Abstract
Projective transformation of spheres onto images produce ellipses, whose centers do not coincide with the projected center of the sphere. This results in an eccentricity error, which must be treated in high precision metrology. This article provides closed formulations for modeling this error in images to enable 3-dimensional (3D) reconstruction of the center of spherical objects. The article also provides a new direct robust method for detecting spherical pattern in point clouds. It was shown that the eccentricity error in an image has only one component in the direction of the major axis of the ellipse. It was also revealed that the eccentricity is zero if and only if the center of the projected sphere lies on the camera’s perspective center. The effectiveness of the robust sphere detection and the eccentricity error modeling method was evaluated on simulated point clouds of spheres and real-world images, respectively. It was observed that the proposed robust sphere fitting method outperformed the popular M-estimator sample consensus in terms of radius and center estimation accuracy by a factor of 13, and 14 on average, respectively. Using the proposed eccentricity adjustment, the estimated 3D center of the sphere using modeled eccentricity was superior to the unmodeled case. It was also observed that the accuracy of the estimated 3D center using modeled eccentricity continuously improved as the number of images increased, whereas the unmodeled eccentricity did not show improvements after eight image views. The results of the investigation show that: (i) the proposed method effectively modeled the eccentricity error, and (ii) the effects of eliminating the eccentricity error in the 3D reconstruction become even more pronounced in a larger number of image views.
Highlights
Spheres in ImagesThe perspective projection of a sphere onto an image is an ellipse
The eccentricity error is an important factor in applications requiring high precision metrology such as tracking balls in sporting events (e.g., FIFA goal-line technology [2]), as well as calibration [3] and registration [4] of optical instruments
The accuracy was quantified as the number of image views increased from two to thirty-two
Summary
Spheres in ImagesThe perspective projection of a sphere onto an image is an ellipse. This is because the camera’s perspective center and a sphere in its field-of-view create a cone, whereby the intersection of the cone and the image’s plane generate an ellipse. The center of the ellipse in the image, even in the absence of measurement errors, does not correspond to the projected center of the sphere onto the image (see Figures 1 and 2). This phenomenon is commonly referred to as the eccentricity error in projective geometry [1]. Modeling the eccentricity error in images is, an important step towards accurate 3D reconstruction of spherical objects from images. To this end, the objectives of Remote Sens.
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