ExTreeM: Scalable Augmented Merge Tree Computation via Extremum Graphs

Over the last decade merge trees have been proven to support a plethora of visualization and analysis tasks since they effectively abstract complex datasets. This paper describes the ExTreeM-Algorithm: A scalable algorithm for the computation of merge trees via extremum graphs. The core idea of ExTreeM is to first derive the extremum graph <inline-formula><tex-math notation="LaTeX">$\mathcal{G}$</tex-math><alternatives><inline-graphic xlink:href="tvcg-lukasczyk-3326526-eqinline-1-small.tif"/></alternatives></inline-formula> of an input scalar field <inline-formula><tex-math notation="LaTeX">$f$</tex-math><alternatives><inline-graphic xlink:href="tvcg-lukasczyk-3326526-eqinline-2-small.tif"/></alternatives></inline-formula> defined on a cell complex <inline-formula><tex-math notation="LaTeX">$\mathcal{K}$</tex-math><alternatives><inline-graphic xlink:href="tvcg-lukasczyk-3326526-eqinline-3-small.tif"/></alternatives></inline-formula>, and subsequently compute the unaugmented merge tree of <inline-formula><tex-math notation="LaTeX">$f$</tex-math><alternatives><inline-graphic xlink:href="tvcg-lukasczyk-3326526-eqinline-4-small.tif"/></alternatives></inline-formula> on <inline-formula><tex-math notation="LaTeX">$\mathcal{G}$</tex-math><alternatives><inline-graphic xlink:href="tvcg-lukasczyk-3326526-eqinline-5-small.tif"/></alternatives></inline-formula> instead of <inline-formula><tex-math notation="LaTeX">$\mathcal{K}$</tex-math><alternatives><inline-graphic xlink:href="tvcg-lukasczyk-3326526-eqinline-6-small.tif"/></alternatives></inline-formula>; which are equivalent. Any merge tree algorithm can be carried out significantly faster on <inline-formula><tex-math notation="LaTeX">$\mathcal{G}$</tex-math><alternatives><inline-graphic xlink:href="tvcg-lukasczyk-3326526-eqinline-7-small.tif"/></alternatives></inline-formula>, since <inline-formula><tex-math notation="LaTeX">$\mathcal{K}$</tex-math><alternatives><inline-graphic xlink:href="tvcg-lukasczyk-3326526-eqinline-8-small.tif"/></alternatives></inline-formula> in general contains substantially more cells than <inline-formula><tex-math notation="LaTeX">$\mathcal{G}$</tex-math><alternatives><inline-graphic xlink:href="tvcg-lukasczyk-3326526-eqinline-9-small.tif"/></alternatives></inline-formula>. To further speed up computation, ExTreeM includes a tailored procedure to derive merge trees of extremum graphs. The computation of the fully augmented merge tree, i.e., a merge tree domain segmentation of <inline-formula><tex-math notation="LaTeX">$\mathcal{K}$</tex-math><alternatives><inline-graphic xlink:href="tvcg-lukasczyk-3326526-eqinline-10-small.tif"/></alternatives></inline-formula>, can then be performed in an optional post-processing step. All steps of ExTreeM consist of procedures with high parallel efficiency, and we provide a formal proof of its correctness. Our experiments, performed on publicly available datasets, report a speedup of up to one order of magnitude over the state-of-the-art algorithms included in the TTK and VTK-m software libraries, while also requiring significantly less memory and exhibiting excellent scaling behavior.