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 α-βswap: for a pair of labels α, β, this move exchanges the labels between an arbitrary set of pixels labeled α and another arbitrary set labeled β. Our first algorithm generates a labeling such that there is no swap move that decreases the energy. The second move we consider is an α-expansion: for a label α, this move assigns an arbitrary set of pixels the label α. 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. 1 Energy minimization in early vision Many early vision problems require estimating some spatially varying quantity (such as intensity or 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 ∈ P must be assigned a label in some set L; for motion or stereo, the labels are disparities, while for image restoration they represent intensities. The goal is to find a labeling f that assigns each pixel p ∈ P a label fp ∈ 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 E(f) = Esmooth(f) + Edata(f). Here Esmooth measures the extent to which f is not piecewise smooth, while Edata measures the disagreement between f and the observed data. Many different energy functions have been proposed in the literature. The form of Edata is typically Edata(f) = ∑