LENS DISTORTION

In real case scenarios there is the need to model the effect of lens distortion on the image

there are 2 principle types of distortion:

flowchart TD
A[Distortions types]
B[radial]
C[distortions due to lens curvature]
D[tangential]
E[due to misalignment \n with the optical sensors]
A --> B & D
B --> C
D --> E

In order to account this types of distortion the model is adjusted accordingly:

$$\left[\begin{array}{c}{x}_{{}^{\prime }}\\ y_{{}^{\prime }}\end{array}\right]=L(r)\times \left[\begin{array}{c}\stackrel{\sim }{x}\\ \stackrel{\sim }{y}\end{array}\right]+\left[\begin{array}{c}d\stackrel{\sim }{x}\\ d\stackrel{\sim }{y}\end{array}\right]$$

Where the non-distorted coordinates ($\left[\begin{array}{c}\stackrel{\sim }{x}\\ \stackrel{\sim }{y}\end{array}\right]$) are mapped to distorted coordinates ($\left[\begin{array}{c}{x}_{{}^{\prime }}\\ y_{{}^{\prime }}\end{array}\right]$) through a non linear function $L(r)$ of the distance from the distortion center $r=\sqrt{(\stackrel{\sim }{x}-{\stackrel{\sim }{x}}_c{)}^2+(\stackrel{\sim }{x}-{\stackrel{\sim }{x}}_c{)}^2}$

The $L(r)$ function is defined for positive values of $r$ only and for $r=0L(r)=1$. This non linear function is approximated by it’s Taylor series:

$$L(r)=1+k_1{r}^2+k_2{r}^4+k_3{r}^6....$$

The tangential distortion is approximated as follows

$$\left[\begin{array}{c}d\stackrel{\sim }{x}\\ d\stackrel{\sim }{y}\end{array}\right]=\left[\begin{array}{c}2p_1\stackrel{\sim }{x}\stackrel{\sim }{y}+p_2(r^2+2{\stackrel{\sim }{x}}^2)\\ p_1(r^2+2{\stackrel{\sim }{x}}^2)+2p_2\stackrel{\sim }{x}\stackrel{\sim }{y}\end{array}\right]$$

The radial coefficients $k_1{k}_2....$ the optical center $(x_c,y_c)$ and the tangential distortion coefficients $p_1,p_2$ are the set of lens distortion parameters that need to be accounted for a realistic modeling of the camera. (for the sake of simplicity the distortion center is taken to coincide with the optical center)

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