Self-calibration using the finite element approach

Theoretical and practical aspects of the finite element approach in self-calibration developed by the author have been investigated in detail. An experimental verification of the proposed technique has been carried out. The experimental results support the conclusion that the reported technique gives a good representation of the systematic errors in photogrammetry. (1) present state-of-the-art in analytical self-calibration is basically as follows: Analytical approaches to camera calibration have a common starting point: compensation of systematic image coordinate errors by analytical models incorporated into the photogrammetric projective equation. The parameters defining systematic errors are then recovered simultaneously with the projective parameters (position, orientation, focal length, principal point) in a least-squares adjustment leading to the minimization of the quadratic form of the residuals of measured quantities. Most reported analytic functions representing film shrinkage, film flatness, and radial lens distortion are assumed to be valid throughout the image plane. A logical extension of this theory is to divide the image plane domain into subdomains or finite elements and then to prescribe a mathematical model for systematic errors over the image plane domain in a piecewise fashion, element by element, thus eliminating the assumption of symmetry. 0099-1112/86/5203-411$02.25/0 ©1986 American Society for Photogrammetry and Remote Sensing where P, are linearly independent selected functions existing over the domain and its boundary and C, are unknown parameters to be determined, not by a differential equation, but by a system of M discrete THE BASIC IDEAS OF THE FINITE ELEMENT METHOD The finite element method can be described in a few words. Suppose that in the problem to be solved it is required to find a function v which minimizes a given expression. This minimizing property leads to a differential equation for v (the Euler Equation), but normally an exact solution is impossible and some approximation is necessary. A trial solution is considered in which v is approximated by a solution V m of linear form Self-Calibration Using the Finite Element Approach PHOTOGRAMMETRIC ENGINEERING AND REMOTE SENSING, Vol. 52, No.3, March 1986, pp. 411-418. INTRODUCTION SINCE ABOUT 1956 the finite element method has developed enormously. It began as a numerical method of stress analysis and is still most widely used for this purpose (Cook, 1974). In addition, it has become popular in many other areas including heat conduction, seepage flow, fluid dynamics, acoustics, electric and magnetic fields, and recently in digital terrain modeling (Ebner et aI., 1978, 1980, 1981). PhotogTammetry can be described as a non-contact measuring technique which extracts reliable geometric measurements from imagery records. Generally, this measuring technique is categorized into two fields: analog and analytical photogrammetry. The last two decades have witnessed a tremendous gTowth for the analytical approach due to developments in high speed digital computer technology, and the growing importance of non-conventional photogrammetry. In the analytical approach, systematic errors in photogrammetry are compensated for by mathematical functions (e.g., orthogonal polynomials). In many cases these mathematical functions yield no insight into the physical or mechanical source of the errors. It will be shown that the finite element approach can play an important role in modeling these errors and in increasing photogrammetric accuracy. In photogrammetry, the finite element method can be used to study and investigate many problems such as camera calibration, film shrinkage, camera stability, flatness of film during exposure, monoand stereo-comparator calibration, distortions in X-ray photogTammetry, and in underwater photogTammetry. It can also be used to correct distortions in satellite imagery. The finite element approach in analytical self-calibration forms the topic of this paper. A search of the current literature (see, for example, Brown (1971), Brown (1972), Kenefick et al. (1972), Ebner (1976), and Fraser (1979)) reveals that the PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING, 1986