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.

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Fast approximate energy minimization via graph cuts

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.

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In the last few years, several new algorithms based on graph cuts have been developed to solve energy minimization problems in computer vision. Each of these techniques constructs a graph such that the minimum cut on the graph also minimizes the energy. Yet, because these graph constructions are complex and highly specific to a particular energy function, graph cuts have seen limited application to date. In this paper, we give a characterization of the energy functions that can be minimized by graph cuts. Our results are restricted to functions of binary variables. However, our work generalizes many previous constructions and is easily applicable to vision problems that involve large numbers of labels, such as stereo, motion, image restoration, and scene reconstruction. We give a precise characterization of what energy functions can be minimized using graph cuts, among the energy functions that can be written as a sum of terms containing three or fewer binary variables. We also provide a general-purpose construction to minimize such an energy function. Finally, we give a necessary condition for any energy function of binary variables to be minimized by graph cuts. Researchers who are considering the use of graph cuts to optimize a particular energy function can use our results to determine if this is possible and then follow our construction to create the appropriate graph. A software implementation is freely available.

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What energy functions can be minimized via graph cuts

1 Defining Network Flow A flow network is a directed graph G = (V,E) in which each edge (u, v) ∈ E has non-negative capacity c(u, v) ≥ 0. We require that if (u, v) ∈ E, then (v, u) / ∈ E. That is, if an edge exists, then the edge between the same vertices going the reverse direction does not exist. Every flow network has a source s and a sink t, and we assume that for every v ∈ V , there is some path s→ · · · → v → · · · → t. Note that this implies that flow networks are connected. Informally, the intuition behind network flow is to think of the edges as pipes and the weights on the edges as the capacity its corresponding pipe per unit of time. The question we are trying to ask is: how many units of water can we send from the source to the sink per unit of time? Formally, a flow in G is a function f : V × V → R that satisfies the following: • Capacity constraint. For all u, v ∈ V , we require 0 ≤ f(u, v) ≤ c(u, v). Our pipe cannot hold more than is allowed as dictated by its capacity. • Flow conservation. For u ∈ V − {s, t}, we require ∑

Network Flows

Tracking new topics, ideas, and "memes" across the Web has been an issue of considerable interest. Recent work has developed methods for tracking topic shifts over long time scales, as well as abrupt spikes in the appearance of particular named entities. However, these approaches are less well suited to the identification of content that spreads widely and then fades over time scales on the order of days - the time scale at which we perceive news and events.We develop a framework for tracking short, distinctive phrases that travel relatively intact through on-line text; developing scalable algorithms for clustering textual variants of such phrases, we identify a broad class of memes that exhibit wide spread and rich variation on a daily basis. As our principal domain of study, we show how such a meme-tracking approach can provide a coherent representation of the news cycle - the daily rhythms in the news media that have long been the subject of qualitative interpretation but have never been captured accurately enough to permit actual quantitative analysis. We tracked 1.6 million mainstream media sites and blogs over a period of three months with the total of 90 million articles and we find a set of novel and persistent temporal patterns in the news cycle. In particular, we observe a typical lag of 2.5 hours between the peaks of attention to a phrase in the news media and in blogs respectively, with divergent behavior around the overall peak and a "heartbeat"-like pattern in the handoff between news and blogs. We also develop and analyze a mathematical model for the kinds of temporal variation that the system exhibits.

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Meme-tracking and the dynamics of the news cycle

In the multiterminal cut problem one is given an edge-weighted graph and a subset of the vertices called terminals, and is asked for a minimum weight set of edges that separates each terminal from all the others. When the number $k$ of terminals is two, this is simply the mincut, max-flow problem, and can be solved in polynomial time. It is shown that the problem becomes NP-hard as soon as $k=3$, but can be solved in polynomial time for planar graphs for any fixed $k$. The planar problem is NP-hard, however, if $k$ is not fixed. A simple approximation algorithm for arbitrary graphs that is guaranteed to come within a factor of $2-2/k$ of the optimal cut weight is also described.

The Complexity of Multiterminal Cuts

In the Multiway Cut problem we are given an edge-weighted graph and a subset of the vertices called terminals, and asked for a minimum weight set of edges that separates each terminal from all the others. When the number k of terminals is two, this is simply the min-cut, max-flow problem, and can be solved in polynomial time. We show that the problem becomes NP-hard as soon as k = 3, but can be solved in polynomial time for planar graphs for any fixed k. The planar problem is NP-hard, however, if k is not fixed. We also describe a simple approximation algorithm for arbitrary graphs that is guaranteed to come within a factor of 2–2/k of the optimal cut weight.

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The complexity of multiway cuts (extended abstract)

We give a complete list of lattices associated to closure systems generated by four maximal cliques and generalize a known characterization to uniform hypergraphs. 

E. Dahlhaus

Papers

The Complexity of Multiterminal Cuts

The complexity of multiway cuts (extended abstract)

Examples of Clique Closure Systems