Cayley Transformation and Numerical Stability of Calibration Equation

Abstract

The application of Cayley transformation to enhance the numerical stability of camera calibration is investigated. First, a new calibration equation, called the standard calibration equation, is introduced using the Cayley transformation and its analytical solution is obtained. The standard calibration equation is equivalent to the classical calibration equation, but it exhibits remarkable better numerical stability. Second, a one-parameter calibration family, called the Cayley calibration family which is equivalent to the standard calibration equation, is obtained using also the Cayley transformation and it is found that this family is composed of those infinite homographies whose rotation has the same axis with the rotation between the two given views. The condition number of equations in the Cayley calibration family varies with the parameter value, and an algorithm to determine the best parameter value is provided. Third, the generalized Cayley calibration families equivalent to the standard calibration equation are also introduced via generalized Cayley transformations. An example of the generalized Cayley transformations is illustrated, called the S-Cayley calibration family. As in the Cayley calibration family, the numerical stability of equations in a generalized Cayley calibration family also depends on the parameter value. In addition, a more generic calibration family is also proposed and it is proved that the standard calibration equation, the Cayley calibration family and the S-Cayley calibration family are all some special cases of this generic calibration family.