Department of Electrical and Computer Engineering

It is highly desirable that an imaging system has a single effective viewpoint. Central catadioptric systems are imaging systems that use mirrors to enhance the field of view while keeping a unique center of projection. A general model for central catadioptric image formation has already been established. The present paper exploits this model to study the catadioptric projection of lines. The equations and geometric properties of general catadioptric line imaging are derived. We show that it is possible to determine the position of both the effective viewpoint and the absolute conic in the catadioptric image plane from the images of three lines. It is also proved that it is possible to identify the type of catadioptric system and the position of the line at infinity without further information. A methodology for central catadioptric system calibration is proposed. Reconstruction aspects are discussed. Experimental results are presented. All the results presented are original and completely new.

An imaging system with a single effective viewpoint is called a central projection system. The conventional perspective camera is an example of central projection system. A catadioptric realization of omnidirectional vision combines reflective surfaces with lenses. Catadioptric systems with an unique projection center are also examples of central projection systems. Whenever an image is acquired, points in 3D space are mapped into points in the 2D image plane. The image formation process represents a transformation from ℜ3 to ℜ2, and mathematical models can be used to describe it. This paper discusses the definition of world coordinate systems that simplify the modeling of general central projection imaging. We show that an adequate choice of the world coordinate reference system can be highly advantageous. Such a choice does not imply that new information will be available in the images. Instead the geometric transformations will be represented in a common and more compact framework, while simultaneously enabling newer insights. The first part of the paper focuses on static imaging systems that include both perspective cameras and catadioptric systems. A systematic approach to select the world reference frame is presented. In particular we derive coordinate systems that satisfy two differential constraints (the “compactness” and the “decoupling” constraints). These coordinate systems have several advantages for the representation of the transformations between the 3D world and the image plane. The second part of the paper applies the derived mathematical framework to active tracking of moving targets. In applications of visual control of motion the relationship between motion in the scene and image motion must be established. In the case of active tracking of moving targets these relationships become more complex due to camera motion. Suitable world coordinate reference systems are defined for three distinct situations: perspective camera with planar translation motion, perspective camera with pan and tilt rotation motion, and catadioptric imaging system rotating around an axis going through the effective viewpoint and the camera center. Position and velocity equations relating image motion, camera motion and target 3D motion are derived and discussed. Control laws to perform active tracking of moving targets using visual information are established.

Perspective camera calibration has been a research subject for a large group of researchers. However, only a small number of those methods base their approaches on the use of monoplane calibration points. We developed one of those methodologies using monoplane calibration points to perform an explicit three-dimensional (3-D) camera calibration. This methodology is based on an iterative approach. To avoid the singularities obtained with the calibration equations when monoplane calibration points are used, this method computes the calibration parameters in a multistep procedure and requires a first-guess solution for the intrinsic parameters. These intrinsic parameters are updated and their accuracy increased during the iterative procedure. All computations required are linear and in addition to the extrinsic parameters the proposed method also computes the first coefficient of the radial distortion (k1) and the skew angle. A first-guess value for the focal length of the lens is required but its value is iteratively updated using the Gauss lens model. This methodology also includes the uncertainty horizontal image scale factor (Sx) on the set of calibration parameters to be computed, which makes this approach independent of the accuracy of the horizontal scale factor. The proposed methodology has the advantage that it can be used with monoplane calibration data with no restrictions for the pose geometry of the camera. In this approach the pose estimation problem is treated separately, computing the pose orientation of the camera in a first step and using this information to compute the pose location. A model for the expected stability of the camera look angles from noisy image data and its stability analysis as a function of the pose geometry of the camera is presented. Several experimental and simulated analyzes were performed and are presented