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Standard convolutional architectures struggle with complex anatomical structures because they restrict image representations to rigid Euclidean grids, discarding critical geometric and topological relationships. To overcome this, the authors introduce CNMTDL, a manifold topological deep learning framework that models images as discrete manifolds and applies combinatorial complex neural networks over Hodge-decomposed components. Evaluated across six 2D and 3D datasets on MedMNIST v2, the approach shows that attention-based higher-order message passing between 0-cells and 2-cells yields superior representation learning for topologically intricate medical data.
Euclidean grids blind standard vision models to complex anatomical topology, but routing Hodge-decomposed features through combinatorial cell complexes unlocks higher-order geometric reasoning across 2D and 3D images.
Medical image analysis remains fundamentally challenging because of the intricate geometric and topological structures present in medical data. Conventional convolutional neural networks model images as regular Euclidean grids, limiting their ability to preserve geometric relationships and higher-order structural information. Recently, manifold topological deep learning (MTDL) has emerged as a promising paradigm that integrates deep learning with geometric and topological representations. Nevertheless, existing methods have not yet fully exploited discrete manifold structures within combinatorial complex neural networks. To bridge this gap, we introduce CNMTDL, a MTDL framework that integrates Hodge decomposition with a combinatorial attention mechanism. In our approach, medical images are represented as discrete manifolds and decomposed into three Hodge components. Features extracted from these components are concatenated and embedded into a combinatorial complex architecture, enabling enhanced higher-order message passing between $0$-cells and $2$-cells through attention-based blocks. We evaluate CNMTDL on six two-dimensional and three-dimensional datasets from the MedMNIST v2 benchmark, demonstrating its effectiveness for medical image analysis.