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 rst move we consider is an --swap: for a pair of labels ; , this move exchanges the labels between an arbitrary set of pixels labeled and another arbitrary set labeled . Our rst algorithm generates a labeling such that there is no swap move that decreases the energy. The second move we consider is an -expansion: for a label , this move assigns an arbitrary set of pixels the label . Our second algorithm, which requires the smoothness term to be a metric, generates a labeling such that there i s n o expansion move that decreases the energy. Moreover, this solution is within a known factor of the global minimum. We experimentally demonstrate the e ectiveness of our approach on image restoration, stereo and motion. Many early vision problems require estimating some spatially varying quantity such a s i n tensity o r disparity from noisy measurements. Such quantities tend to be piecewise smooth; they vary smoothly at most points, but change dramatically at object boundaries. Every pixel p 2 P must be assigned a label in some set L; for motion or stereo, the labels are disparities, while for image restoration they represent i n tensities. The goal is to nd a labeling f that assigns each pixel p 2 P a label f p 2 L , where f is both piecewise smooth and consistent with the observed data. These vision problems can be naturally formulated in terms of energy minimization. In this framework, one seeks the labeling f that minimizes the energy Ef = E smooth f + E data f: Here E smooth measures the extent to which f is not piecewise smooth, while E data measures the disagreement b e t ween f and the observed data. Many di erent energy functions have been proposed in the literature. , where i p is the observed intensity of the pixel p. The choice of E smooth is a critical issue, and many di erent functions have been proposed. For example, in standard regularization-based vision 6 , E smooth makes f smooth everywhere. This leads to poor results at object boundaries. Energy functions that do not have this problem are called discontinuity-preserving. A large number of discontinuity-preserving energy functions have been proposed see for example 7 . Geman and Geman's seminal paper 3 gave a B a yesian interpretation of many energy functions, and proposed a discontinuitypreserving energy function based on Markov Random Fields MRF's. The major di culty with energy minimization for early vision lies in the enormous computational costs. Typically these energy functions have many local minima i.e., they are non-convex. Worse still, the space of possible labelings has dimension jPj, which i s m a n y thousands. There have been numerous attempts to design fast algorithms for energy minimization. Simulated annealing was popularized in computer vision by 3 , and is widely used since it can optimize an arbitrary energy function. Unfortunately, minimizing an arbitrary energy function requires exponential time, and as a consequence simulated annealing is very slow. In practice, annealing is ine cient partly because at each step it changes the value of a single pixel. The energy functions that we consider in this paper arise in a variety of di erent contexts, including the Bayesian labeling of MRF's. We allow D p to be
The main objective of this paper is to present an extension of the previously defined work towards its embedding in a Markov-Random-Field (MRF) framework, while optimization is carried out through graph cut REF .
Fast approximate energy minimization via graph cuts
Proceedings of the Seventh IEEE International Conference on Computer Vision
Proceedings of the Seventh IEEE International Conference on Computer Vision
Computer Science