Deviations from the camera model

The perspective camera model describes relatively well the image formation process for most cameras. However, when high accuracy is required or when low-end cameras are used, additional effects have to be taken into account.

The failures of the optical system to bring all light rays received from a point object to a single image point or to a prescribed geometric position should then be taken into account. These deviations are called aberrations. Many types of aberrations exist (e.g. astigmatism, chromatic aberrations, spherical aberrations, coma aberrations, curvature of field aberration and distortion aberration). It is outside the scope of this work to discuss them all. The interested reader is referred to the work of Willson [219] and to the photogrammetry literature [179].

Many of these effects are negligible under normal acquisition circumstances. Radial distortion, however, can have a noticeable effect for shorter focal lengths. Radial distortion is a linear displacement of image points radially to or from the center of the image, caused by the fact that objects at different angular distance from the lens axis undergo different magnifications.

It is possible to cancel most of this effect by warping the image. The coordinates in undistorted image plane coordinates $({ x, y})$ can be obtained from the observed image coordinates $(x_o, y_o)$ by the following equation:

$$\begin{array}{l} x = x_o + (x_o - c_x)(K_1 r^2 + K_2 r^4 + \... ... y = y_o + (y_o - c_y)(K_1 r^2 + K_2 r^4 + \ldots ) \end{array}$$ (C21)

where $K_1$ and $K_2$ are the first and second parameters of the radial distortion and

$$r= (x_o - c_x)^2 + (y_o - c_y)^2 \enspace .$$

Note that it can sometimes be necessary to allow the center of radial distortion to be different from the principal point [220].

When the focal length of the camera changes (through zoom or focus) the parameters $K_1$ and $K_2$ will also vary. In a first approximation this can be modeled as follows:

$$\begin{array}{l} x=x_o + (x_o - c_x)(K_{f1} \frac{r^2}{f^2} ... ...\frac{r^2}{f^2} + K_{f2} \frac{r^4}{f^4} + \ldots ) \end{array}$$ (C22)

Due to the changes in the lens system this is only an approximation, except for digital zooms where (3.22) is exact.