Kalle Åström

Euclidean reconstruction from image sequences with varying and unknown focal length and principal point

The special case of reconstruction from image sequences taken by cameras with skew equal to 0 and aspect ratio equal to 1 has been treated. These type of cameras, here called cameras with Euclidean image planes, represent rigid projections where neither the principal point nor the focal length is known, it is shown that it is possible to reconstruct an unknown object from images taken by a camera with Euclidean image plane up to similarity transformations, i.e., Euclidean transformations plus changes in the global scale. An algorithm, using bundle adjustment techniques, has been implemented. The performance of the algorithm is shown on simulated data.

Motion from the frontier of curved surfaces

The frontier of a curved surface is the envelope of contour generators showing the boundary, at least locally, of the visible region swept out under viewer motion. In general, the outlines of curved surfaces (apparent contours) from different viewpoints are generated by different contour generators on the surface and hence do not provide a constraint on viewer motion. We show that frontier points, however, have projections which correspond to a real point on the surface and can be used to constrain viewer motion by the epipolar constraint. We show how to recover viewer motion from frontier points for both continuous and discrete motion, calibrated and uncalibrated cameras. We present preliminary results of an iterative scheme to recover the epipolar line structure from real image sequences using only the outlines of curved surfaces. A statistical evaluation as also performed to estimate the stability of the solution.<<ETX>>

Robust factorization

Factorization algorithms for recovering structure and motion from an image stream have many advantages, but they usually require a set of well-tracked features. Such a set is in generally not available in practical applications. There is thus a need for making factorization algorithms deal effectively with errors in the tracked features. We propose a new and computationally efficient algorithm for applying an arbitrary error function in the factorization scheme. This algorithm enables the use of robust statistical techniques and arbitrary noise models for the individual features. These techniques and models enable the factorization scheme to deal effectively with mismatched features, missing features, and noise on the individual features. The proposed approach further includes a new method for Euclidean reconstruction that significantly improves convergence of the factorization algorithms. The proposed algorithm has been implemented as a modification of the Christy-Horaud factorization scheme, which yields a perspective reconstruction. Based on this implementation, a considerable increase in error tolerance is demonstrated on real and synthetic data. The proposed scheme can, however, be applied to most other factorization algorithms.

Critical Motions for Auto-Calibration When Some Intrinsic Parameters Can Vary

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.

Flexible calibration: minimal cases for auto-calibration

This paper deals with the concept of auto-calibration, i.e. methods to calibrate a camera on-line. In particular we deal with minimal conditions on the intrinsic parameters needed to make a Euclidean reconstruction, called flexible calibration. The main theoretical results are that it is only needed to know that one intrinsic parameter is constant. The method is based on an initial projective reconstruction, which is upgraded to a Euclidean one. The number of images needed increases with the complexity of the constraints, but the number of points needed is only the number needed in order to obtain a projective reconstruction. The theoretical results are exemplified in a number of experiments. An algorithm, based on bundle adjustments and a linear initialization method are presented and experiments are performed on both synthetic and real data.

Minimal Conditions on Intrinsic Parameters for Euclidean Reconstruction

We investigate the constraints on the intrinsic parameters that are needed in order to reconstruct an unknown scene from a number of its projective images. Two such minimal cases are studied in detail. Firstly, it is shown that it is sufficient to know the skew parameter, even if all other parameters are unknown and varying, to obtain an Euclidean reconstruction. Secondly, the same thing can be done for known aspect ratio, again when all other intrinsic parameters are unknown and varying. In fact, we show that it is sufficient to know any of the 5 intrinsic parameters to make Euclidean reconstruction. An algorithm, based upon bundle adjustment techniques, to obtain Euclidean reconstruction in the above mentioned cases are presented. Experiments are shown on the slightly simpler case of both known aspect ratio and skew (Less)

Motion estimation in image sequences using the deformation of apparent contours

The problem of determining the camera motion from apparent contours or silhouettes of a priori unknown curved 3D surfaces is considered. In a sequence of images, it is shown how to use the generalized epipolar constraint on apparent contours. One such constraint is obtained for each epipolar tangency point in each image pair. An accurate algorithm for computing the motion is presented based on a maximum likelihood estimate. It is shown how to generate initial estimates on the camera motion using only the tracked contours. It is also shown that in theory the motion can be calculated from the deformation of a single contour. The algorithm has been tested on several real image sequences, for both Euclidean and projective reconstruction. The resulting motion estimate is compared to motion estimates calculated independently using standard feature-based methods. The motion estimate is also used to classify the silhouettes as curves or apparent contours. The statistical evaluation shows that the technique gives accurate and stable results.

Fast and Stable Polynomial Equation Solving and Its Application to Computer Vision

This paper presents several new results on techniques for solving systems of polynomial equations in computer vision. Gröbner basis techniques for equation solving have been applied successfully to several geometric computer vision problems. However, in many cases these methods are plagued by numerical problems. In this paper we derive a generalization of the Gröbner basis method for polynomial equation solving, which improves overall numerical stability. We show how the action matrix can be computed in the general setting of an arbitrary linear basis for ℂ[x]/I. In particular, two improvements on the stability of the computations are made by studying how the linear basis for ℂ[x]/I should be selected. The first of these strategies utilizes QR factorization with column pivoting and the second is based on singular value decomposition (SVD). Moreover, it is shown how to improve stability further by an adaptive scheme for truncation of the Gröbner basis. These new techniques are studied on some of the latest reported uses of Gröbner basis methods in computer vision and we demonstrate dramatically improved numerical stability making it possible to solve a larger class of problems than previously possible.

Conjugate gradient bundle adjustment

Bundle adjustment for multi-view reconstruction is traditionally done using the Levenberg-Marquardt algorithm with a direct linear solver, which is computationally very expensive. An alternative to this approach is to apply the conjugate gradients algorithm in the inner loop. This is appealing since the main computational step of the CG algorithm involves only a simple matrix-vector multiplication with the Jacobian. In this work we improve on the latest published approaches to bundle adjustment with conjugate gradients by making full use of the least squares nature of the problem. We employ an easy-to-compute QR factorization based block preconditioner and show how a certain property of the preconditioned system allows us to reduce the work per iteration to roughly half of the standard CG algorithm.

Minimizing the description length using steepest descent

Recently there has been much attention to MDL and its effectiveness in automatic shape modelling. One problem of this technique has been the slow convergence of the optimization step. In this paper the Jacobian of the objective function is derived. Being able to calculate the Jacobian, a variety of optimisation techniques can be considered. In this paper we apply steepest descent and show that it is more efcient than the previously proposed Nelder-Mead Simplex optimisation.