In this paper we address a classification problem that has not been considered before, namely motion segmentation given pairwise matches only. Our contribution to this unexplored task is a novel formulation of motion segmentation as a two-step process. First, motion segmentation is performed on image pairs independently. Secondly, we combine independent pairwise segmentation results in a robust way into the final globally consistent segmentation. Our approach is inspired by the success of averaging methods. We demonstrate in simulated as well as in real experiments that our method is very effective in reducing the errors in the pairwise motion segmentation and can cope with large number of mismatches.
We propose a geometric approach to 3D motion segmentation from point correspondences in three perspective views. We demonstrate that after applying a polynomial embedding to the point correspondences, they become related by the so-called multibody trilinear constraint and its associated multibody trifocal tensor, which are natural generalizations of the trilinear constraint and the trifocal tensor to multiple motions. We derive a rank constraint on the embedded correspondences from which one can estimate the number of independent motions, as well as linearly solve for the multibody trifocal tensor. We then show how to compute the epipolar lines associated with each image point from the common root of a set of univariate polynomials and the epipoles by solving a pair of plane clustering problems using Generalized Principal Component Analysis (GPCA). The individual trifocal tensors are then obtained from the second-order derivatives of the multibody trilinear constraint. Given epipolar lines and epipoles or trifocal tensors, one can immediately obtain an initial clustering of the correspondences. We use this clustering to initialize an iterative algorithm that alternates between the computation of the trifocal tensors and the segmentation of the correspondences. We test our algorithm on various synthetic and real scenes and compare it with other algebraic and iterative algorithms. Given point correspondences in multiple perspective views of a scene containing multiple rigid-body motions, we present an algorithm for segmenting the correspondences according to the multiple motions. We exploit the fact that when the depths of the points are known, the point trajectories associated with a single motion live in a subspace of dimension at most four. Thus motion segmentation with known depths can be achieved by methods of subspace separation, such as GPCA or LSA. When the depths are unknown, we proceed iteratively. Given the segmentation, we compute the depths using standard techniques. Given the depths, we use GPCA or LSA to segment the scene into multiple motions. Experiments on the Hopkins 155 motion segmentation database show that our method compares favorably against existing affine motion segmentation methods in terms of segmentation error and execution time. Given an image sequence of a scene consisting of multiple rigidly moving objects, multi-body structure-and-motion (MSaM) is the task to segment the image feature tracks into the different rigid objects and compute the multiple-view geometry of each object. We present a framework for multibody structure-and-motion based on model selection. In a recover-and-select procedure, a redundant set of hypothetical scene motions is generated. Each subset of this pool of motion candidates is regarded as a possible explanation of the image feature tracks, and the most likely explanation is selected with model selection. The framework is generic and can be used with any parametric camera model, or with a combination of different models. It can deal with sets of correspondences, which change over time, and it is robust to realistic amounts of outliers. The framework is demonstrated for different camera and scene models. We present a novel optimisation framework for the estimation of the multi-body motion segmentation and 3D reconstruction of a set of point trajectories in the presence of missing data. The proposed solution not only assigns the trajectories to the correct motion but it also solves for the 3D location of multi-body shape and it fills the missing entries in the measurement matrix. Such a solution is based on two fundamental principles: each of the multi-body motions is controlled by a set of metric constraints that are given by the specific camera model, and the shape matrix that describes the multi-body 3D shape is generally sparse. We jointly include such constraints in a unique optimisation framework which, starting from an initial segmentation, iteratively enforces these set of constraints in three stages. First, metric constraints are used to estimate the 3D metric shape and to fill the missing entries according to an orthographic camera model. Then, wrongly segmented trajectories are detected by using sparse optimisation of the shape matrix. A final reclassification strategy assigns the detected points to the right motion or discards them as outliers. We provide experiments that show consistent improvements to previous approaches both on synthetic and real data.  Motion segmentation involves clustering features together that belong to independently moving objects. The image features on each of these objects conform to one of several putative motion models, but the number and type of motion is unknown a priori. In order to cluster these features, the problems of model selection, robust estimation and clustering must all be addressed simultaneously. Within this paper we place the three problems into a common statistical framework; investigating the use of information criteria and robust mixture models as a principled way for motion segmentation of images. The final result is a general fully automatic algorithm for clustering that works in the presence of noise and outliers. In the last few decades, Structure from Motion (SfM) and visual Simultaneous Localization and Mapping (visual SLAM) techniques have gained significant interest from both the computer vision and robotic communities. Many variants of these techniques have started to make an impact in a wide range of applications, including robot navigation and augmented reality. However, despite some remarkable results in these areas, most SfM and visual SLAM techniques operate based on the assumption that the observed environment is static. However, when faced with moving objects, overall system accuracy can be jeopardized. In this article, we present for the first time a survey of visual SLAM and SfM techniques that are targeted toward operation in dynamic environments. We identify three main problems: how to perform reconstruction (robust visual SLAM), how to segment and track dynamic objects, and how to achieve joint motion segmentation and reconstruction. Based on this categorization, we provide a comprehensive taxonomy of existing approaches. Finally, the advantages and disadvantages of each solution class are critically discussed from the perspective of practicality and robustness. We want to deduce, from a sequence of noisy two-dimensional images of a scene of several rigid bodies moving independently in three dimensions, the number of bodies and the grouping of given feature points in the images to the bodies. Prior processing is assumed to have identified features or points common to all frames and the images are assumed to be created by orthographic projection (i.e., perspective effects are minimal). We describe a computationally inexpensive algorithm that can determine which points or features belong to which rigid body using the fact that, with exact observations in orthographic projection, points on a single body lie in a three or less dimensional linear manifold of frame space. If there are enough observations and independent motions, these manifolds can be viewed as a set linearly independent, four or less dimensional subspaces. We show that the row echelon canonical form provides direct information on the grouping of points to these subspaces. Treatment of the noise is the most difficult part of the problem. This paper uses a statistical approach to estimate the grouping of points to subspaces in the presence of noise by computing which partition has the maximum likelihood. The input data is assumed to be contaminated with independent Gaussian noise. The algorithm can base its estimates on a user-supplied standard deviation of the noise, or it can estimate the noise from the data. The algorithm can also be used to estimate the probability of a user-specified partition so that the hypothesis can be combined with others using Bayesian statistics. The problem of simultaneous structure from motion estimation for multiple independently moving objects from a monocular image sequence is addressed. Two Bayesian algorithms are presented for solving this problem using the sequential importance sampling (SIS) technique. The empirical posterior distribution of object motion and feature separation parameters is approximated by weighted samples. The first algorithm addresses the problem when only two moving objects are present. A singular value decomposition (SVD)-based sample clustering algorithm is shown to be capable of separating samples related to different objects. A pair of SIS procedures is used to track the posterior distribution of the motion parameters. In the second algorithm, a balancing step is added into the SIS procedure to preserve samples of low weights so that all objects have enough samples to propagate empirical motion distributions. By using the proposed algorithms, the relative motions of all the moving objects with respect to the camera can be simultaneously estimated. Both algorithms have been tested on synthetic and real- image sequences. Improved results have been achieved. An efficient and robust framework is proposed for two-view multiple structure-and-motion segmentation of unknown number of rigid objects. The segmentation problem has three unknowns, namely the object memberships, the corresponding fundamental matrices, and the number of objects. To handle this otherwise recursive problem, hypotheses for fundamental matrices are generated through local sampling. Once the hypotheses are available, a combinatorial selection problem is formulated to optimize a model selection cost which takes into account the hypotheses likelihoods and the model complexity. An explicit model for outliers is also added for robust segmentation. The model selection cost is minimized through the branch-and-bound technique of combinatorial optimization. The proposed branch-and-bound approach efficiently searches the solution space and guaranties optimality over the current set of hypotheses. The efficiency and the guarantee of optimality of the method is due to its ability to reject solutions without explicitly evaluating them. The proposed approach was validated with synthetic data, and segmentation results are presented for real images. The structure-from-motion problem has been extensively studied in the field of computer vision. Yet, the bulk of the existing work assumes that the scene contains only a single moving object. The more realistic case where an unknown number of objects move in the scene has received little attention, especially for its theoretical treatment. In this paper we present a new method for separating and recovering the motion and shape of multiple independently moving objects in a sequence of images. The method does not require prior knowledge of the number of objects, nor is dependent on any grouping of features into an object at the image level. For this purpose, we introduce a mathematical construct of object shapes, called the shape interaction matrix, which is invariant to both the object motions and the selection of coordinate systems. This invariant structure is computable solely from the observed trajectories of image features without grouping them into individual objects. Once the matrix is computed, it allows for segmenting features into objects by the process of transforming it into a canonical form, as well as recovering the shape and motion of each object. The theory works under a broad set of projection models (scaled orthography, paraperspective and affine) but they must be linear, so it excludes projective “cameras”. Multibody structure from motion (SfM) is the extension of classical SfM to dynamic scenes with multiple rigidly moving objects. Recent research has unveiled some of the mathematical foundations of the problem, but a practical algorithm which can handle realistic sequences is still missing. In this paper, we discuss the requirements for such an algorithm, highlight theoretical issues and practical problems, and describe how a static structure-from-motion framework needs to be extended to handle real dynamic scenes. Theoretical issues include different situations in which the number of independently moving scene objects changes: Moving objects can enter or leave the field of view, merge into the static background (e.g., when a car is parked), or split off from the background and start moving independently. Practical issues arise due to small freely moving foreground objects with few and short feature tracks. We argue that all of these difficulties need to be handled online as structure-from-motion estimation progresses, and present an exemplary solution using the framework of probabilistic model-scoring.  In this paper, we tackle the problem of mapping multiple 3D rigid structures and estimating their motions from perspective views through a car-mounted camera. The proposed method complements conventional localization and mapping algorithms (such as Visual Odometry and SLAM) to estimate motions of other moving objects in addition to the vehicle's motion. We present a theoretical framework for robust estimation of multiple motions and structures from perspective images. The method is based on the factorization of the projective trajectory matrix without explicit estimation of projective depth values. We exploit the epipolar geometry of calibrated cameras to generate several hypotheses for motion segments. Once the hypotheses are obtained, they are evaluated in an iterative scheme by alternating between estimation of 3D structures and estimation of multiple motions. The proposed framework does not require any knowledge about the number of motions and is robust to noisy image measurements. The method is evaluated on street-level sequences from a car-mounted camera. A benchmark dataset is also used to compare the results with previous works, although most of the related works use synthetic scenes simulating desktop environments.