The α-expansion algorithm [7] has had a significant impact in computer vision due to its generality, effectiveness, and speed. Thus far it can only minimize energies that involve unary, pairwise, and specialized higher-order terms. Our main contribution is to extend α-expansion so that it can simultaneously optimize “label costs ” as well. An energy with label costs can penalize a solution based on the set of labels that appear in it. The simplest special case is to penalize the number of labels in the solution. Our energy is quite general, and we prove optimality bounds for our algorithm. A natural application of label costs is multi-model fitting, and we demonstrate several such applications in vision: homography detection, motion segmentation, and unsupervised image segmentation. Our C++/MATLAB implementation is publicly available. 1. Some Useful Regularization Energies In a labeling problem we are given a set of observations P (pixels, features, data points) and a set of labels L (categories, geometric models, disparities). The goal is to assign each observation p ∈ P a label fp ∈ L such that the joint labeling f minimizes some objective function E(f). Most labeling problems in computer vision are ill-posed and in need of regularization, but the most useful regularizers often make the problem NP-hard. Our work is about how to effectively optimize two such regularizers: a preference for fewer labels in the solution, and a preference for spatial smoothness. Figure 1 suggests how these criteria cooperate to give clean results. Surprisingly, there is no good algorithm to optimize their combination. 1 Our main contribution is a way to simultaneously optimize both of these criteria inside the powerful α-expansion algorithm [7]. Label costs. Start from a basic (unregularized) energy E(f) = ∑ pDp(fp), where optimal fp can each be determined independently from the ‘data costs’. Suppose, however, that we wish to explain the observations using as few unique labels as necessary. We can introduce label costs into E(f) to penalize each unique label that appears in f: E(f) = ∑

We develop a convex relaxation of maximum a posteriori estimation of a mixture of regression models. Although our relaxation involves a semidefinite matrix variable, we reformulate the problem to eliminate the need for general semidefinite programming. In particular, we provide two reformulations that admit fast algorithms. The first is a max-min spectral reformulation exploiting quasi-Newton descent. The second is a min-min reformulation consisting of fast alternating steps of closed-form updates. We evaluate the methods against Expectation-Maximization in a real problem of motion segmentation from video data. 1

The Uncapacitated Facility Location Problem (UFLP) is one of the most widely studied discrete location problems, whose applications arise in a variety of settings. We tackle the UFLP using probabilistic inference in a graphical model- an approach that has received little attention in the past. We show that the fixed points of max-product linear programming (MPLP), a convexified version of the max-product algorithm, can be used to construct a solution with a 3-approximation guarantee for metric UFLP instances. In addition, we characterize some scenarios under which the MPLP solution is guaranteed to be globally optimal. We evaluate the performance of both max-sum and MPLP empirically on metric and non-metric problems, demonstrating the advantages of the 3-approximation construction and algorithm applicability to non-metric instances.

We present a novel Quadratic Program (QP) formulation for robust multi-model fitting of geometric structures in vision data. Our objective function enforces both the fidelity of a model to the data and the similarity between its associated inliers. Departing from most previous optimizationbased approaches, the outcome of our method is a ranking of a given set of putative models, instead of a pre-specified number of “good ” candidates (or an attempt to decide the right number of models). This is particularly useful when the number of structures in the data is a priori unascertainable due to unknown intent and purposes. Another key advantage of our approach is that it operates in a unified optimization framework, and the standard QP form of our problem formulation permits globally convergent optimization techniques. We tested our method on several geometric multi-model fitting problems on both synthetic and real data. Experiments show that our method consistently achieves state-of-the-art results. 1.

Abstract. We report on preliminary work in the application of lowlevel visual motion cues to identify potential hazards during on-road driving. In conjunction with a clinical study of hazard perception in older age drivers, we consider the detection of a range of hazardous scenarios identified as particularly challenging for older drivers in video sequences used in the clinical study. Central to our approach is the use of visual motion as a means of estimating self motion, from which we identify optical flow due to other motions in the scene. We report results obtained using the same hazard perception test used in clinical trials. 1

Multi-structure model fitting has traditionally taken a two-stage approach: First, sample a (large) number of model hypotheses, then select the subset of hypotheses that optimise a joint fitting and model selection criterion. This disjoint two-stage approach is arguably suboptimal and inefficient — if the random sampling did not retrieve a good set of hypotheses, the optimised outcome will not represent a good fit. To overcome this weakness we propose a new multi-structure fitting approach based on Reversible Jump MCMC. Instrumental in raising the effectiveness of our method is an adaptive hypothesis generator, whose proposal distribution is learned incrementally and online. We prove that this adaptive proposal satisfies the diminishing adaptation property crucial for ensuring ergodicity in MCMC. Our method effectively conducts hypothesis sampling and optimisation simultaneously, and yields superior computational efficiency over previous two-stage methods. 1

Computer vision is full of problems elegantly expressed in terms of energy minimization. We characterize a class of energies with hierarchical costs and propose a novel hierarchical fusion algorithm. Hierarchical costs are natural for modeling an array of difficult problems. For example, in semantic segmentation one could rule out unlikely object combinations via hierarchical context. In geometric model estimation, one could penalize the number of unique model families in a solution, not just the number of models—a kind of hierarchical MDL criterion. Hierarchical fusion uses the well-known α-expansion algorithm as a subroutine, and offers a much better approximation bound in important cases.

This paper proposes a simple “prior-free ” method for solving non-rigid structure-from-motion factorization problems. Other than using the basic low-rank condition, our method does not assume any extra prior knowledge about the nonrigid scene or about the camera motions. Yet, it runs reliably, produces optimal result, and does not suffer from the inherent basis-ambiguity issue which plagued many conventional nonrigid factorization techniques. Our method is easy to implement, which involves solving no more than an SDP (semi-definite programming) of small and fixed size, a linear Least-Squares or trace-norm minimization. Extensive experiments have demonstrated that it outperforms most of the existing linear methods of nonrigid factorization. This paper offers not only new theoretical insight, but also a practical, everyday solution, to non-rigid structure-from-motion. 1 1.