Knud Henriksen

Virtual trackballs revisited

Rotation of three-dimensional objects by a two-dimensional mouse is a typical task in computer-aided design, operation simulations, and desktop virtual reality. The most commonly used rotation technique is a virtual trackball surrounding the object and operated by the mouse pointer. We review and provide a mathematical foundation for virtual trackballs. The first, but still popular, virtual trackball was described by Chen et al. (1998). We show that the virtual trackball by Chen et al. does not rotate the object along the intended great circular arc on the virtual trackball and we give a correction. Another popular virtual trackball is Shoemakes quaternion implementation (1992), which we show to be a special case of the virtual trackball by Chen et al.. Shoemake extends the scope of the virtual trackball to the full screen. Unfortunately, Shoemakes virtual trackball is inhomogeneous and discontinuous with consequences for usability. Finally, we review Bells virtual trackball (1998) and discuss studies of the usability of virtual trackballs.

Stereo ranging with verging cameras

A method of computing absolute range from stereo disparities by using verging cameras is presented. The approach differs from others by concentrating, through both analysis and experiment, on the effects caused by convergence, rather than on the general camera calibration problem. To compute stereo disparities, linear image features are extracted and matched using a hypothesize-and-verify method. To compute range, the relationship between object distance, vergence angle, and disparity is derived. Experimental results show the precision of the range computation, excluding mistaken matches, to be approximately 5% for object distances up to three meters and a baseline distance of 13 cm. Including mistaken matches results in performance that is an order of magnitude worse, leading the authors to suggest methods to identify and model them. >

Using Mirror Cameras for Estimating Depth

A mirror camera arrangement attached to a conventional perspective camera is suggested for the purpose of computing depth of spatial points. The arrangement simulates a multi camera set up, where all internal parameters are equal and a common cyclopedian system is well defined. A linear system, from which spatial point coordinates may be determined, is derived. Computational experiments with actual mirror camera data are reported. Average relative error on estimated depth values was 0.6 %.

Relating Scene Depth to Image Ratios

An alternative to the classic depth from stereo disparities is presented. In this new approach two scene points with different and finite depths are viewed by two identical cameras with parallel optic axes. Both the case of frontal views and of side views, where both cameras are rotated by the same angle is addressed. A simple construction for the vanishing point of the line connecting the two scene points is presented. Both for frontal views and side views it is shown that the relative scene depth of two points equals the reciprocal of the ratio of the image plane distances from the image points to the vanishing point of the line they define. For side views it is furthermore shown how the lens plane separation and ratios of image plane distances to vanishing points directly determine the absolute depths to the scene points. Neither camera focal length, image plane optic center, image coordinate scale nor coordinate disparities are used in the calculations of the absolute scene depths.

Estimating Time to Contact with Curves, avoiding Calibration and Aperture Problem

A set of simple time to contact estimators are derived, using isolated points or curve segments. For this purpose the use of both optic flow and optic acceleration is suggested. For curves it is pointed out, that there is no aperture problem present, since normal flow and acceleration of the curve segment is sufficient for estimating time to contact. Time to contact with a curve segment may be calculated without calibrating camera focal length and camera coordinate system, without computing spatial velocity and depth maps and without computing the complete optic flow field for the curve segment. Computational illustrations with actual camera data are reported.

Direct determination of the orientation of a translating 3D straight line

Abstract A new method for determination of spatial orientation of a straight line is derived. The system is linear and involve orientation of the projected line and optic flow of one feature point, but neither camera position nor motion.

Coordinate-Free Camera Calibration

A method for calibration of the optic center and the focal length of a pin-hole camera is suggested, where the camera parameters are constructed geometrically in the image plane. Box shaped objects of unknown dimensions, like furniture, buildings or a textured ground plane with trees, lamp poles or people are examples of sufficient calibration objects. Experiments are reported with relative errors in the range of one percent for suitable calibration views. Jens Arnspang can be reached by Phone: +45 35 32 14 00, email: arnspangC@diku.dk, or http://www.diku.dk/users/arnspang/