Calibration of the Cameras of Photogrammetric Measurement Systems by Means of the Method of Least Squares

Abstract

The use of the method of least squares in calibration of cameras in contactless spatial measurement systems is investigated.The technology of contactless spatial measurements has become increasingly more common in recent years both in machine construction and in related fields of industry [1]. By comparison to contact measurement systems, contactless measurement systems exhibit a host of advantages, chief among which are the absence of any need for mechanical contact with the subject of measurement; high rate at which measurement information is obtained (down to several tens of thousands of points per minute); and the fact that it is possible to perform spatial measurements of objects of complex form.Contactless spatial measurement systems are based on such technologies as photogrammetry, triangulation, and laser interferometry; moreover, still or film cameras (depending on the nature of the problems that have to be solved, whether static or dynamic) are most often used to implement the technologies. The cameras constitute an optical system ( Fig. 1) consisting of several lenses that determine the magnification, solid angle, and the guides of the light flux 1 to the digital array 3 through a semitransparent plate 2 that transmits a portion of the light flux through the optical system 4, 5 into the view finder 6. Thus, a digital image which, from the geometric point of view, is a central projection of the three-dimensional subject of measurement into the camera's plane of projection is usually formed at the camera output. Schematically, a model of a still camera may be depicted in accordance with Fig. 2. Upon projection to some point S, the polyhedron ABCD in three-dimensional space leaves a track abcd in the plane of projection (in the CCD array of the camera). Measurements by means of cameras are related directly or indirectly with reconstruction of the lines of the projection link connecting the center of projection S, a point in the test object, and its track in the plane of the image [2, 3]. The use of photogrammetric technologies to produce measurement information makes it possible to investigate surfaces of complex form with sufficient precision (down to 0.01 mm) [4].On the basis of mathematical models, a camera is described by nine parameters, among which may be distinguished internal and external parameters [2,3]. The internal parameters are those which determine the solid angle of the camera as well as the deviation of the principal optical axis from the nominal position. The parameters of external orientation in the projective model of the camera include the coordinates of the center of projection of the camera S(x S , y S , z S ) and the angles α, ω, and χ, which specify the direction of its principal optical axis. Basically, these parameters determine the matrices of transformation from some absolute (global) coordinate system X a , Y a , Z a into a coordinate system bound with the camera X, Y, Z in which the axes X and Y are respectively parallel to the axes X p and Y p of the planar coordinate syste...

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“…Estimate the position of the principal point on the image with a typical offline calibration procedure for cameras. 17–19…”

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Algorithm of correction of error caused by perspective distortions of measuring mark images

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