Building practical filtrations on objects to detect topological and geometric features is an important task in the field of Topological Data Analysis (TDA). In this paper, leveraging the ability of the Laplacian of Gaussian operator to enhance the boundaries of medical images, we define the G-LoG (Gaussian-Laplacian of Gaussian) bi-filtration to generate the features more suitable for multi-parameter persistence module. By modeling volumetric images as bounded functions, then we prove the interleaving distance on the persistence modules obtained from our bi-filtrations on the bounded functions is stable with respect to the maximum norm of the bounded functions. Finally, we conduct experiments on the MedMNIST dataset, comparing our bi-filtration against single-parameter filtration and the established deep learning baselines, including Google AutoML Vision, ResNet, AutoKeras and auto-sklearn. Experiments results demonstrate that our bi-filtration significantly outperforms single-parameter filtration. Notably, a simple Multi-Layer Perceptron (MLP) trained on the topological features generated by our bi-filtration achieves performance comparable to complex deep learning models trained on the original dataset.

翻译：在拓扑数据分析领域，构建对象上的实用滤过结构以检测拓扑与几何特征是一项重要任务。本文利用高斯-拉普拉斯算子增强医学图像边界的能力，定义了G-LoG（高斯-高斯-拉普拉斯）双滤过，以生成更适合多参数持久性模块的特征。通过将体数据图像建模为有界函数，我们证明了从有界函数上的双滤过获得的持久性模块之间的交错距离，相对于有界函数的极大范数是稳定的。最后，我们在MedMNIST数据集上进行实验，将我们的双滤过与单参数滤过以及已建立的深度学习基线（包括Google AutoML Vision、ResNet、AutoKeras和auto-sklearn）进行比较。实验结果表明，我们的双滤过显著优于单参数滤过。值得注意的是，基于我们双滤过生成的拓扑特征训练的简单多层感知机，其性能可与在原始数据集上训练的复杂深度学习模型相媲美。