Geometric Properties of Central Catadioptric Line Images and Their Application in Calibration

In central catadioptric systems, lines in a scene are projected to conic curves in the image. This work studies the geometry of the central catadioptric projection of lines and its use in calibration. It is shown that the conic curves where the lines are mapped possess several projective invariant properties. From these properties, it follows that any central catadioptric system can be fully calibrated from an image of three or more lines. The image of the absolute conic, the relative pose between the camera and the mirror, and the shape of the reflective surface can be recovered using a geometric construction based on the conic loci where the lines are projected. This result is valid for any central catadioptric system and generalizes previous results for paracatadioptric sensors. Moreover, it is proven that systems with a hyperbolic/elliptical mirror can be calibrated from the image of two lines. If both the shape and the pose of the mirror are known, then two line images are enough to determine the image of the absolute conic encoding the camera’s intrinsic parameters. The sensitivity to errors is evaluated and the approach is used to calibrate a real camer

Plane-Based Calibration of Central Catadioptric Cameras

International audienceWe present a novel calibration technique for all central catadioptric cameras using images of planar grids. We adopted the well-known sphere camera model to describe the catadioptric projection. We show that, using the so-called lifted coordinates, a linear relation mapping the grid points to the corresponding points on the image plane can be written as a 6 × 6 matrix Hcata , which acts like the classical 3 × 3 ho- mography for perspective cameras. We show how to compute the image of the absolute conic (IAC) from at least 3 homo- graphies and how to recover from it the intrinsic parameters of the catadioptric camera. In the case of paracatadioptric cameras one such homography is enough to estimate the IAC, thus allowing the calibration from a single image

Applications of Conformal Geometric Algebra in Computer Vision and Graphics

Abstract. This paper introduces the mathematical framework of con-formal geometric algebra (CGA) as a language for computer graphics and computer vision. Specifically it discusses a new method for pose and position interpolation based on CGA which firstly allows for existing interpolation methods to be cleanly extended to pose and position in-terpolation, but also allows for this to be extended to higher-dimension spaces and all conformal transforms (including dilations). In addition, we discuss a method of dealing with conics in CGA and the intersection and reflections of rays with such conic surfaces. Possible applications for these algorithms are also discussed.

A Full Scale Camera Calibration Technique with Automatic Model Selection – Extension and Validation

This thesis presents work on the testing and development of a complete camera calibration approach which can be applied to a wide range of cameras equipped with normal, wide-angle, fish-eye, or telephoto lenses. The full scale calibration approach estimates all of the intrinsic and extrinsic parameters. The calibration procedure is simple and does not require prior knowledge of any parameters. The method uses a simple planar calibration pattern. Closed-form estimates for the intrinsic and extrinsic parameters are computed followed by nonlinear optimization. Polynomial functions are used to describe the lens projection instead of the commonly used radial model. Statistical information criteria are used to automatically determine the complexity of the lens distortion model. In the first stage experiments were performed to verify and compare the performance of the calibration method. Experiments were performed on a wide range of lenses. Synthetic data was used to simulate real data and validate the performance. Synthetic data was also used to validate the performance of the distortion model selection which uses Information Theoretic Criterion (AIC) to automatically select the complexity of the distortion model. In the second stage work was done to develop an improved calibration procedure which addresses shortcomings of previously developed method. Experiments on the previous method revealed that the estimation of the principal point during calibration was erroneous for lenses with a large focal length. To address this issue the calibration method was modified to include additional methods to accurately estimate the principal point in the initial stages of the calibration procedure. The modified procedure can now be used to calibrate a wide spectrum of imaging systems including telephoto and verifocal lenses. Survey of current work revealed a vast amount of research concentrating on calibrating only the distortion of the camera. In these methods researchers propose methods to calibrate only the distortion parameters and suggest using other popular methods to find the remaining camera parameters. Using this proposed methodology we apply distortion calibration to our methods to separate the estimation of distortion parameters. We show and compare the results with the original method on a wide range of imaging systems

Paracatadioptric Camera Calibration Using Lines

Paracatadioptric sensors combine a parabolic shaped mirror and a camera inducing an orthographic projection. Such a configuration provides a wide field of view while keeping a single effective viewpoint. Previous work in central catadioptric sensors proved that a line projects into a conic curve and that three line images are enough to calibrate the system. However the estimation of the conic curves where lines are mapped is hard to accomplish. In general only a small arc of the conic is visible in the image and conventional conic fitting techniques are unable to correctly estimate the curve. The present work shows that a set of conic curves corresponds to paracatadioptric line images if, and only if, certain properties are verified. These properties are used to constraint the search space and correctly estimate the curves. The accurate estimation of a minimum of three line images allows the complete calibration of the paracatadioptric camera. If the camera is skewless and the aspect ratio is known then the conic fitting problem is solved naturally by an eigensystem. For the general situation the conic curves are estimated using non-linear optimization.