Globally Optimal Coupled Surfaces for Semi-automatic Segmentation of Medical Images

Manual delineations are of paramount importance in medical imaging, for instance to train supervised methods and evaluate automatic segmentation algorithms. In volumetric images, manually tracing regions of interest is an excruciating process in which much time is wasted labeling neighboring 2D slices that are similar to each other. Here we present a method to compute a set of discrete minimal surfaces whose boundaries are specified by user-provided segmentations on one or more planes. Using this method, the user can for example manually delineate one slice every n and let the algorithm complete the segmentation for the slices in between. Using a discrete framework, this method globally minimizes a cost function that combines a regularizer with a data term based on image intensities, while ensuring that the surfaces do not intersect each other or leave holes in between. While the resulting optimization problem is an integer program and thus NP-hard, we show that the equality constraint matrix is totally unimodular, which enables us to solve the linear program (LP) relaxation instead. We can then capitalize on the existence of efficient LP solvers to compute a globally optimal solution in practical times. Experiments on two different datasets illustrate the superiority of the proposed method over the use of independent, labelwise optimal surfaces (similar to 5% mean increase in Dice when one every six slices is labeled, with some structures improving up to similar to 10% in Dice).