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radial distortion

We represent the camera model with the follouing equation:

\[ (u,v,1) = K * (R|t)* (x_w, y_w, z_w, 1)\]

In this equation, $ (u,v)$ is not the actually observed image point since virtually all imaging devices introduce certain amount of nonlinear distortions. Among the nonlinear distortions, radial distortion, wich is performed along the radial direction from the center of distortion is the most severe paprt. The removal or alleviation of the radial distortion is commonly performed by first applaying a parametric radial distortion model, estimating the distortion coeficients, and then correcting the distortion.

Lens distortion is very important for acurate 3-D measurement. Let $ (u_d, v_d)$ be the actually observed image point and assume that the center of distortion is at the principal point. The relationship between the unddistored and the distored radial distances is given by $ r_d = r + \delta_r$, where $ r_d$ is the distored radial distance and $\delta_r$ the radial distortion.

Notation of some other variables used throughout this document:

Remarq: In this chapter $(x,y)$ and $ (x_d, y_d)$ are in the retine reference frame. In the Introduction we note it$ (x_r, y_r)$.

Most of the existing works on radial distortion models can be trace back to an early study in pohotogrammetry where the radial distortion is governed by the following polynomial equation:

\[ r_d = r* f(r) = r(1+ k_1 r^2+ k_2 r^4 + k_3 r^6 + ...) \]

where $ k_1$, $k_2$,$ k_3 $, ... are the distortion coefficients.

It follows that

\[ x_d = x*f(r) , y_d = y*f(r)\]

which is equivalent to

\[ \left\{ \begin{array}{cc} u_d-u_0=(u-u_0)f(r)\\v_d-v_0=(v-v_0)f(r) \end{array}\right. \]

(we can see this resuilt using the intrinsec parameters matrix)

For the polinomial radial distortion model and its variations, the distortion is specially dominated by the first term and it has also be found that too hight an order my cause numerical inestability. To use two terms of radial distortion is a good choice. When using two coefficients, the relationship between the distored ans undistored radial distances becomes

\[ r_d = r*(1+k_1 r^2+k_2 r^4) \]

The inverse of this polinomial function is difficult to perform analytically but can be obtained numerically via an iterative scheme. There are also ways to approximate it without iterations, where $ r $ can be calculated from $ r_d $ by

\[ r = r_d *(1-k_1 r^2_d - k^2 r^4_d) \]

The fitting results given by the above model can be satisfactory when the distortion coefficients are small values (this is our case). However, the equation introduces another source of error that will inevitably degrade the calibration accuracy. For more details refer the paper A New Analytical Radial Distortion Model for Camera Calibration by Lili Ma, YangQuan Chen and Kevin L. Moore.

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