Constrained FEM Self-Calibration

Polynomial models have been adopted as standard analytical tools in imaging metrology for attenuating image distortions for which corrections are applied to image point measurements. A n alternative correction model, the finite element approach, stems from the fact that the displacement of a point in the image plane is projectively equivalent to a proportional change in the principal distance. By dividing the image plane into smaller entities, distortions can be modeled by piecewise principal distance correction using the finite element method (EM) of self-calibration. Problems can arise, however, when a large number of finite elements is employed. The application of shape function constraints i s proposed to overcome rank defects arising from sparse point distribution for those cases. The adequacy of this modified FEM approach is tested relative to an accepted polynomial model for three different data sets. The evaluation criteria include the degree of compensation for distortions, object space precision and accuracy, and parameter correlation. Introduction In the past 30 years or so, analytical photogrammetry has evolved into a reliable tool for making accurate close-range industrial measurements. A survey of relevant literature reveals a vast range of engineering applications such as automobile deformation mapping (Beyer, 1995), radio antenna calibration (Kenefick, 1971; Fraser, 1986), machinery deformation monitoring (Fraser, 1985), and industrial tool measurement (Fraser and Shortis, 1995). Positional accuracy requirements stipulated for such close-range industrial measurement projects can be very demanding-typically at the sub-millimetre level in all three dimensions. In order to meet such stringent requirements, precise image point mensuration must be exercised along with strong network design and proper analytical treatment of interior geometry. While network design in photogrammetry involves the process of selecting an optimum imaging configuration or exterior geometry, calibration is the task of determining the interior geometry of a camera. Although the principal distance and principal point constitute the framework of the inner geometry, distortions that are inherent in any imaging system must also be considered. Distortions are systematic phenomena that degrade the geometric quality of acquired imagery. That is, points are displaced from their theoretical location in the image plane due to deviations from the fundamental condition of collinearity. The approach most frequently utilized by the photogrammetric community for dealing with distortions is to augment the collinearity equations with polynomial correction terms. These expressions attempt to describe the behavior of systematic phenomena as a function of position in the image plane. Various forms of polynomials have been proposed over the years by many different authors such as Brown (1976), Ebner (1976), and El-Hakim and Faig (1977). HowDepartment of Geomatics Engineering, The University of Calgary, 2500 University Drive NW, Calgary, Alberta T2N 1N4, Canada. PE&RS September 1997 ever, great care must be exercised in selecting a set of additional parameters. Parameters that are statistically insignificant and/or highly correlated with other parameters can cause ill-conditioned systems of normal equations and degrade the accuracy of estimated object points (Ebner, 1976). Moreover, a particular set of additional parameters employed for one sensor may not be appropriate for another. An alternative approach to distortion modeling selfcalibration by the finite element method was proposed by Munjy in 1986. This method is based upon the projective equivalence of a point displacement in the image plane and a proportional change in principal distance. Figure 1 illustrates this concept with the case of accentuated focal plane unflatness. Here it can be seen that the in-plane correction term, dr, is projectively equivalent to the out-of-plane term, dc. The finite element method (FEM) is a versatile modeling technique utilized in many disciplines of engineering for solving complex problems. A familiar photogrammetric use of the FEM is digital terrain modeling in which a surface can be represented by a mesh of three-dimensional splines that describe topographic variation throughout the model. Similarly, the FEM is used in the self-calibration problem to model the variation in principal distance due to distortions throughout the image plane. Principal distances (or corrections to the principal distance) estimated at each element node enable compensation of distortions within elements through the use of a shape function. The shape function, therefore, serves to describe the variation in principal distance due to distortions within a particular finite element. This calibration approach allows for modeling of localized biases that may arise due to focal plane deformation in addition to more global effects such as radial lens distortion. Furthermore, the method does not rely upon the assumption of distortion symmetry about the principal point (Munjy, 1986a). The extension of the concept in Figure 1 to three dimensions is illustrated in Figure 2. This diagram depicts a four finite element principal distance correction surface in the form of a contour plot with the element boundaries denoted by bold lines. Similarly, Figure 3 depicts a correction surface with 36 elements that is clearly smoother than that of Figure 2. Given this concept, one could potentially use as many finite elements as required to model any given distortion pattern. Aside from the adverse effects of overparameterization on object space positional accuracy, another problem arises from the effect of point distribution. When the number of elements becomes large, favorable image point distribution can become quite sparse (depending upon the data set), most notably at the format periphery. An unsampled peripheral element creates a rank deficient normal equations matrix. A means to overcome this problem through the use of conPhotogrammetric Engineering & Remote Sensing, Vol. 63, No. 9, September 1997, pp. 1111-1119. 0099-1112/97/6309-1111$3.00/0