Implementation and complexity of the watershed-from-markers algorithm computed as a minimal cost forest

The watershed algorithm belongs to classical algorithms in mathematical morphology. Lotufo et al. published a principle of the watershed computation by means of an Image Foresting Transform (IFT), which computes a shortest path forest from given markers. The algorithm itself was described for a 2D case (image) without a detailed discussion of its computation and memory demands for real datasets. As IFT cleverly solves the problem of plateaus and as it gives precise results when thin objects have to be segmented, it is obvious to use this algorithm for 3D datasets taking in mind the minimizing of a higher memory consumption for the 3D case without loosing low asymptotical time complexity of O(m+C) (and also the real computation speed). The main goal of this paper is an implementation of the IFT algorithm with a priority queue with buckets and careful tuning of this implementation to reach as minimal memory consumption as possible. The paper presents five possible modifications and methods of implementation of the IFT algorithm. All presented implementations keep the time complexity of the standard priority queue with buckets but the best one minimizes the costly memory allocation and needs only 19-45% of memory for typical 3D medical imaging datasets. Memory saving was reached by an IFT algorithm simplification, which stores more elements in temporary structures but these elements are simpler and thus need less memory. The best presented modification allows segmentation of large 3D medical datasets (up to 512x512x680 voxels) with 12-or 16-bits per voxel on currently available PC based workstations.Comment: v1: 10 pages, 6 figures, 7 tables EUROGRAPHICS conference, Manchester, UK, 2001. v2: 12 pages, reformated for letter, corrected IFT to "Image Foresting Tranform

Insights into human genetic variation and population history from 929 diverse genomes.

Genome sequences from diverse human groups are needed to understand the structure of genetic variation in our species and the history of, and relationships between, different populations. We present 929 high-coverage genome sequences from 54 diverse human populations, 26 of which are physically phased using linked-read sequencing. Analyses of these genomes reveal an excess of previously undocumented common genetic variation private to southern Africa, central Africa, Oceania, and the Americas, but an absence of such variants fixed between major geographical regions. We also find deep and gradual population separations within Africa, contrasting population size histories between hunter-gatherer and agriculturalist groups in the past 10,000 years, and a contrast between single Neanderthal but multiple Denisovan source populations contributing to present-day human populations.Wellcome grants 098051 and 206194, and S.A.M. and R.D. also by Wellcome grant 207492. A.B. and P.S. were supported by the Francis Crick Institute (FC001595) which receives its core funding from Cancer Research UK, the UK Medical Research Council and the Wellcome Trust. P.S. was also supported by the European Research Council (grant no. 852558) and the Wellcome Trust (217223/Z/19/Z). R.H. was supported by a Gates Cambridge scholarship. P.H. was supported by Estonian Research Council Grant PUT1036. D.R. is an Investigator of the Howard Hughes Medical Institute