Open Access

Vertebral labelling and segmentation are two fundamental tasks in an automated spine processing pipeline. Reliable and accurate processing of spine images is expected to benefit clinical decision support systems for diagnosis, surgery planning, and population-based analysis of spine and bone health. However, designing automated algorithms for spine processing is challenging predominantly due to considerable variations in anatomy and acquisition protocols and due to a severe shortage of publicly available data. Addressing these limitations, the Large Scale Vertebrae Segmentation Challenge (VerSe) was organised in conjunction with the International Conference on Medical Image Computing and Computer Assisted Intervention (MICCAI) in 2019 and 2020, with a call for algorithms tackling the labelling and segmentation of vertebrae. Two datasets containing a total of 374 multi-detector CT scans from 355 patients were prepared and 4505 vertebrae have individually been annotated at voxel level by a human-machine hybrid algorithm (https://osf.io/nqjyw/, https://osf.io/t98fz/). A total of 25 algorithms were benchmarked on these datasets. In this work, we present the results of this evaluation and further investigate the performance variation at the vertebra level, scan level, and different fields of view. We also evaluate the generalisability of the approaches to an implicit domain shift in data by evaluating the top-performing algorithms of one challenge iteration on data from the other iteration. The principal takeaway from VerSe: the performance of an algorithm in labelling and segmenting a spine scan hinges on its ability to correctly identify vertebrae in cases of rare anatomical variations. The VerSe content and code can be accessed at: https://github.com/anjany/verse.

Mobile telecommunication systems establish a large number of communication links with a limited number of available frequencies; reuse of the same or adjacent frequencies on neighboring links causes interference. The task to find an assignment of frequencies to channels with minimal interference is the frequency assignment problem. The frequency assignment problem is usually treated as a graph coloring problem where the number of colors is minimized, but this approach does not model interference minimization correctly. We give in this paper a new integer programming formulation of the frequency assignment problem, the orientation model, and develop a heuristic two-stage method to solve it. The algorithm iteratively solves an outer and an inner optimization problem. The outer problem decides for each pair of communication links which link gets the higher frequency and leads to an acyclic subdigraph problem with additional longest path restrictions. The inner problem to find an optimal assignment respecting an orientation leads to a min-cost flow problem.

We present a graph-theoretic model for the \emph{frequency assignment problem} in Cellular Phone Networks: Obeying several technical and legal restrictions, frequencies have to be assigned to transceivers so that interference is as small as possible. This optimization problem is NP-hard. Good approximation cannot be guaranteed, unless P = NP. We describe several assignment heuristics. These heuristics are simple and not too hard to implement. We give an assessment of the heuristics' efficiency and practical usefulness. For this purpose, typical instances of frequency assignment problems with up to 4240 transceivers and 75 frequencies of a German cellular phone network operator are used. The results are satisfying from a practitioner's point of view. The best performing heuristics were integrated into a network planning system used in practice.