Mathematical morphology (MM) is a powerful and widely used framework in image processing. Through set-theoretic and discrete geometric principles, MM operations such as erosion, dilation, opening, and closing effectively manipulate digital images by modifying local structures via structuring elements (SEs), while cubical homology captures global topological features such as connected components and loop structures within images. Building on the GUDHI package for persistent homology (PH) computation on cubical complexes, we propose the MMPersistence library, which integrates MM operations with diverse SEs and PH computation to extract multiscale persistence information. By employing SEs of different shapes to construct topological filtrations, the proposed MM-based PH framework encodes both spatial and morphological characteristics of digital images, providing richer local geometric information than conventional cubical homology alone and establishing a unified foundation for analyzing digital images that integrates topological insight with morphological image processing techniques.

翻译：数学形态学（MM）是图像处理中一个强大且广泛应用的框架。通过集合论和离散几何原理，诸如腐蚀、膨胀、开运算和闭运算等MM操作，借助结构元素（SEs）修改局部结构，从而有效处理数字图像；而立方同调则捕捉图像内部的全局拓扑特征，如连通分量和环状结构。基于GUDHI包在立方复形上进行持久同调（PH）计算的基础，我们提出了MMPersistence库，该库将MM操作与多样化的SEs及PH计算相结合，以提取多尺度持久性信息。通过采用不同形状的SEs构建拓扑过滤，所提出的基于MM的PH框架编码了数字图像的空间与形态特征，提供了比传统立方同调更丰富的局部几何信息，并为分析数字图像建立了一个统一的基础，该基础将拓扑洞察与形态学图像处理技术相结合。