Image Distortion Non-Linear Distortion and Calibration Hans Georg Schaathun
NTNU, Noregs Teknisk-Naturvitskaplege Universitet
September 2023
Start the sound player below, please. Image Distortion
Non-Linear Distortion and Calibration
This is a talk for the taught module in machine vision, AIS2204 at NTNU.
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
We see the same problem when we draw flat maps of a round Earth.
Parts of the map has to be stretched, or compressed, or fragmented.
This is not an issue if you map a small area, of if you take pictures
with a narrow angle lens, because a sufficiently small section of the
spherical surface will appear approximately flat.
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
$$
\begin{align}
x_d &= x(1 + a_1r^2 + a_2r^4 + a_3r^6) \\
y_d &= y(1 + a_1r^2 + a_2r^4 + a_3r^6)
\end{align}
$$
Radial distortion is non-linear. The light that should have appeared in
pixel (x,y) is shifted to another position (xd,yd) which can be described
by these polynomial equations.
The simplest effective model as given by Ma is these fourth degree expressions.
OpenCV uses a sixth order model, given slightly differently with (xd,yd) as a function
of (x,y) instead of the other way around.
Calibration algorithms are used to determined the coeffients a1, a2, and a3.
Given the availability of calibration software, we can usually ignore
non-linear distortion in our analysis.
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
$$
\begin{align}
x_d &= x+ [2p_1xy+p_2(r^2+2x^2)] \\
y_d &= y+[p_1(r^2+2y^2)+2p_2xy]
\end{align}
$$
Tangential distortion occurs when the lens and image plane are not parallel.
Like radial distortion, tangential distortion is non-linear, so that the pixel that should have had co-ordinates (x,y) is shifted to position (xd,yd).