Anders Heyden

Computer Vision — ECCV 2002

We present a novel algorithm for recovering a smooth manifold of unknown dimension and topology from a set of points known to belong to it. Numerous applications in computer vision can be naturally interpreted as instanciations of this fundamental problem. Recently, a non-iterative discrete approach, tensor voting, has been introduced to solve this problem and has been applied successfully to various applications. As an alternative, we propose a variational formulation of this problem in the continuous setting and derive an iterative algorithm which approximates its solutions. This method and tensor voting are somewhat the differential and integral form of one another. Although iterative methods are slower in general, the strength of the suggested method is that it can easily be applied when the ambient space is not Euclidean, which is important in many applications. The algorithm consists in solving a partial differential equation that performs a special anisotropic diffusion on an implicit representation of the known set of points. This results in connecting isolated neighbouring points. This approach is very simple, mathematically sound, robust and powerful since it handles in a homogeneous way manifolds of arbitrary dimension and topology, embedded in Euclidean or non-Euclidean spaces, with or without border. We shall present this approach and demonstrate both its benefits and shortcomings in two different contexts: (i) data visual analysis, (ii) skin detection in color images.

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.

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 from Image Sequences by Means of Relative Depths

This paper deals with the problem of reconstructing the locations of n points in space from m different images without camera calibration. It shows how these problems can be put into a similar theoretical framework.A new concept, the reduced fundamental matrix, is introduced. It contains just 4 parameters and can be used to predict locations of points in the images and to make reconstruction. We also introduce the concept of reduced fundamental tensor, which describes the relations between points in 3 images. It has 15 components and depends on 9 parameters. Necessary and sufficient conditions for a tensor to be a reduced fundamental tensor are derived. This framework can be generalised to a sequence of images. The dependencies between the different representations are investigated. Furthermore a canonical form of the camera matrices in a sequence are presented.

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.

Using conic correspondences in two images to estimate the epipolar geometry

In this paper it is shown hour corresponding conics in two images can be used to estimate the epipolar geometry in terms of the fundamental/essential matrix. The corresponding conics can, be images of either planar celtics or silhouettes of quadrics. It is shown that one conic correspondence gives two independent constraints on the fundamental matrix and a method to estimate the fundamental matrix from at least four corresponding conics is presented. Furthermore, a new type of fundamental matrix for describing conic correspondences is introduced. Finally, it is shown that the problem of estimating the fundamental matrix from 5 point correspondences and 1 conic correspondence in general has 10 different solutions. A method to calculate these solutions is also given together with an experimental validation.

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)

Affine Structure and Motion from Points, Lines and Conics

In this paper several new methods for estimating scene structure and camera motion from an image sequence taken by affine cameras are presented. All methods can incorporate both point, line and conic features in a unified manner. The correspondence between features in different images is assumed to be known.Three new tensor representations are introduced describing the viewing geometry for two and three cameras. The centred affine epipoles can be used to constrain the location of corresponding points and conics in two images. The third order, or alternatively, the reduced third order centred affine tensors can be used to constrain the locations of corresponding points, lines and conics in three images. The reduced third order tensors contain only 12 components compared to the 16 components obtained when reducing the trifocal tensor to affine cameras.A new factorization method is presented. The novelty lies in the ability to handle not only point features, but also line and conic features concurrently. Another complementary method based on the so-called closure constraints is also presented. The advantage of this method is the ability to handle missing data in a simple and uniform manner. Finally, experiments performed on both simulated and real data are given, including a comparison with other methods.

A Common Framework for Multiple View Tensors

In this paper, we will introduce a common framework for the definition and operations on the different multiple view tensors. The novelty of the proposed formulation is to not fix any parameters of the camera matrices, but instead letting a group act on them and look at the different orbits. In this setting the multiple view geometry can be viewed as a four-dimensional linear manifold in IR3 m, where m denotes the number of images. The Grassman coordinates of this manifold are the epipoles, the components of the fundamental matrices, the components of the trifocal tensor and the components of the quadfocal tensor. All relations between these Grassman coordinates can be expressed using the so called quadratic p-relations, which are quadratic polynomials in the Grassman coordinates. Using this formulation it is evident that the multiple view geometry is described by four different kinds of projective invariants; the epipoles, the fundamental matrices, the trifocal tensors and the quadfocal tensors.

A fast algorithm for level set-like active contours

This paper describes a fast algorithm for topology independent tracking of moving interfaces under curvature- and velocity field-dependent speed laws. This is usually done in the level set framework using the narrow-band algorithm, which accurately solves the level set equation but is too slow to use in real-time or near real-time image segmentation applications. In this paper we introduce a fast algorithm for tracking moving interfaces in a level set-like manner. The algorithm relies on two key components: First, it tracks the interface by scheduling point-wise propagation events using a heap sorted queue. Second, the local geometric properties of the interface are defined so that they can be efficiently updated in an incremental manner and so that they do not require the presence of the signed distance function. Finally examples are given that indicate that the algorithm is fast and accurate enough for near real-time segmentation applications.