Globally Optimal Closed-Surface Segmentation for Connectomics Bjoern Andres 1,⋆ , Thorben Kroeger 1,⋆ , Kevin L. Briggman 2 , Winfried Denk 3 , Natalya Korogod 4 , Graham Knott 4 , Ullrich Koethe 1 , and Fred A. Hamprecht 1 1 HCI, University of Heidelberg 2 NIH, Bethesda 3 MPI for Medical Research, Heidelberg 4 EPFL, Lausanne Abstract. We address the problem of partitioning a volume image into a previ- ously unknown number of segments, based on a likelihood of merging adjacent supervoxels. Towards this goal, we adapt a higher-order probabilistic graphical model that makes the duality between supervoxels and their joint faces explicit and ensures that merging decisions are consistent and surfaces of final segments are closed. First, we propose a practical cutting-plane approach to solve the MAP inference problem to global optimality despite its NP-hardness. Second, we ap- ply this approach to challenging large-scale 3D segmentation problems for neu- ral circuit reconstruction (Connectomics), demonstrating the advantage of this higher-order model over independent decisions and finite-order approximations. 1 Introduction This paper studies the problem of partitioning a volume image into a previously un- known number of segments, based on a likelihood of merging adjacent supervoxels. We choose a graphical model approach in which binary variables are associated with the joint faces of supervoxels, indicating for each face whether the two adjacent super- voxels should belong to the same segment (0) or not (1). Models of low order can lead to inconsistencies where a face is labeled as 1 even though there exists a path from one of the adjacent segments to the other along which all faces are labeled as 0. As a result, the union of all faces labeled as 1 need not form closed surfaces. Such inconsistencies can be excluded by a higher-order conditional random field (CRF) [1] that constrains the binary labelings to the multicut polytope [2], thus ensuring closed surfaces. While the number of multicut constraints can be exponential [3], constraints that are violated by a given labeling can be found in quadratic time [4]. The MAP inference problem can therefore be addressed by the cutting-plane method, i.e. by solving a sequence of relaxed problems to global optimality until no more constraints are violated [4]. Here, we show that the optimization scheme described in [1] is unsuitable for large 3D segmentations where the supervoxel adjacency graph is denser and non-planar. We therefore extend the cutting-plane approach by adding only constraints which are facet- defining by a property of the multicut polytope (Section 4.3), a double-ended, paral- lel search to find violated constraints (Section 4.2) and a problem-specific warm-start ⋆ Contributed equally. A. Fitzgibbon et al. (Eds.): ECCV 2012, Part III, LNCS 7574, pp. 778–791, 2012. © Springer-Verlag Berlin Heidelberg 2012