研究图像尺度空间度量的稳定性，提升对形变和噪声的鲁棒性。
On the stability of scale-space metrics
- 基于高斯尺度空间构建度量，分析其对几何变形的稳定性。
- 提出旋转不变版本，有效应对断层投影角度变化的影响。
- 算法高效且抗加性噪声，适用于实际图像比较任务。
我们研究了一类经典度量在函数高斯尺度空间表示上的稳定性，重点关注二维图像（双变量函数）的比较问题。这些度量在调和分析（特别是Besov空间理论）和传统图像处理方法中均有先例，部分情形与特定Wasserstein距离度量等价。本文量化了这些度量对几何形变的鲁棒性，并引入旋转不变版本，使其在比较断层投影时对角度变化保持稳定。同时，我们提出了从有限样本中高效计算度量的算法，并证明其对抗加性噪声的稳健性。数值实验验证了理论结果。
原文摘要 · Abstract (English)
We study the stability of a classical family of metrics defined over functions' Gaussian scale-space representations, focusing on the comparison of images (functions of two variables). These metrics have precedents both in harmonic analysis, specifically the theory of Besov spaces, and in classical methods of image processing; special cases are also known to be metrically equivalent to certain Wasserstein distances. We quantify these metrics' robustness to geometric deformations, and introduce rotationally-invariant versions that are stable to changes in angle when comparing tomographic projections. We also describe computationally efficient algorithms for evaluating the metrics from finite samples, and prove their robustness to additive noise. The results are illustrated through numerical experiments.
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