A Nonlinear Gauss–Seidel Algorithm for Noncoplanar and Coplanar Camera Calibration with Convergence Analysis

摘要

In this study, we discuss the nonlinear structure of the camera calibration problem and present new and provably convergent algorithms for noncoplanar and coplanar cases. From the perspective of optimization theory, we have included the following features in solving this nonlinear problem: (1) An initialization algorithm that computes an approximate solution as a starting value close to the global minimum. (2) A main estimation method that partitions the parameter space and uses a Gauss–Seidel optimization procedure for block components. For the noncoplanar case, the extrinsic and lens distortion parameters are computed by linear iterations or in closed form in each iteration. Nonlinear optimization is performed on a reduced parameter space of dimension three. For the coplanar case, the lens distortion parameters are computed by linear iterations. In both cases, the orthonormality condition of the camera vectors is satisfied. Thus, while performing nonlinear optimization over all parameters, we still retain many advantages of the linear methods, and in the process obtain an optimal solution. (3) A Lyapunov type convergence analysis is given for the algorithms. The structure of the objective function is analyzed in each iteration. In addition, for the coplanar case, we discuss new methods for obtaining starting estimates of image center and scale factor parameters. Furthermore, we consider lens distortion with radial, decentering, and thin prism distortion models.