发表机构
Institute of Technology, University of Tartu（塔尔图大学技术学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文为量子启发式无监督分割的谱表示建立了噪声稳定性与旋转等变性的理论保证，推导了有限样本扰动界并验证了其有效性。
AI 中文摘要
无监督分割方法将图像嵌入为薛定谔型哈密顿量的势能，并从其低能本征态中读取对象结构，在自然图像和体积图像上表现出强大的经验性能，然而这种谱表示对传感器噪声和几何变换的鲁棒性从未在理论上得到确立。本文填补了这一空白。采用离散图哈密顿公式，与现有的量子启发分割框架一致，并推导出显式的有限样本扰动界，该界将加性噪声（已知方差）下特征值和基态特征向量的漂移与算子的谱间隙联系起来，将最初为半经典信号分析开发的一维噪声误差分析扩展到用于图像分割的二维图设置。特征向量级界进一步传播到诱导的二值分割掩码，首次给出了噪声下分割稳定性的定量保证，而非经验观察。补充的等变性分析刻画了算子谱在平面刚体运动群下精确变换的条件，并量化了实践中常用的各向异性势项引入的等变性差距。理论预测在自然图像和噪声扰动基准上进行了数值验证，显示推导的界与观察到的谱和分割掩码偏差之间高度一致，为实践者在已知噪声条件下选择算子参数提供了原则性准则。
英文摘要
Unsupervised segmentation methods that embed an image as the potential of a Schrödinger-type Hamiltonian and read out object structure from its low-lying eigenstates have demonstrated strong empirical performance across natural and volumetric imagery, yet the robustness of this spectral representation to sensor noise and geometric transformation has never been established theoretically. This paper closes that gap. A discrete graph-Hamiltonian formulation is adopted, consistent with existing quantum-inspired segmentation frameworks, and an explicit finite-sample perturbation bound is derived that links the drift of its eigenvalues and ground-state eigenvector under additive noise of known variance to the operator's spectral gap, extending a one-dimensional noise-error analysis originally developed for semiclassical signal analysis to the two-dimensional graph setting used in image segmentation. The eigenvector-level bound is further propagated to the induced binary segmentation mask, giving, for the first time, a quantitative guarantee on segmentation stability under noise rather than an empirical observation of it. A complementary equivariance analysis characterizes the conditions under which the operator's spectrum transforms exactly under the planar rigid-motion group, and quantifies the equivariance gap introduced by anisotropic potential terms commonly used in practice. Theoretical predictions are validated numerically on natural-image and noise-perturbed benchmarks, showing close agreement between the derived bounds and observed spectral and segmentation-mask deviation, and offering practitioners a principled criterion for selecting operator parameters under known noise conditions.
