Many tasks in computer vision involve assigning a label (such as disparity) to every pixel. A common constraint is that the labels should vary smoothly almost everywhere while preserving sharp discontinuities that may exist, e.g., at object boundaries. These tasks are naturally stated in terms of energy minimization. The authors consider a wide class of energies with various smoothness constraints. Global minimization of these energy functions is NP-hard even in the simplest discontinuity-preserving case. Therefore, our focus is on efficient approximation algorithms. We present two algorithms based on graph cuts that efficiently find a local minimum with respect to two types of large moves, namely expansion moves and swap moves. These moves can simultaneously change the labels of arbitrarily large sets of pixels. In contrast, many standard algorithms (including simulated annealing) use small moves where only one pixel changes its label at a time. Our expansion algorithm finds a labeling within a known factor of the global minimum, while our swap algorithm handles more general energy functions. Both of these algorithms allow important cases of discontinuity preserving energies. We experimentally demonstrate the effectiveness of our approach for image restoration, stereo and motion. On real data with ground truth, we achieve 98 percent accuracy.

/pdf/fast-approximate-energy-minimization-via-graph-cuts-3usqtddnkx.pdf

Fast approximate energy minimization via graph cuts

We consider the general problem of learning from labeled and unlabeled data, which is often called semi-supervised learning or transductive inference. A principled approach to semi-supervised learning is to design a classifying function which is sufficiently smooth with respect to the intrinsic structure collectively revealed by known labeled and unlabeled points. We present a simple algorithm to obtain such a smooth solution. Our method yields encouraging experimental results on a number of classification problems and demonstrates effective use of unlabeled data.

/pdf/learning-with-local-and-global-consistency-2vi1gqygbg.pdf

Learning with Local and Global Consistency

In this paper we address the problem of minimizing a large class of energy functions that occur in early vision. The major restriction is that the energy function's smoothness term must only involve pairs of pixels. We propose two algorithms that use graph cuts to compute a local minimum even when very large moves are allowed. The first move we consider is an /spl alpha/-/spl beta/-swap: for a pair of labels /spl alpha/,/spl beta/, this move exchanges the labels between an arbitrary set of pixels labeled a and another arbitrary set labeled /spl beta/. Our first algorithm generates a labeling such that there is no swap move that decreases the energy. The second move we consider is an /spl alpha/-expansion: for a label a, this move assigns an arbitrary set of pixels the label /spl alpha/. Our second algorithm, which requires the smoothness term to be a metric, generates a labeling such that there is no expansion move that decreases the energy. Moreover, this solution is within a known factor of the global minimum. We experimentally demonstrate the effectiveness of our approach on image restoration, stereo and motion.

/pdf/fast-approximate-energy-minimization-via-graph-cuts-5cbspd4tux.pdf

Since the subject of traffic dynamics has captured the interest of physicists, many surprising effects have been revealed and explained. Some of the questions now understood are the following: Why are vehicles sometimes stopped by ``phantom traffic jams'' even though drivers all like to drive fast? What are the mechanisms behind stop-and-go traffic? Why are there several different kinds of congestion, and how are they related? Why do most traffic jams occur considerably before the road capacity is reached? Can a temporary reduction in the volume of traffic cause a lasting traffic jam? Under which conditions can speed limits speed up traffic? Why do pedestrians moving in opposite directions normally organize into lanes, while similar systems ``freeze by heating''? All of these questions have been answered by applying and extending methods from statistical physics and nonlinear dynamics to self-driven many-particle systems. This article considers the empirical data and then reviews the main approaches to modeling pedestrian and vehicle traffic. These include microscopic (particle-based), mesoscopic (gas-kinetic), and macroscopic (fluid-dynamic) models. Attention is also paid to the formulation of a micro-macro link, to aspects of universality, and to other unifying concepts, such as a general modeling framework for self-driven many-particle systems, including spin systems. While the primary focus is upon vehicle and pedestrian traffic, applications to biological or socio-economic systems such as bacterial colonies, flocks of birds, panics, and stock market dynamics are touched upon as well.

Traffic and related self-driven many-particle systems

This is a tutorial review on the Potts model aimed at bringing out in an organized fashion the essential and important properties of the standard Potts model. Emphasis is placed on exact and rigorous results, but other aspects of the problem are also described to achieve a unified perspective. Topics reviewed include the mean-field theory, duality relations, series expansions, critical properties, experimental realizations, and the relationship of the Potts model with other lattice-statistical problems.

The Potts model

In considering the statistics of the ‘no-field’ square Ising lattice in which each unit is capable of two configurations and only nearest neighbours interact, Kramers and Wannier (3) were able to deduce an inversion transformation under which the partition function of the lattice is invariant when the temperature is transformed from a low to a high (‘inverted’) value. The important property of this inversion transformation is that its fixed point gives the transition point of the lattice.

Some generalized order - disorder transformations

The manner in which vehicles follow each other on a highway without passing and the propagation of disturbances down a line of vehicles has been investigated further. Criteria are derived for both local and asymptotic stability in a chain of vehicles. The influence of next nearest neighbors as well as a statistical theory of stability is discussed. “Acceleration noise” is proposed as a parameter that might be employed to characterize the driver-car-road complex under various conditions. Some preliminary experimental measurements of acceleration noise are discussed.

Traffic Dynamics: Analysis of Stability in Car Following

The steady-state flow is examined for a car-following model in which the acceleration at time t of a car attempting to follow a lead car is proportional to the relative velocity at a time t − Δ and in which the sensitivity λ is no longer taken constant as in previous work but is inversely proportional to the car spacing. The characteristics of the steady-state flow for this model are described and compared with experimental data.

Car-Following Theory of Steady-State Traffic Flow

In this paper we rederive, by simpler methods, the Onsager‐Kaufman formulas for the correlations and the Onsager formula for the spontaneous magnetization of the rectangular two‐dimensional Ising lattice. The Pfaffian approach is used to derive the correlations in terms of Pfaffians, and for the correlations in a row a single Toeplitz determinant is obtained which is proved equivalent to the Onsager‐Kaufman result. The spontaneous magnetization is obtained as the limiting value of an infinite Toeplitz determinant and its evaluation is facilitated by use of the generalization of a result first published by Szego.

Correlations and Spontaneous Magnetization of the Two‐Dimensional Ising Model

DESCRIPTIONS ARE PRESENTED OF A VARIETY OF TRANSPORTATION NETWORKS, AND FLOW CONCEPTS USED IN TRANSPORTATION PROBLEMS ARE SUMMARISED. EXTREMAL PRINCIPLES, EQUIPLIBRIUM FLOW PATTERNS, AND THEIR APPLICATION TO TREE-BUILDING ALGORITHMS AND TRAFFIC ASSIGMENT PROCEDURES ARE DISCUSSED IN DETAIL. A LARGE NUMBER OF PROBLEMS, SOLUTIONS TO PROBLEMS, AND REVIEWS OF PUBLISHED LITERATURE ARE INCLUDED. THE BOOK CONCLUDES WITH AN ANALYSIS OF THE IMPORTANT TRIP DISTRIBUTION MODELS.

Renfrey B. Potts

Papers

Some generalized order - disorder transformations

Traffic Dynamics: Analysis of Stability in Car Following

Car-Following Theory of Steady-State Traffic Flow

Correlations and Spontaneous Magnetization of the Two‐Dimensional Ising Model

Flows in transportation networks