## Abstract

We present a simple method to determine the relative distortion of axially symmetric lens systems. This method uses graphs to determine every parametric value instead of nonlinear minimization computation and is composed of an LCD screen to display a square grid pattern of pixel-wide spots and a set of analyzing processes for the spots in the image. The two Cartesian components of the spot locations are processed by a two-step linear least-square fitting to third-order polynomials. The graphs for the coefficients enable us to determine the amount of decentering of the camera lens axis with respect to the center of the image array and the tip/tilt of the screen, which in turn gives the relative distortion coefficient. We present experimental results to demonstrate the utility of the method by comparing our results with the corresponding values determined by open source software available online.

© 2012 Optical Society of America

Full Article |

PDF Article
### Equations (7)

Equations on this page are rendered with MathJax. Learn more.

(1)
$$\frac{{r}^{\prime}-r}{r}={K}_{1}{r}^{2},$$
(2)
$$\begin{array}{l}{x}^{\prime}(i,j)=[1+{K}_{1}{\mathrm{\Delta}}^{2}({i}^{2}+{j}^{2})]\mathrm{\Delta}i\\ {y}^{\prime}(i,j)=[1+{K}_{1}{\mathrm{\Delta}}^{2}({i}^{2}+{j}^{2})]\mathrm{\Delta}j\end{array}\},$$
(3)
$$\begin{array}{l}{x}^{\prime}(i,j)=[1+{K}_{1}{\mathrm{\Delta}}^{2}({i}^{2}+{j}^{2})]\mathrm{\Delta}i+{x}_{o}\\ {y}^{\prime}(i,j)=[1+{K}_{1}{\mathrm{\Delta}}^{2}({i}^{2}+{j}^{2})]\mathrm{\Delta}j+{y}_{o}\end{array}\},$$
(4)
$${y}^{\prime}(i,{j}_{o})=\mathrm{\Delta}[1+{K}_{1}{\mathrm{\Delta}}^{2}({i}^{2}+{j}_{o}^{2})]{j}_{o}+{y}_{o},$$
(5)
$${y}^{\prime}(i,{j}_{o})={a}_{2T}({j}_{o}){i}^{2}+{a}_{0T}({j}_{o}),$$
(6)
$$\begin{array}{l}{a}_{2T}({j}_{o})={K}_{1}{\mathrm{\Delta}}^{3}{j}_{o}\\ {a}_{0T}({j}_{o})={K}_{1}{\mathrm{\Delta}}^{3}{j}_{o}^{3}+\mathrm{\Delta}{j}_{o}+{y}_{o}\end{array}\}.$$
(7)
$$\begin{array}{l}x(i,j)=\mathrm{\Delta}(i-\frac{\mathrm{\Delta}d}{q}\mathrm{\theta}{i}^{2})\\ y(i,j)=\mathrm{\Delta}(j-\frac{\mathrm{\Delta}d}{q}\varphi {j}^{2})\end{array}\},$$