coefficients $(p_1,p_2)$.

OpenCV equivalence: This is identical to OpenCV's distortPoints with the 5-parameter model (k1, k2, p1, p2, k3). Note OpenCV's parameter ordering differs from the mathematical ordering.

Physical Interpretation

• $k_1>0$: barrel distortion (lines bow outward from center)
• $k_1<0$: pincushion distortion (lines bow inward)
• $k_2,k_3$: higher-order radial corrections
• $p_1,p_2$: tangential distortion from imperfect lens-sensor alignment (lens elements not perfectly centered)

Typical values for industrial cameras: $|k_1|\sim 0.01\text{-}0.3$, $|k_2|\sim 0.001\text{-}0.1$, $|p_1|,|p_2|\sim 0.0001\text{-}0.01$.

When to Fix $k_3$

The sixth-order radial term $k_3{r}^6$ is only significant for wide-angle lenses where $r$ reaches large values. For typical industrial cameras with moderate field of view:

• $k_3$ is poorly constrained by the data and often absorbs noise
• Estimating $k_3$ can cause overfitting, leading to worse extrapolation outside the calibration region
• The library defaults to fix_k3: true in most problem configurations

Recommendation: Only estimate $k_3$ with high-quality data covering the full image area, or for lenses with field of view > 90°.

Undistortion (Inverse)

Given distorted coordinates ${\mathbf{n}}_d$, recovering undistorted coordinates ${\mathbf{n}}_u$ requires inverting the distortion model. Since the forward model is a polynomial without a closed-form inverse, calibration-rs uses iterative fixed-point refinement:

$${\mathbf{n}}_u^{(0)}={\mathbf{n}}_d$$

$${\mathbf{n}}_u^{(k+1)}={\mathbf{n}}_d-\left(D({\mathbf{n}}_u^{(k)})-{\mathbf{n}}_u^{(k)}\right)$$

This rearranges the distortion equation to isolate ${\mathbf{n}}_u$ and iterates to convergence. The default is 8 iterations, which provides accuracy well below sensor noise for typical distortion magnitudes.

Convergence

The fixed-point iteration converges when the Jacobian of the distortion residual has spectral radius less than 1, which holds for physically realistic distortion values (small $k_i$, $p_i$ relative to $r$). For extreme distortion, more iterations may be needed via the iters parameter.

OpenCV equivalence: cv::undistortPoints performs the same iterative inversion.

The DistortionModel Trait

#![allow(unused)]
fn main() {
pub trait DistortionModel<S: RealField> {
    fn distort(&self, n_undist: &Point2<S>) -> Point2<S>;
    fn undistort(&self, n_dist: &Point2<S>) -> Point2<S>;
}
}

The BrownConrady5<S> struct:

#![allow(unused)]
fn main() {
pub struct BrownConrady5<S: RealField> {
    pub k1: S, pub k2: S, pub k3: S,
    pub p1: S, pub p2: S,
    pub iters: u32,   // undistortion iterations (default: 8)
}
}

NoDistortion is the identity implementation for distortion-free cameras.

Distortion in the Projection Pipeline

Distortion operates in normalized image coordinates (after pinhole projection, before intrinsics):

$$\text{pixel}=K(\underset{\text{distorted normalized}}{\underbrace{D(\Pi ({\mathbf{P}}_c))}})$$

This is the standard convention: distortion is defined on the $Z=1$ plane, not in pixel space. The advantage is that distortion parameters are independent of image resolution and focal length.