In this paper, we introduce a shape descriptor that we call "interior function". This is a Topological Data Analysis (TDA) based descriptor that refines previous descriptors for image analysis. Using this concept, we define subcomplex lacunarity, a new index that quantifies geometric characteristics of necrosis in tumors such as conglomeration. Building on this framework, we propose a set of indices to analyze necrotic morphology and construct a diagram that captures the distinct structural and geometric properties of necrotic regions in tumors. We present an application of this framework in the study of MRIs of Glioblastomas (GB). Using cluster analysis, we identify four distinct subtypes of Glioblastomas that reflect geometric properties of necrotic regions.

翻译：本文提出了一种称为"内部函数"的形状描述子。这是一种基于拓扑数据分析（TDA）的描述子，它改进了先前用于图像分析的描述子。利用这一概念，我们定义了亚复形空隙度——一种量化肿瘤坏死几何特征（如聚集程度）的新指标。基于该框架，我们提出了一套分析坏死形态的指标，并构建了能够捕捉肿瘤坏死区域独特结构和几何特征的图谱。我们将该框架应用于胶质母细胞瘤（GB）的MRI研究，通过聚类分析识别出反映坏死区域几何特征的四种胶质母细胞瘤亚型。