Mathematical Methods for Camera Self-Calibration in Photogrammetry and Computer Vision

Abstract

Camera calibration is a central subject in photogrammetry and geometric computer vision. Selfcalibration is a most flexible and highly useful technique, and it plays a significant role in camera automatic interior/exterior orientation and image-based reconstruction. This thesis study is to provide a mathematical, intensive and synthetic study on the camera self-calibration techniques in aerial photogrammetry, close range photogrammetry and computer vision. In aerial photogrammetry, many self-calibration additional parameters (APs) are used increasingly without evident mathematical or physical foundations, and moreover they may be highly correlated with other correction parameters. In close range photogrammetry, high correlations exist between different terms in the ‘standard’ Brown self-calibration model. The negative effects of those high correlations on self-calibration are not fully clear. While distortion compensation is essential in the photogrammetric self-calibration, geometric computer vision concerns auto-calibration (known as selfcalibration as well) in calibrating the internal parameters, regardless of distortion and initial values of internal parameters. Although camera auto-calibration from N 3 views has been studied extensively in the last decades, it remains quite a difficult problem so far. The mathematical principle of self-calibration models in photogrammetry is studied synthetically. It is pointed out that photogrammetric self-calibration (or building photogrammetric self-calibration models) can – to a large extent – be considered as a function approximation problem in mathematics. The unknown function of distortion can be approximated by a linear combination of specific mathematical basis functions. With algebraic polynomials being adopted, a whole family of Legendre self-calibration model is developed on the base of the orthogonal univariate Legendre polynomials. It is guaranteed by the Weierstrass theorem, that the distortion of any frame-format camera can be effectively calibrated by the Legendre model of proper degree. The Legendre model can be considered as a superior generalization of the historical polynomial models proposed by Ebner and Grun, to which the Legendre models of second and fourth orders should be preferred, respectively. However, from a mathemtical viewpoint, the algebraic polynomials are undesirable for self-calibration purpose due to high correlations between polynomial terms. These high correlations are exactly those occurring in the Brown model in close range photogrammetry. They are factually inherent in all selfcalibration models using polynomial representation, independent of block geometry. According to the correlation analyses, a refined model of the in-plane distortion is proposed for close range camera calibration. After examining a number of mathematical basis functions, the Fourier series are suggested to be the theoretically optimal basis functions to build the self-calibration model in photogrammetry. Another family of Fourier self-calibration model is developed, whose mathematical foundations are the Laplace’s equation and the Fourier theorem. By considering the advantages and disvantages of the physical and the mathematical self-calibration models, it is recommended that the Legendre or the Fourier model should be combined with the radial distortion parameters in many calibration applications. A number of simulated and empirical tests are performed to evaluate the new self-calibration models. The airborne camera tests demonstrate that, both the Legendre and the Fourier self-calibration models are rigorous, flexible, generic and effective to calibrate the distortion of digital frame airborne cameras of large-, mediumand small-formats, mounted in singleand multi-head systems (including the DMC, DMC II, UltraCamX, UltraCamXp, DigiCAM cameras and so on). The advantages of the Fourier model result from the fact that it usually needs fewer APs and obtains more reliable distortion calibration. The tests in close range photogrammetry show that, although it is highly correlated with