The Euler characteristic (EC) is a powerful topological descriptor that can be used to quantify the shape of data objects that are represented as fields/manifolds. Fast methods for computing the EC are required to enable processing of high-throughput data and real-time implementations. This represents a challenge when processing high-resolution 2D field data (e.g., images) and 3D field data (e.g., video, hyperspectral images, and space-time data obtained from fluid dynamics and molecular simulations). In this work, we present parallel algorithms (and software implementations) to enable fast computations of the EC for 2D and 3D fields using vertex contributions. We test the proposed algorithms using synthetic data objects and data objects arising in real applications such as microscopy, 3D molecular dynamics simulations, and hyperspectral images. Results show that the proposed implementation can compute the EC a couple of orders of magnitude faster than ${\tt GUDHI}$ (an off-the-shelf and state-of-the art tool) and at speeds comparable to ${\tt CHUNKYEuler}$ (a tool tailored to scalable computation of the EC). The vertex contributions approach is flexible in that it compute the EC as well as other topological descriptors such as perimeter, area, and volume (${\tt CHUNKYEuler}$ can only compute the EC). Scalability with respect to memory use is also addressed by providing low-memory versions of the algorithms; this enables processing of data objects beyond the size of dynamic memory. All data and software needed for reproducing the results are shared as open-source code.

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