Calibration

The camera calibration phase allows for the extraction of (some or all) parameters that enable the pin-hole model to project points from world coordinates to camera coordinates. In English, camera calibration, which involves deriving intrinsic and/or extrinsic parameters, is referred to as Camera resectioning, as the concept of Camera Calibration can also refer to the problem of the photometric calibration of the system.

Calibration techniques can be divided into two categories depending on which pin-hole camera model is to be derived:

implicit

where the elements of the projection matrix $\mathbf{P}$ or the homography matrix $\mathbf{H}$ are extracted in order to project points from one coordinate system to another, disregarding the internal structure of the sensor;

explicit

where the physical parameters of the system involved in the perspective projection are extracted.

Implicit calibration is usually a faster process and, with a sufficient number of points, represents reality quite well. However, as we will see shortly, the linear version that minimizes an algebraic quantity is not the maximum likelihood estimator. Implicit calibration also overlooks some non-linearities that are always present in physical systems.

Explicit calibration, on the other hand, not only accurately represents the model with the minimum number of parameters but also provides greater flexibility in the use of the obtained parameters. This allows for operations on images or for dynamically varying certain parameters of the system, and it enables the implementation of the maximum likelihood estimator.

To enable the application of the calibration techniques presented in this section, it is essential to establish constraints between the involved projective spaces, such as image coordinates and their corresponding world coordinates. Through these relationships, it is possible to derive the parameters that represent the projective model being utilized.

The boundary that separates implicit calibration from explicit calibration sometimes tends to blur: under certain conditions, it is possible to transition from one mode to the other.

• With Direct Linear Transformation, as discussed in section 8.5.1, it is possible to implicitly calibrate the system by knowing the positions of points in world coordinates and image coordinates, deriving the projection matrix $\mathbf{P}$, or the projection matrix of a single plane $\mathbf{H}$, without explicitly stating any camera parameters. Subsequently, using equation (8.29), one can derive the matrix $[\mathbf{R} \mathbf{t} ]$ of the extrinsic parameters, although information about the intrinsic parameters is still required.
• It has been previously mentioned (see section 8.2.2) how it is possible to derive the rotation matrix given the knowledge of the intrinsic parameter matrix and the positions of the vanishing points.
• If the rotation matrix $\mathbf{R}$ is known, it is possible to explicitly obtain the values of the angles that generated it (there may be multiple solutions in this case).
• If one can obtain the intrinsic parameter matrix $\mathbf{K}$, it is straightforward to explicitly derive the intrinsic parameters of the camera.
• Zhang, in section 8.5.4, proposes a method to derive the intrinsic parameters of the camera if the relative positions of points belonging to the same plane are known, observed from multiple viewpoints.