Topological integral transforms have found many applications in shape analysis, from prediction of clinical outcomes in brain cancer to analysis of barley seeds. Using Euler characteristic as a measure, these objects record rich geometric information on weighted polytopal complexes. While some implementations exist, they only enable discretized representations of the transforms, and they do not handle weighted complexes (such as for instance images). Moreover, recent hybrid transforms lack an implementation. In this paper, we introduce Eucalc, a novel implementation of three topological integral transforms -- the Euler characteristic transform, the Radon transform, and hybrid transforms -- for weighted cubical complexes. Leveraging piecewise linear Morse theory and Euler calculus, the algorithms significantly reduce computational complexity by focusing on critical points. Our software provides exact representations of transforms, handles both binary and grayscale images, and supports multi-core processing. It is publicly available as a C++ library with a Python wrapper. We present mathematical foundations, implementation details, and experimental evaluations, demonstrating Eucalc's efficiency.

翻译：拓扑积分变换在形状分析中具有广泛应用，从脑癌临床结果预测到大麦种子分析。以欧拉示性数为测度，这些变换记录了加权多面体复形上的丰富几何信息。尽管已有部分实现方案，但现有方法仅能获取变换的离散化表示，且无法处理加权复形（例如图像）。此外，近年提出的混合变换仍缺乏实现。本文提出Eucalc——一种面向加权立方复形的三种拓扑积分变换（欧拉示性数变换、拉东变换及混合变换）的新型实现方法。通过利用分段线性莫尔斯理论与欧拉积分的特性，该算法聚焦于临界点，显著降低了计算复杂度。我们的软件可提供变换的精确表示，支持二值图与灰度图处理，并具备多核运算能力。该工具以C++库形式公开并提供Python封装。本文阐述了其数学基础、实现细节与实验评估，验证了Eucalc的高效性。