Three-Dimensional Data from Images

Reinhard Klette, Karsten Schlüns, and Andreas Koschan

Springer Singapore, 1998

ISBN 981-3083-71-9


page 60ff:

In the subsection (iii) Transformation of undistorted into distorted image coordinates the indices $u$ and $v$ are misprinted. They have to be changed from $u$ to $v$ and from $v$ to $u$ in the way given (in Latex notation) as follows:

The distortedly projected, valid image coordinates $(x_v , y_v)$ can be determined from undistorted image coordinates $(x_u, y_u)$ using the equations
$x_v = x_u + D_x $ and $y_v = y_u + D_y$
where it holds that
$D_x = x_u (\kappa_1 r^2 + \kappa_2 r^4)$,
$D_y = y_u (\kappa_1 r^2 + \kappa_2 r^4)$, and
$r = \sqrt{x_u^2 + y_u^2}$ .

The distortion coefficients $\kappa_1$ and $\kappa_2$ have to be calibrated. Starting with the image coordinates $(x_v , y_v)$ a correction of locations of image points can be calculated using these equations as specified above. However, for the inverse calculation (from ideal to valid coordinates) the algebraic transformation of these equations of parameters $\kappa_1$ and $\kappa_2$ leads to a system of nonlinear equations. Based on these equations,
$x_{ui} = \fract{x_v}{1 + \kappa_1 r^2_{i-1} + \kappa_2 r^4_{i-1}}$,
$y_{ui} = \fract{y_v}{1 + \kappa_1 r^2_{i-1} + \kappa_2 r^4_{i-1}}$
with $ r_i = \sqrt{ x_{ui}^2 + y_{ui}^2}$ ,for $i \in \{1,\ldots, n\}$
the distortion of ideal coordinates $(x_u, y_u)$ can be determined iteratively ...
CITR: last update: 27 April 2000