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 3-D motion segmentation algorithms for affine cameras on a benchmark of 155 motion sequences of checkerboard, traffic, and articulated scenes. 1.

This paper describes methods for recovering time-varying shape and motion of non-rigid 3D objects from uncalibrated 2D point tracks. For example, given a video recording of a talking person, we would like to estimate the 3D shape of the face at each instant, and learn a model of facial deformation. Time-varying shape is modeled as a rigid transformation combined with a non-rigid deformation. Reconstruction is ill-posed if arbitrary deformations are allowed, and thus additional assumptions about deformations are required. We first suggest restricting shapes to lie within a lowdimensional subspace, and describe estimation algorithms. However, this restriction alone is insufficient to constrain reconstruction. To address these problems, we propose a reconstruction method using a Probabilistic Principal Components Analysis (PPCA) shape model, and an estimation algorithm that simultaneously estimates 3D shape and motion for each instant, learns the PPCA model parameters, and robustly fills-in missing data points. We then extend the model to model temporal dynamics in object shape, allowing the algorithm to robustly handle severe cases of missing data.

This paper considers the problem of factorizing a matrix with missing components into a product of two smaller matrices, also known as principal component analysis with missing data (PCAMD). The Wiberg algorithm is a numerical algorithm developed for the problem in the community of applied mathematics. We argue that the algorithm has not been correctly understood in the computer vision community. Although there are many studies in our com-munity, almost every one of which refers to the Wiberg study, as far as we know, there is no literature in which the performance of the Wiberg algorithm is investigated or the detail of the algorithm is presented. In this paper, we present derivation of the algorithm along with a problem in its implementation that needs to be carefully considered, and then examine its performance. The experimental results demonstrate that the Wiberg algorithm shows a consid-erably good performance, which should contradict the conventional view in our community, namely that minimization-based algorithms tend to fail to converge to a global minimum rel-atively frequently. The performance of the Wiberg algorithm is such that even starting with random initial values, it converges in most cases to a correct solution, even when the matrix has many missing components and the data are contaminated with very strong noise. Our con-clusion is that the Wiberg algorithm can also be used as a standard algorithm for the problems of computer vision. 3 1

Non-negative tensor factorization (NTF) has recently been proposed as sparse and efficient image representation (Welling and Weber, Patt. Rec. Let., 2001). Until now, sparsity of the tensor factorization has been empirically observed in many cases, but there was no systematic way to control it. In this work, we show that a sparsity measure recently proposed for non-negative matrix factorization (Hoyer, J. Mach. Learn. Res., 2004) applies to NTF and allows precise control over sparseness of the resulting factorization. We devise an algorithm based on sequential conic programming and show improved performance over classical NTF codes on artificial and on real-world data sets.

Random Sample Consensus (RANSAC) is the most widely used robust regression algorithm in computer vision. However, RANSAC has a few drawbacks which make it difficult to use for practical applications. Some of these problems have been addressed through improved sampling algorithms or better cost functions, but an important difficulty still remains. The algorithm is not user independent, and requires knowledge of the scale of the inlier noise. We propose a new robust regression algorithm, the projection based M-estimator (pbM). The pbM algorithm is derived by building a connection to the theory of kernel density estimation and this leads to an improved cost function, which gives better performance. Furthermore, pbM is user independent and does not require any knowledge of the scale of noise corrupting the inliers. We propose a general framework for the pbM algorithm which can handle heteroscedastic data and multiple linear constraints on each data point through the use of Grassmann manifold theory. The performance of pbM is compared with RANSAC and M-Estimator Sample Consensus (MSAC) on various real problems. It is shown that pbM gives better results than RANSAC and MSAC in spite of being user independent.

Abstract—We give the first complete theoretical convergence analysis for the iterative extensions of the Sturm/Triggs algorithm. We show that the simplest extension, SIESTA, converges to nonsense results. Another proposed extension has similar problems, and experiments with “balanced ” iterations show that they can fail to converge or become unstable. We present CIESTA, an algorithm that avoids these problems. It is identical to SIESTA except for one simple extra computation. Under weak assumptions, we prove that CIESTA iteratively decreases an error and approaches fixed points. With one more assumption, we prove it converges uniquely. Our results imply that CIESTA gives a reliable way of initializing other algorithms such as bundle adjustment. A descent method such as Gauss-Newton can be used to minimize the CIESTA error, combining quadratic convergence with the advantage of minimizing in the projective depths. Experiments show that CIESTA performs better than other iterations.

In this paper we address the problem of recovering structure and motion from a large number of intrinsically calibrated perspective cameras. We describe a method that combines (1) weak-perspective reconstruction in the presence of noisy and missing data and (2) an algorithm that updates weakperspective reconstruction to perspective reconstruction by incrementally estimating the projective depths. The method also solves for the reversal ambiguity associated with affine factorization techniques. The method has been successfully applied to the problem of calibrating the external parameters (position and orientation) of several multiple-camera setups. Results obtained with synthetic and experimental data compare favourably with results obtained with nonlinear minimization such as bundle adjustment. 1.

The calculation of a low-rank approximation of a matrix is a fundamental operation in many computer vision applications. The workhorse of this class of problems has long been the Singular Value Decomposition. However, in the presence of missing data and outliers this method is not applicable, and unfortunately, this is often the case in practice. In this paper we present a method for calculating the low-rank factorization of a matrix which minimizes the L1 norm in the presence of missing data. Our approach represents a generalization the Wiberg algorithm of one of the more convincing methods for factorization under the L2 norm. By utilizing the differentiability of linear programs, we can extend the underlying ideas behind this approach to include this class of L1 problems as well. We show that the proposed algorithm can be efficiently implemented using existing optimization software. We also provide preliminary experiments on synthetic as well as real world data with very convincing results. 1.

In this paper we study singular values of a matrix whose one entry varies while all other entries are prescribed. In particular, we find the possible pth singular value of such a matrix, and we define explicitly the unknown entry such that the completed matrix has the minimal possible pth singular value. This in turn determines possible pth singular value of a matrix under rank one perturbation. Moreover, we determine the possible value of pth singular value of a partially prescribed matrix whose set of unknown entries has a form of a Young diagram. In particular, we give a fast algorithm for defining the completion that minimizes the pth singular value of such matrix.

The problem of missing data is ubiquitous in domains such as biomedical signal processing, network traffic analysis, bibliometrics, social network analysis, chemometrics, computer vision, and communication networks---all domains in which data collection is subject to occasional errors. Moreover, these data sets can be quite large and have more than two axes of variation, e.g., sender, receiver, time. Many applications in those domains aim to capture the underlying latent structure of the data; in other words, they need to factorize data sets with missing entries. If we cannot address the problem of missing data, many important data sets will be discarded or improperly analyzed. Therefore, we need a robust and scalable approach for factorizing multi-way arrays (i.e., tensors) in the presence of missing data. We focus on one of the most well-known tensor factorizations, CANDECOMP/PARAFAC (CP), and formulate the CP model as a weighted least squares problem that models only the known entries. We develop an algorithm called CP-WOPT (CP Weighted OPTimization) using a first-order optimization approach to solve the weighted least squares problem. Based on extensive numerical experiments, our algorithm is shown to successfully factor tensors with noise and up to 70% missing data. Moreover, our approach is significantly faster than the leading alternative and scales to larger problems. To show the real-world usefulness of CP-WOPT, we illustrate its applicability on a novel EEG (electroencephalogram) application where missing data is frequently encountered due to disconnections of electrodes.