Roberto Tron

A Benchmark for the Comparison of 3-D Motion Segmentation Algorithms

Over the past few years, several methods for segmenting a scene containing multiple rigidly moving objects have been proposed. However, most existing methods have been tested on a handful of sequences only, and each method has been often tested on a different set of sequences. Therefore, the comparison of different methods has been fairly limited. In this paper, we compare four 3D motion segmentation algorithms for affine cameras on a benchmark of 155 motion sequences of checkerboard, traffic, and articulated scenes.

Motion Segmentation in the Presence of Outlying, Incomplete, or Corrupted Trajectories

In this paper, we study the problem of segmenting tracked feature point trajectories of multiple moving objects in an image sequence. Using the affine camera model, this problem can be cast as the problem of segmenting samples drawn from multiple linear subspaces. In practice, due to limitations of the tracker, occlusions, and the presence of nonrigid objects in the scene, the obtained motion trajectories may contain grossly mistracked features, missing entries, or corrupted entries. In this paper, we develop a robust subspace separation scheme that deals with these practical issues in a unified mathematical framework. Our methods draw strong connections between lossy compression, rank minimization, and sparse representation. We test our methods extensively on the Hopkins155 motion segmentation database and other motion sequences with outliers and missing data. We compare the performance of our methods to state-of-the-art motion segmentation methods based on expectation-maximization and spectral clustering. For data without outliers or missing information, the results of our methods are on par with the state-of-the-art results and, in many cases, exceed them. In addition, our methods give surprisingly good performance in the presence of the three types of pathological trajectories mentioned above. All code and results are publicly available at http://perception.csl.uiuc.edu/coding/motion/.

Motion segmentation via robust subspace separation in the presence of outlying, incomplete, or corrupted trajectories

We examine the problem of segmenting tracked feature point trajectories of multiple moving objects in an image sequence. Using the affine camera model, this motion segmentation problem can be cast as the problem of segmenting samples drawn from a union of linear subspaces. Due to limitations of the tracker, occlusions and the presence of nonrigid objects in the scene, the obtained motion trajectories may contain grossly mistracked features, missing entries, or not correspond to any valid motion model. In this paper, we develop a robust subspace separation scheme that can deal with all of these practical issues in a unified framework. Our methods draw strong connections between lossy compression, rank minimization, and sparse representation. We test our methods extensively and compare their performance to several extant methods with experiments on the Hopkins 155 database. Our results are on par with state-of-the-art results, and in many cases exceed them. All MATLAB code and segmentation results are publicly available for peer evaluation at http://perception.csl.uiuc.edu/coding/motion/.

Multiframe Motion Segmentation with Missing Data Using PowerFactorization and GPCA

Abstract We consider the problem of segmenting multiple rigid-body motions from point correspondences in multiple affine views. We cast this problem as a subspace clustering problem in which point trajectories associated with each motion live in a linear subspace of dimension two, three or four. Our algorithm involves projecting all point trajectories onto a 5-dimensional subspace using the SVD, the PowerFactorization method, or RANSAC, and fitting multiple linear subspaces representing different rigid-body motions to the points in ℝ5 using GPCA. Unlike previous work, our approach does not restrict the motion subspaces to be four-dimensional and independent. Instead, it deals gracefully with all the spectrum of possible affine motions: from two-dimensional and partially dependent to four-dimensional and fully independent. Our algorithm can handle the case of missing data, meaning that point tracks do not have to be visible in all images, by using the PowerFactorization method to project the data. In addition, our method can handle outlying trajectories by using RANSAC to perform the projection. We compare our approach to other methods on a database of 167 motion sequences with full motions, independent motions, degenerate motions, partially dependent motions, missing data, outliers, etc. On motion sequences with complete data our method achieves a misclassification error of less that 5% for two motions and 29% for three motions.

Distributed pose averaging in camera networks via consensus on SE(3)

In this paper, we propose distributed algorithms for estimating the average pose of an object viewed by a localized network of camera motes. To this effect, we propose distributed averaging consensus algorithms on the group of 3D rigid-body transformations, SE(3). We rigorously analyze the convergence of the proposed algorithms, and show that naive generalizations of Euclidean consensus algorithms fail to converge to the correct solution. We also provide synthetic experiments that confirm our analysis and validate our approach.

Distributed image-based 3-D localization of camera sensor networks

We consider the problem of distributed estimation of the poses of N cameras in a camera sensor network using image measurements only. The relative rotation and translation (up to a scale factor) between pairs of neighboring cameras can be estimated using standard computer vision techniques. However, due to noise in the image measurements, these estimates may not be globally consistent. We address this problem by minimizing a cost function on SE(3)N in a distributed fashion using a generalization of the classical consensus algorithm for averaging Euclidean data. We also derive a condition for convergence, which relates the step-size of the consensus algorithm and the degree of the camera network graph. While our methods are designed with the camera sensor network application in mind, our results are applicable to other localization problems in a more general setting. We also provide synthetic simulations to test the validity of our approach.

Distributed computer vision algorithms through distributed averaging

Traditional computer vision and machine learning algorithms have been largely studied in a centralized setting, where all the processing is performed at a single central location. However, a distributed approach might be more appropriate when a network with a large number of cameras is used to analyze a scene. In this paper we show how centralized algorithms based on linear algebraic operations can be made distributed by using simple distributed averages. We cover algorithms such as SVD, least squares, PCA, GPCA, 3-D point triangulation, pose estimation and affine SfM.

Distributed 3-D Localization of Camera Sensor Networks From 2-D Image Measurements

In this paper we propose distributed algorithms that use 2-D image measurements to estimate the absolute 3-D poses of the nodes in a camera network, with the purpose of enabling higher-level tasks such as tracking and recognition. We assume that pairs of cameras with overlapping fields of view can estimate their relative 3-D pose (rotation and translation direction) using standard computer vision techniques. The solution we propose combines these local, noisy estimates into a single consistent localization. We derive our algorithms from optimization problems on the manifold of poses. We provide theoretical results on the convergence of the algorithms (choice of the step-size, initialization) and on the properties of their solutions (sensitivity, uniqueness). We also provide experiments on synthetic and real data. Interestingly, our algorithm for estimating the rotation part of the poses shows some degree of robustness to outliers.

On the Convergence of Gradient Descent for Finding the Riemannian Center of Mass

We study the problem of finding the global Riemannian center of mass of a set of data points on a Riemannian manifold. Specifically, we investigate the convergence of constant step-size gradient descent algorithms for solving this problem. The challenge is that often the underlying cost function is neither globally differentiable nor convex, and despite this one would like to have guaranteed convergence to the global minimizer. After some necessary preparations we state a conjecture which we argue is the best (in a sense described) convergence condition one can hope for. The conjecture specifies conditions on the spread of the data points, step-size range, and the location of the initial condition (i.e., the region of convergence) of the algorithm. These conditions depend on the topology and the curvature of the manifold and can be conveniently described in terms of the injectivity radius and the sectional curvatures of the manifold. For manifolds of constant nonnegative curvature (e.g., the sphere and the rotation group in

Distributed Computer Vision Algorithms

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