Calibration

The camera calibration stage makes it possible to recover some or all of the parameters that allow the pin-hole model to be used to project points from world coordinates into camera coordinates. In English, camera calibration—namely, recovering the intrinsic and/or extrinsic parameters—is called Camera resectioning, since the term Camera Calibration may also refer to the problem of photometric calibration of the system.

Calibration techniques can be divided into two categories according to which pin-hole camera model is to be recovered:

implicit

where the elements of the projective matrix $\mathbf{P}$ or the homography matrix $\mathbf{H}$ are extracted so that points can be projected from one coordinate system to another, without considering 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 reasonably well; however, as we will see shortly, the linear version, which minimizes an algebraic quantity, is not the maximum-likelihood estimator. Implicit calibration also ignores some nonlinearities that are always present in physical systems. In addition to correctly representing the model with the minimum number of parameters, explicit calibration provides greater flexibility in using the recovered parameters, making it possible to perform operations on images or dynamically vary certain system parameters, and allows implementation of the maximum-likelihood estimator.

To apply the calibration techniques presented in this section, constraints between the projective spaces involved are required, such as points in image coordinates and the corresponding points in world coordinates. These relationships make it possible to recover the parameters representing the projective model used.

The boundary separating implicit from explicit calibration sometimes tends to disappear: under suitable conditions, it is possible to move from one to the other.

• With the Direct Linear Transformation, Section 9.5.1, it is possible to calibrate the system implicitly by knowing the positions of points in world and image coordinates, recovering the projection matrix $\mathbf{P}$, or the projection matrix of a single plane $\mathbf{H}$, without explicitly specifying any camera parameters. Using Equation (9.38), it is then possible to recover the matrix $[\mathbf{R} \mathbf{t} ]$ of the extrinsic parameters, provided that information about the intrinsic parameters is available.
• It was mentioned earlier (see Section 9.2.2) that the rotation matrix can be recovered given 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 (multiple solutions may exist in this case).
• If the intrinsic parameter matrix $\mathbf{K}$ can be obtained, the camera's intrinsic parameters can be readily recovered explicitly.
• Zhang, in Section 9.5.4, proposes a method for recovering the camera's intrinsic parameters when the relative positions of points belonging to the same plane are known, with the plane observed from multiple viewpoints.