lens distortion model

topics/01_cameras/index.md

@@ -224,6 +224,27 @@ As lenses don't exactly bend light rays following the pinhole camera model, they
 
 ![](img/lens_distortion.png)
 
+There are plenty of lens distortion models, with varying complexity and number of parameters. One model that is surprisingly simple and effective is the division model:
+
+$$
+\begin{cases}
+u_u = u_{cd} + (u - u_{cd})\alpha
+v_u = v_{cd} + (v - v_{cd})\alpha
+\end{cases}
+$$
+
+with $(u_{cd}, v_{cd})$ being the center of distortion, $(u, v)$ the distorted sensor coordinate, and $(u_u, v_u)$ its undistorted counterpart, matching a pinhole model. The distortion coefficient $\alpha$ is a function of the radial distance from the optical center:
+
+$$
+\alpha = \frac{1}{1+k_1 r^2 + k_2 r^4}
+$$
+
+with:
+
+$$
+r = \sqrt{(u-u_{cd})^2 + (v-v_{cd})^2}
+$$
+
 ## Chromatic aberration
 
 The refractive index of the material used in the lens can slightly differ as a function of the wavelength of the incoming light, causing separation of colors: