. This paper extends the recovery of structure and motion to image sequences with several independently moving objects. The motion, structure, and camera calibration are all a-priori unknown. The fundamental constraint that we introduce is that multiple motions must share the same camera parameters. Existing work on independent motions has not employed this constraint, and therefore has not gained over independent static-scene reconstructions. We show how this constraint leads to several new results in structure and motion recovery, where Euclidean reconstruction becomes possible in the multibody case, when it was underconstrained for a static scene. We show how to combine motions of high-relief, low-relief and planar objects. Additionally we show that structure and motion can be recovered from just 4 points in the uncalibrated, fixed camera, case. Experiments on real and synthetic imagery demonstrate the validity of the theory and the improvement in accuracy obtained usin...

Abstract. In this paper we describe the theory and practice ofself-calibration ofcameras which are fixed in location and may freely rotate while changing their internal parameters by zooming. The basis ofour approach is to make use ofthe so-called infinite homography constraint which relates the unknown calibration matrices to the computed inter-image homographies. In order for the calibration to be possible some constraints must be placed on the internal parameters ofthe camera. We present various self-calibration methods. First an iterative non-linear method is described which is very versatile in terms ofthe constraints that may be imposed on the camera calibration: each ofthe camera parameters may be assumed to be known, constant throughout the sequence but unknown, or free to vary. Secondly, we describe a fast linear method which works under the minimal assumption of zero camera skew or the more restrictive conditions ofsquare pixels (zero skew and known aspect ratio) or known principal point. We show experimental results on both synthetic and real image sequences (where ground truth data was available) to assess the accuracy and the stability ofthe algorithms and to compare the result ofapplying different constraints on the camera parameters. We also derive an optimal Maximum Likelihood estimator for the calibration and the motion parameters. Prior knowledge about the distribution ofthe estimated parameters (such as the location ofthe principal point) may also be incorporated via Maximum a Posteriori estimation. We then identify some near-ambiguities that arise under rotational motions showing that coupled changes ofcertain parameters are barely observable making them indistinguishable. Finally we study the negative effect ofradial distortion in the self-calibration process and point out some possible solutions to it. 1.

Abstract. Auto-calibration is the recovery of the full camera geometry and Euclidean scene structure from several images of an unknown 3D scene, using rigidity constraints and partial knowledge of the camera intrinsic parameters. It fails for certain special classes of camera motion. This paper derives necessary and sufficient conditions for unique auto-calibration, for several practically important cases where some of the intrinsic parameters are known (e.g. skew, aspect ratio) and others can vary (e.g. focal length). We introduce a novel subgroup condition on the camera calibration matrix, which helps to systematize this sort of auto-calibration problem. We show that for subgroup constraints, criticality is independent of the exact values of the intrinsic parameters and depends only on the camera motion. We study such critical motions for arbitrary numbers of images under the following constraints: vanishing skew, known aspect ratio and full internal calibration modulo unknown focal lengths. We give explicit, geometric descriptions for most of the singular cases. For example, in the case of unknown focal lengths, the only critical motions are: (i) arbitrary rotations about the optical axis and translations, (ii) arbitrary rotations about at most two centres, (iii) forward-looking motions along an ellipse and/or a corresponding hyperbola in an orthogonal plane. Some practically important special cases are also analyzed in more detail.

We discuss how to automatically obtain the metric calibration of an ad-hoc network of cameras with no centralized processor. We model the set of uncalibrated cameras as nodes in a communication network, and propose a distributed algorithm in which each camera performs a local, robust bundle adjustment over the camera parameters and scene points of its neighbors in an overlay “vision graph”. We analyze the performance of the algorithm on both simulated and real data, and show that the distributed algorithm results in a fairer allocation of messages per node while achieving comparable calibration accuracy to centralized bundle adjustment.

This paper shows how to upgrade the projective reconstruction of a scene to a metric one in the case where the only assumption made about the cameras observing that scene is that they have rectangular pixels (zero-skew cameras). The proposed approach is based on a simple characterization of zero-skew projection matrices in terms of line geometry, and it handles zero-skew cameras with arbitrary or known aspect ratios in a unified framework. The metric upgrade computation is decomposed into a sequence of linear operations, including linear leastsquares parameter estimation and eigenvalue-based symmetric matrix factorization, followed by an optional non-linear least-squares refinement step. A few classes of critical motions for which a unique solution cannot be found are spelled out. A MATLAB implementation has been constructed and preliminary experiments with real data are presented.

In this paper we address the problem of using quaternions in unconstrained nonlinear optimization of 3-D rotations. Quaternions representing rotations have four elements but only three degrees of freedom, since they must be of norm one. This constraint has to be taken into account when applying e. g. the Levenberg-Marquardt algorithm, a method for unconstrained nonlinear optimization widely used in computer vision. We propose an easy to use method for achieving this. Experiments using our parametrization in photogrammetric bundle-adjustment are presented at the end of the paper.

Zoom lenses appear to fit very naturally into the framework of active vision --- controlling a zoom lens allows an adjustment of the image, enabling either an analysis of a wide scene or a close look at a region or object of particular interest. However, their integration into vision systems is not without difficulty since zoom interacts insidiously with both low and high level processes. This thesis concerns developing and analyzing algorithms that function in spite of zoom; algorithms for visual tracking, camera calibration and Euclidean reconstruction. An approach grounded in visual geometry is adopted, motivated by the notion that the geometric descriptions of point (corner) and line (straight edge) features are zoom-invariant.

We present a batch method for recovering Euclidian camera motion from sparse image data. The main purpose of the algorithm is to recover the motion parameters using as much of the available information and as few computational steps as possible. The algorithm thus places itself in the gap between factorisation schemes, which make use of all available information in the initial recovery step, and sequential approaches which are able to handle sparseness in the image data. Euclidian camera matrices are approximated via the affine camera model, thus making the recovery direct in the sense that no intermediate projective reconstruction is made. Using a little known closure constraint, the FA-closure, we are able to formulate the camera coefficients linearly in the entries of the affine fundamental matrices. The novelty of the presented work is twofold: Firstly the presented formulation allows for a particularly good conditioning of the estimation of the initial motion parameters but also for an unprecedented diversity in the choice of possible regularisation terms. Secondly, the new autocalibration scheme presented here is in practice guaranteed to yield a Least Squares Estimate of the calibration parameters. As a bi-product, the affine camera model is rehabilitated as a useful model for most cameras and scene con-

This paper deals with the problem of incorporating natural regularity conditions on the motion in an MAP estimator for structure and motion recovery from uncalibrated image sequences. The purpose of incorporating these constraints is to increase performance and robustness. Autocalibration and structure and motion algorithms are known to have problems with (i) the frequently occurring critical camera motions, (ii) local minima in the non-linear optimization and (iii) the high correlation between different intrinsic and extrinsic parameters of the camera, e.g. the coupling between focal length and camera position. The camera motion (both intrinsic and extrinsic parameters) is modelled as a random walk process, where the inter-frame motions are assumed to be independently normally distributed. The proposed scheme is demonstrated on both simulated and real data showing the increased performance. 1.

In this paper we develop a theory of non-parametric self-calibration. Recently, schemes have been devised for non-parametric laboratory calibration, but not for selfcalibration. We allow an arbitrary warp to model the intrinsic mapping, with the only restriction that the camera is central and that the intrinsic mapping has a well-defined non-singular matrix derivative at a finite number of points under study.