Calibration according to Tsai

Camera calibration for various applications requires a comprehensive understanding of both intrinsic and extrinsic parameters. One of the most widely used methods is undoubtedly that of Tsai (Tsa87) from 1985. The merit of Tsai's work was to bring order to the previously discussed state of the art and to provide a unique and accepted nomenclature for the camera parameters as presented here.

The Tsai camera model is based on the perspective projection of the pin-hole camera and consists (in its classical form) of 11 parameters:

f

Focal length of the camera

k

First-order radial distortion coefficient

Cx,Cy

Coordinates of the optical center of the lens

Sx

A horizontal scale factor

Rx, Ry, Rz

Rotation angles for the transformation between world coordinates and camera coordinates

Tx, Ty, Tz

Translation vector for the transformation between world coordinates and camera coordinates

Tsai conducts a comprehensive analysis of all the techniques developed thus far for calibration, and ultimately proposes a modular system, where each module allows for the extraction of a series of these parameters.

It mainly points out that if the camera is distorted but the principal point is aligned with the center of distortion, the following relationships hold:

$$\frac{u_{d}}{v_{d}} = \frac{u_{u}}{v_{u}}$$ (8.61)

Consequently, it is possible to establish constraints under this condition using the distorted coordinates rather than the undistorted ones. This method is therefore also referred to as the radial alignment constraint (RAC).

Initially, using the camera parameters provided by the manufacturer, calculate the translation and rotation vector from a grid with coplanar points $z_i=0$ of known coordinates, leveraging the constraint

$$(r_0 x_i + r_1 y_i + \tilde{t}_x) u'_i = (r_3 x_i + r_4 y_i + \tilde{t}_y) v'_i$$ (8.62)

with $(u'_i, v'_i)$ normalized camera coordinates using the parameters of the camera and lens supplied by the manufacturer. From this constraint, a large overdetermined linear system of the type can be created.

$$\begin{bmatrix} x_i u'_i & y_i u'_i & u'_i & - v'_i x_i & ... ...}{\tilde{t}_y} \\ \frac{r_4}{\tilde{t}_y} \end{bmatrix} = v'_i$$ (8.63)

having set $\tilde{t}_y \neq 0$ (that is, the grid should not intersect the optical axis). The remaining parameters of the matrix $\mathbf{R}$ are obtained using equation (8.20).

Subsequently, it proceeds to derive the correct intrinsic parameters using these values for the rotation and translation matrix.