用黏性解理论稳定图像重建，提升医学影像分类效果。
A Viscosity Semigroup Framework for Stable Image Reconstruction
- 基于黏性解框架构建多尺度图像表示，保证解的唯一性和稳定性。
- 在肺癌分类任务中实现0.875 AUC，训练过程无明显波动。
- 适合需要高稳定性的医学图像重建场景，尤其适用于低信噪比数据。
从尺度空间理论的公理化出发，我们提出一种针对退化椭圆-抛物型偏微分方程生成的多尺度图像表示的黏性解框架。不引入新半群理论，而是在标准黏性解设定下，利用比较原理获得一致、唯一性和上确界范数下的压缩性。该视角启发了一种混合重建算子：先通过学习的逆映射，再经非线性扩散演化。连续层面的扩散算子满足非扩张性，从而保障重建过程的稳定性。该框架在基于CT的间皮瘤分类任务中表现优异，达到0.875 AUC，且各轮次间变化极小；相比之下，基线模型的AUC在0.49至0.80之间波动，无明确收敛趋势。这些结果与黏性理论所预测的稳定作用一致。
原文摘要 · Abstract (English)
Starting from the axiomatic formulation of scale-space theory, we develop a viscosity-solution framework for multiscale image representations arising from degenerate elliptic-parabolic partial differential equations. Rather than introducing a new semigroup theory, we work within the standard viscosity-solution setting, using comparison principles to obtain well-posedness, uniqueness, and contraction in the supremum norm. This perspective is used to motivate a hybrid reconstruction operator in which a learned inverse map is followed by a nonlinear diffusion evolution. At the continuous level, the diffusion operator satisfies non-expansiveness, which provides stability for the reconstruction process; this framework is then evaluated on a CT-based mesothelioma classification task, where it attains an AUC of 0.875 with negligible variation across epochs, while the baseline model acquires AUC values from 0.49 to 0.80 without a clear convergence pattern. These observations are consistent with the stabilizing role suggested by the discussed viscosity theory.
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