Three‐dimensional reconstruction of seed implants by randomized rounding and visual evaluation

Abstract

The development of efficient 3D seed reconstruction algorithms is an ongoing and vivid research topic. Since the 1980s many publications about seed assignment were published. In this paper a novel mathematical approach is described to solve the 3D assignment problem for the reconstruction of seeds with radiographs: we present a fast linear programming approach together with afterwards applying the so-called randomized rounding scheme to compute good (possibly partial) assignments. We apply a visualization software that allows user interaction to check the solution given by the algorithm and to augment partial assignments. The second step is justified as the randomized algorithm already returns optimal solutions is many cases, and in cases with partial assignments it fails to match only a very small number of seed images. Our algorithm transfers ideas from recent breakthrough research work on the design of efficient randomized algorithms in discrete optimization and computer science to the seed reconstruction problem.