Tsai calibration

One of the methods that has played the greatest historical role in the development of camera-calibration techniques is the one proposed by Tsai (Tsa87).

The method organizes the calibration problem by distinguishing between the intrinsic parameters, which describe the camera's internal geometry and optics, and the extrinsic parameters, which describe the camera's position and orientation with respect to the world reference frame.

The classical formulation considers parameters such as focal length, the optical center, a scale factor associated with the acquisition system, and a radial-distortion coefficient, in addition to the six parameters describing the rotation and translation between the world and camera reference frames.

A characteristic aspect of Tsai's method is the use of the Radial Alignment Constraint (RAC). Assuming that the distortion center coincides with the principal point, radial distortion changes the distance of a point from the distortion center but not its direction. Letting $(u_d,v_d)$ denote the distorted coordinates and $(u_u,v_u)$ the undistorted coordinates, we therefore have

$$\frac{u_d}{v_d} = \frac{u_u}{v_u},$$ (9.71)

where the coordinates are expressed relative to the distortion center.

This constraint makes it possible to partially separate the determination of the extrinsic parameters from the estimation of the intrinsic and distortion parameters. The method proceeds in two stages: an initial estimate of the camera's position and orientation is followed by the determination of the effective focal length and distortion parameters.

Tsai's method is primarily of historical importance, since it represents one of the first systematic approaches to geometric camera calibration and formalizes several concepts that are still used today. Modern techniques, however, make it possible to formulate the problem more generally and jointly estimate the camera parameters by directly minimizing the reprojection error.