发表机构
University of Sydney（悉尼大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究提出融入数学先验的M-Net，在LiTS、KiTS、BraTS数据集上提升U-Net分割性能，证实数学归纳偏置可增强医学图像分割效果。
AI 中文摘要
目的：基于深度学习的医学图像分割已取得显著成功，但纯数据驱动方法往往无法利用医学图像固有的丰富数学结构。我们研究显式数学归纳偏置，具体为矩阵谱分析和向量微积分算子，是否能在纯数据驱动学习之外提升分割性能。方法：我们提出M-Net（数学增强网络），将三种互补数学先验融入U-Net：（1）由中心化局部像素矩阵的条件数导出的连续谱特征，提供纹理病态性的可微度量；（2）从图像梯度场计算的物理场算子（散度及类离散旋度的边界不规则性算子），捕捉焦点强度极值与边缘非光滑性；（3）数学注意力门（MAG），在跳跃连接处自适应融合数学特征与CNN提取的深度特征。结果：在三个基准数据集（LiTS、KiTS、BraTS）上的实验显示，M-Net在肝脏、肾脏、脑肿瘤分割任务中分别取得78.42%、76.15%、83.67%的Dice分数，较基线U-Net分别提升12.37%、3.52%、5.55%。 ablation实验表明，条件数特征较二元可逆性特征贡献2.14%的增益，而MAG较简单拼接额外提升1.45%。结论：M-Net证实数学归纳偏置为医学图像分割提供有效互补信息，连续条件数特征较离散替代方案提供更优梯度信息，MAG在整个网络中保留这些先验。本研究为将线性代数与向量微积分融入医学成像深度架构开辟了途径。
英文摘要
Purpose: Deep learning-based medical image segmentation has achieved remarkable success, yet purely data-driven approaches often fail to exploit the rich mathematical structure inherent in medical images. We investigate whether explicit mathematical inductive biases, specifically matrix spectral analysis and vector calculus operators, can enhance segmentation beyond data-driven learning alone. Methods: We propose M-Net (Math-Augmented Network), which integrates three complementary mathematical priors into U-Net: (1) continuous spectral features derived from the condition number of centered local pixel matrices, providing a differentiable measure of texture ill-conditioning; (2) physical field operators (divergence and a discrete curl-like boundary irregularity operator) computed from image gradient fields, capturing focal intensity extrema and edge non-smoothness; and (3) a Math-Attention Gate (MAG) that adaptively fuses mathematical features with CNN-extracted deep features at skip connections. Results: Experiments on three benchmarks (LiTS, KiTS, and BraTS) show that M-Net achieves Dice scores of 78.42%, 76.15%, and 83.67%, outperforming baseline U-Net by 12.37%, 3.52%, and 5.55% on liver, kidney, and brain tumor segmentation, respectively. Ablations reveal that the condition-number feature contributes a 2.14% gain over binary invertibility features, while MAG adds 1.45% over simple concatenation. Conclusion: M-Net establishes that mathematical inductive biases provide effective complementary information for medical image segmentation. The continuous condition-number feature offers superior gradient information over discrete alternatives, and MAG preserves these priors throughout the network. This work opens avenues for integrating linear algebra and vector calculus into deep architectures for medical imaging.
