Distortion

Assuming the camera model is pinhole, according to Brown's distortion model, there are two types of common distortions: radial distortion and tangential distortion.

Radial distortion

• The most commonly encountered distortions are radially symmetric, due to the symmetry of a photographic lens. These radial distortions can be classified as either barrel distortions or pincushion distortions.

  • Barrel distortion: image magnification decreases with distance from the optical axis.

  • Pincushion distortion: image magnification increases with distance from the optical axis.

• Mathematically, barrel and pincushion distortion are quadratic, meaning they increase as the square of distance from the center. We can model it with three parameters: $k_1, k_2, k_3$.

$$x_{\text {distorted }}=x\left(1+k_{1} r^{2}+k_{2} r^{4}+k_{3} r^{6}\right) \\ y_{\text {distorted }}=y\left(1+k_{1} r^{2}+k_{2} r^{4}+k_{3} r^{6}\right)$$

Tangential distortion

• Tangential distortion can happen when the lens is not fully parallel to the image plane.

• Mathematically, we can model this distortion with two parameters: $p_1, p_2$.

$$x_{\text {distorted }}=x+2 p_{1} x y+p_{2}\left(r^{2}+2 x^{2}\right) \\ y_{\text {distorted }}=y+p_{1}\left(r^{2}+2 y^{2}\right)+2 p_{2} x y$$

References

• https://en.wikipedia.org/wiki/Distortion_(optics)

• 14 Lectures in Visual SLAM