A New Algorithm for Energy Minimization with Discontinuities

Many tasks in computer vision involve assigning a label (such as disparity) to every pixel. These tasks can be formulated as energy minimization problems. In this paper, we consider a natural class of energy functions that permits discontinuities. Computing the exact minimum is NP-hard. We have developed two approximation algorithms based on graph cuts. These algorithms find a local minimum of the energy function with respect to a certain move space. One of these methods was described in [6]. Here we describe a new algorithm based on α-expansion moves. This algorithm generates a labeling where there is no subset of pixels S and label α such that assigning the label α to every pixel in S will reduce the energy. We apply our method to the stereo matching problem, and obtain promising experimental results. Empirically, the new technique outperforms our previous algorithm both in terms of running time and output quality. 1 Energy minimization in early vision Many early vision problems require estimating some spatially varying quantity (such as intensity, disparity or texture) 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 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 ∗The authors are with the Computer Science Department, Cornell University, Ithaca, NY 14853; email: [email protected], [email protected], [email protected]; telephone 607-2558413 (voice) 607-255-4428 (fax).