Persistent homology is perhaps the most popular and useful tool offered by topological data analysis, with point-cloud data being the most common setup. Its older cousin, the Euler characteristic curve (ECC) is less expressive, but far easier to compute. It is particularly suitable for analyzing imaging data, and is commonly used in fields ranging from astrophysics to biomedical image analysis. These fields are embracing GPU computations to handle increasingly large datasets. We therefore propose an optimized GPU implementation of ECC computation for 2D and 3D grayscale images. The goal of this paper is twofold. First, we offer a practical tool, illustrating its performance with thorough experimentation, but also explain its inherent shortcomings. Second, this simple algorithm serves as a perfect backdrop for highlighting basic GPU programming techniques that make our implementation so efficient, and some common pitfalls we avoided. This is intended as a step towards a wider usage of GPU programming in computational geometry and topology software. We find this is particularly important as geometric and topological tools are used in conjunction with modern, GPU-accelerated machine learning frameworks.

翻译：常态同族体也许是由地形数据分析提供的最受欢迎和最有用的工具, 点球数据是最常用的设置。 它的老表弟 Euler 特征曲线(ECC) 不太直观, 但比较容易计算。 它特别适合分析成像数据, 并且通常用于从天体物理学到生物医学图像分析等各个领域。 这些领域包含 GPU 计算, 以处理越来越庞大的数据集。 因此, 我们提议对 2D 和 3D 灰度图像优化 GPU 计算 ECC 。 本文的目标是双重的。 首先, 我们提供了一个实用工具, 用彻底的实验来说明其性能, 同时解释其内在缺陷。 其次, 这个简单算法是突出基本 GPU 编程技术的完美背景, 使得我们的实施效率很高, 以及我们避免的一些常见的陷阱。 这是朝着在计算几何和地形软件中更广泛地使用 GPU 程序迈出的一步。 我们发现这一点特别重要, 因为测量和地形工具是结合现代的, GPUPU- 加速机器学习框架一起使用的。</s>