证明了图像噪声下Wasserstein距离误差随噪声标准差平方根增长，优于欧氏距离。
Quantifying the noise sensitivity of the Wasserstein metric for images
- 推导高斯噪声下有限样本的期望误差界
- 发现有符号2-Wasserstein偏差误差与噪声标准差平方根成正比
- 适合关注图像相似性度量在噪声中鲁棒性的研究者
Wasserstein度量被越来越多地用作图像相似性度量。本文研究当图像被视为像素网格上的离散测度时，其对像素级加性噪声的敏感性。针对高斯噪声模型，推导了有限样本的期望误差上界。结果表明，有符号2-Wasserstein偏差的误差与噪声标准偏差的平方根成正比，优于欧氏距离的线性增长，为最优传输距离在噪声环境中的优势提供了理论依据。实验验证了理论结果，并揭示了一个奇特现象：增加噪声水平反而降低Wasserstein距离。对冷冻电镜图像的案例研究显示，在高噪声条件下，Wasserstein度量仍能捕捉数据流形的几何结构，而欧氏度量已失效。
原文摘要 · Abstract (English)
Wasserstein metrics are increasingly adopted as similarity scores for images. We consider the sensitivity of Wasserstein metrics with respect to pixel-wise additive noise when the images are treated as discrete measures on the pixel grid. We derive finite-sample expectation bounds for a Gaussian noise model. Among other results, we prove that the error in the signed 2-Wasserstein discrepancy scales with the square root of the noise standard deviation. This is favorable compared to the Euclidean metric that scales linearly, and thus provides a theoretical basis for the benefits of optimal transport distances in noisy settings. We present experiments that support our theoretical findings and point to a peculiar phenomenon where increasing the level of noise can decrease the Wasserstein distance. A case study on cryo-electron microscopy images demonstrates that the Wasserstein metric can capture the geometry of the data manifold in high noise settings even when the Euclidean metric fails.
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