Two Non-Linear Distortion Effects

• Radial
• Tangential

Radial Distortion

An image of the Louvre museum entry taken with the 7.5 mm f/5.6 circular fisheye Nikkor lens,by Dan Lindsay;under CC BY 3.0 (Source)

Canon 8–15 mm zoom at 8 mm of BMW M3,CC BY-SA 3.0 (Source)

A Goode projection of a Visible Earth image collected by the Earth Observatory experiment of the U.S. Government's NASA space agency. The reticle is 15 degrees in latitude and longitude. (Public domain, courtesy of NASA) Source

Distortion Examples from theOpenCV documentation

Radial Distortion

• Simplest effective model:

$$ \begin{align} x &= x_d(1 + a_1r^2 + a_2r^4) \\ y &= y_d(1 + a_1r^2 + a_2r^4) \\ r &= \sqrt{x_d^2+y_d^2} \end{align} $$

• $(x,y)$ are the true co-ordinate of some point
• $(x_d,y_d)$ are the co-ordinates in the distorted image

OpenCV uses a sixth order model

$$ \begin{align} x_d &= x(1 + a_1r^2 + a_2r^4 + a_3r^6) \\ y_d &= y(1 + a_1r^2 + a_2r^4 + a_3r^6) \end{align} $$

Tangential Distortion

Tangential Distortion

Tangential Distortion occurs when the lens and image plane are not parallel.

A pixel $(x,d)$ is moved to $(x_d,y_d)$ defined by

$$ \begin{align} x_d &= x+ [2p_1xy+p_2(r^2+2x^2)] \\ y_d &= y+[p_1(r^2+2y^2)+2p_2xy] \end{align} $$

Closure

Time for exercises

• Calibrate your own camera
• Use OpenCV
• Checkerboards in the class set