Cayley Transformation and Numerical Stability of Calibration Equation

摘要

The application of Cayley transformation to en_hance the numerical stability of camera calibration is inves_tigated. First, a new calibration equation, called the standard calibration equation, is introduced using the Cayley transfor_mation and its analytical solution is obtained. The standard calibration equation is equivalent to the classical calibration equation, but it exhibits remarkable better numerical stabil_ity. Second, a one-parameter calibration family, called the Cayley calibration family which is equivalent to the stan_dard 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 give views. The con_dition number of equations in the Cayley calibration family varies with the parameter value, and an algorithm to deter_mine the best parameter value is provided. Third, the gen_eralized Cayley calibration families equivalent to the stan_dard calibration equation are also introduced via general_ized 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 cal_ibration family also depends on the parameter value. In addi_tion, a more generic calibration family is also proposed and it is proved that the standard calibration equation, the Cay_ley calibration family and the S-Cayley calibration family are all some special cases of this generic calibration family.