Rikard Berthilsson

An Iterative Factorization Method for Projective Structure and Motion from Image Sequences

Abstract In this article a novel recursive method for estimating structure and motion from image sequences is presented. The novelty lies in the fact that the output of the algorithm is independent of the chosen coordinate systems in the images as well as the ordering of the points. It relies on subspace and factorization methods and is derived from both ordinary coordinate representations and camera matrices and from a so-called depth and shape analysis. In addition, no initial phase is needed to start the algorithm. It starts directly with the first two images and incorporates new images as soon as new corresponding points are obtained. The performance of the algorithm is shown on both simulated and real data. Moreover, the two different approaches, one using camera matrices and the other using the concepts of affine shape and depth, are unified into a general theory of structure and motion from image sequences.

Reconstruction of General Curves, Using Factorization and Bundle Adjustment

In this paper, we extend the notion of affine shape, introduced by Sparr, from finite point sets to curves. The extension makes it possible to reconstruct 3D-curves up to projective transformations, from a number of their 2D-projections. We also extend the bundle adjustment technique from point features to curves.The first step of the curve reconstruction algorithm is based on affine shape. It is independent of choice of coordinates, is robust, does not rely on any preselected parameters and works for an arbitrary number of images. In particular this means that, except for a small set of curves (e.g. a moving line), a solution is given to the aperture problem of finding point correspondences between curves. The second step takes advantage of any knowledge of measurement errors in the images. This is possible by extending the bundle adjustment technique to curves.Finally, experiments are performed on both synthetic and real data to show the performance and applicability of the algorithm.

Recursive structure and motion from image sequences using shape and depth spaces

A novel recursive method for estimating structure and motion from image sequences is presented. The novelty lies in the fact that the output of the algorithm is independent of the chosen coordinate systems in the images as well as the ordering of the points. It relies on subspace methods and is derived from both ordinary coordinate representations and camera matrices and from a so called depth and shape analysis. Furthermore, no initial phase is needed to start up the algorithm. It starts directly with the first two images and incorporates new images as soon as new corresponding points are obtained. The performance of the algorithm is shown on simulated data. Moreover, the two different approaches, one using camera matrices and the other using the concepts of affine shape and depth, are unified into a general theory of structure and motion from image sequences.

Reconstruction of 3D-curves from 2D-images using affine shape methods for curves

In this paper, we propose an algorithm for doing reconstruction of general 3D-curves from a number of 2D-images taken by uncalibrated cameras. No point correspondences between the images are assumed. The curve and the view points are uniquely reconstructed, modulo common projective transformations and the point correspondence problem is solved. Furthermore, the algorithm is independent of the choice of coordinates, as it is based on orthogonal projections and aligning subspaces. The algorithm is based on an extension of affine shape of finite point configurations to more general objects.

Real Time Viterbi Optimization of Hidden Markov Models for Multi Target Tracking

In this paper the problem of tracking multiple objects in im- age sequences is studied. A Hidden Markov Model describ- ing the movements of multiple objects is presented. Previ- ously similar models have been used, but in real time sys- tem the standard dynamic programming Viterbi algorithm is typically not used to find the global optimum state se- quence, as it requires that all past and future observations are available. In this paper we present an extension to the Viterbi algorithm that allows it to operate on infinite time sequences and produce the optimum with only a finite de- lay. This makes it possible to use the Viterbi algorithm in real time applications. Also, to handle the large state spaces of these models another extension is proposed. The global optimum is found by iteratively running an approximative algorithm with higher and higher precision. The algorithm can determine when the global optimum is found by main- taining an upper bound on all state sequences not evalu- ated. For real time performance some approximations are needed and two such approximations are suggested. The theory has been tested on three real data experiments, all with promising results.

Extension of Affine Shape

In this paper, we extend the notion of affine shape, introduced by Sparr, from finite point sets to more general sets. It turns out to be possible to generalize most of the theory. The extension makes it possible to reconstruct, for example, 3D-curves up to projective transformations, from a number of their 2D-projections. An algorithm is presented, which is independent of choice of coordinates, is robust, does not rely on any preselected parameters and works for an arbitrary number of images. In particular this means that a solution is given to the aperture problem of finding point correspondences between curves.

Reconstruction of curves in R/sup 3/, using factorization and bundle adjustment

In this paper we extend the notion of affine shape, introduced by Sparr (1995, 1996), from finite point sets to curves. The extension makes it possible to reconstruct 3D-curves up to projective transformations, from a number of their 2D-projections. We also extend the bundle adjustment technique from point features to curves. The first step of the curve reconstruction algorithm is based on affine shape, is independent of choice of coordinates, robust, does not rely on any preselected parameters and works for an arbitrary number of images. In particular this means that a solution is given to the aperture problem of finding point correspondences between curves. The second step takes advantage of any knowledge of measurement errors in the images. This is possible by extending the bundle adjustment technique to curves. Finally, experiments are performed on both synthetic and real data to show the performance and applicability of the algorithm.

A Statistical Theory of Shape

In this paper, we will study the statistical theory of shape for ordered finite point configurations, or otherwise stated, the uncertainty of geometric invariants. A general approach for defining shape and finding its density, expressed in the densities for the individual points, is developed. Some examples, that can be computed analytically, are given, including both affine and positive similarity shape. Projective shape and projective invariants are important topics in computer vision and are discussed at the end of the paper.

Recognition of Planar Objects Using the Density of Affine Shape

In this paper, we will study the recognition problem for finite point configurations, in a statistical manner. We study the statistical theory of shape for ordered finite point configurations, or otherwise stated, the uncertainty of geometric invariants. Here, a general approach for defining shape and finding its density, expressed in the densities for the individual points, is developed. No approximations are made, resulting in an exact expression of the uncertainty region. In particular, we will concentrate on the affine shape, where often analytical computations is possible. In this case confidence intervals for invariants can be obtained from a priori assumptions on the densities of the detected points in the images. However, the theory is completely general and can be used to compute the density of any invariant (Euclidean, affine, similarity, projective, etc.) from arbitrary densities of the individual points. These confidence intervals can be used in such applications as geometrical hashing, recognition of ordered point configurations, and error analysis of reconstruction algorithms. Finally, an example will be given, illustrating the theory for the problem of recognizing planar point configurations from images taken by an affine camera. This case is of particular importance in applications, where details on a conveyor belt are captured by a camera, with image plane parallel to the conveyor belt and extracted feature points from the images are used to sort the objects.

Recognition of Planar Point Configurations Using Density of Affine Shape

In this paper, we study the statistical theory of shape for ordered finite point configurations, or otherwise stated, the uncertainty of geometric invariants. Such studies have been made for affine invariants in e.g. [GHJ92], [Wer93], where in the former case a bound on errors are used instead of errors described by density functions, and in the latter case a first order approximation gives an ellipsis as uncertainty region. Here, a general approach for defining shape and finding its density, expressed in the densities for the individual points, is developed. No approximations are made, resulting in an exact expression of the uncertainty region. Similar results have been obtained for the special case of the density of the cross ratio, see [May95,ast96].