摘要翻译：
本文提出了一种鲁棒有效的时变标量数据拓扑特征跟踪方法。基于持久化图之间相对于Wasserstein度量的最优匹配来跟踪结构。这从根本上依赖于解决所有连续时间步骤的分配问题，这是最优运输的一个特例。我们的方法依赖于两个主要贡献。首先，我们回顾了Kuhn和Munkres的开创性分配算法，该算法特别适用于持久图匹配问题。其次，我们提出了Wasserstein度量的一个扩展，它显著地提高了域嵌入持久对匹配的几何稳定性。我们证明了这种几何提升对提高分配矩阵的稀疏性和计算时间有额外的积极的副作用。全局框架通过计算持久性图和为每两个连续的时间步骤并行地寻找最优匹配来实现粗粒度并行。临界轨迹是通过将随时间连续匹配的持久性对相关联来构造的。在后处理阶段用几何阈值检测合并和分裂事件。在实际数据集上的大量实验表明，我们的匹配方法比开创性的Munkres算法快了一个数量级。此外，与现代近似方法相比，我们的方法提供了有竞争力的运行时间，同时产生了精确的结果。我们通过从各种模拟的时变数据集中提取临界点轨迹来证明我们的全局框架的有效性，并将其与现有的基于相关体重叠的方法进行比较。对噪声和时间分辨率下采样的鲁棒性进行了经验证明。
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英文标题：
《Lifted Wasserstein Matcher for Fast and Robust Topology Tracking》
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作者：
Maxime Soler, M\'elanie Plainchault, Bruno Conche, Julien Tierny
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最新提交年份：
2019
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分类信息：

一级分类：Electrical Engineering and Systems Science        电气工程与系统科学
二级分类：Image and Video Processing        图像和视频处理
分类描述：Theory, algorithms, and architectures for the formation, capture, processing, communication, analysis, and display of images, video, and multidimensional signals in a wide variety of applications. Topics of interest include: mathematical, statistical, and perceptual image and video modeling and representation; linear and nonlinear filtering, de-blurring, enhancement, restoration, and reconstruction from degraded, low-resolution or tomographic data; lossless and lossy compression and coding; segmentation, alignment, and recognition; image rendering, visualization, and printing; computational imaging, including ultrasound, tomographic and magnetic resonance imaging; and image and video analysis, synthesis, storage, search and retrieval.
用于图像、视频和多维信号的形成、捕获、处理、通信、分析和显示的理论、算法和体系结构。感兴趣的主题包括：数学，统计，和感知图像和视频建模和表示；线性和非线性滤波、去模糊、增强、恢复和重建退化、低分辨率或层析数据；无损和有损压缩编码；分割、对齐和识别；图像渲染、可视化和打印；计算成像，包括超声、断层和磁共振成像；以及图像和视频的分析、合成、存储、搜索和检索。
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一级分类：Computer Science        计算机科学
二级分类：Computational Geometry        计算几何
分类描述：Roughly includes material in ACM Subject Classes I.3.5 and F.2.2.
大致包括ACM课程I.3.5和F.2.2中的材料。
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一级分类：Computer Science        计算机科学
二级分类：Computer Vision and Pattern Recognition        计算机视觉与模式识别
分类描述：Covers image processing, computer vision, pattern recognition, and scene understanding. Roughly includes material in ACM Subject Classes I.2.10, I.4, and I.5.
涵盖图像处理、计算机视觉、模式识别和场景理解。大致包括ACM课程I.2.10、I.4和I.5中的材料。
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一级分类：Computer Science        计算机科学
二级分类：Graphics        图形学
分类描述：Covers all aspects of computer graphics. Roughly includes material in all of ACM Subject Class I.3, except that I.3.5 is is likely to have Computational Geometry as the primary subject area.
涵盖了计算机图形学的各个方面。大致包括所有ACM课程I.3的材料，除了I.3.5可能有计算几何作为主要的学科领域。
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英文摘要：
  This paper presents a robust and efficient method for tracking topological features in time-varying scalar data. Structures are tracked based on the optimal matching between persistence diagrams with respect to the Wasserstein metric. This fundamentally relies on solving the assignment problem, a special case of optimal transport, for all consecutive timesteps. Our approach relies on two main contributions. First, we revisit the seminal assignment algorithm by Kuhn and Munkres which we specifically adapt to the problem of matching persistence diagrams in an efficient way. Second, we propose an extension of the Wasserstein metric that significantly improves the geometrical stability of the matching of domain-embedded persistence pairs. We show that this geometrical lifting has the additional positive side-effect of improving the assignment matrix sparsity and therefore computing time. The global framework implements a coarse-grained parallelism by computing persistence diagrams and finding optimal matchings in parallel for every couple of consecutive timesteps. Critical trajectories are constructed by associating successively matched persistence pairs over time. Merging and splitting events are detected with a geometrical threshold in a post-processing stage. Extensive experiments on real-life datasets show that our matching approach is an order of magnitude faster than the seminal Munkres algorithm. Moreover, compared to a modern approximation method, our method provides competitive runtimes while yielding exact results. We demonstrate the utility of our global framework by extracting critical point trajectories from various simulated time-varying datasets and compare it to the existing methods based on associated overlaps of volumes. Robustness to noise and temporal resolution downsampling is empirically demonstrated.
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PDF链接：
https://arxiv.org/pdf/1808.0587