The layered net surface problems in discrete geometry and medical image segmentation

Abstract

Efficient detection of multiple inter-related surfaces representing the boundaries of objects of interest in d-D images (d ≥ 3) is important and remains challenging in many medical image analysis applications. In this paper, we study several layered net surface (LNS) problems captured by an interesting type of geometric graphs called ordered multi-column graphs in the d-D discrete space (d ≥ 3). The LNS problems model the simultaneous detection of multiple mutually related surfaces in three or higher dimensional medical images. Although we prove that the d-D LNS problem (d ≥ 3) on a general ordered multi-column graph is NP-hard, the (special) ordered multi-column graphs that model medical image segmentation have the self-closure structures, and admit polynomial time exact algorithms for solving the LNS problems. Our techniques also solve the related net surface volume (NSV) problems of computing well-shaped geometric regions of an optimal total volume in a d-D weighted voxel grid. The NSV problems find applications in medical image segmentation and data mining. Our techniques yield the first polynomial time exact algorithms for several high dimensional medical image segmentation problems. The practical efficiency and accuracy of the algorithms are showcased by experiments based on real medical data. © Springer-Verlag Berlin Heidelberg 2005.