论文标题
数字图像的价值偏移分叉
Value-Offset Bifiltrations for Digital Images
论文作者
论文摘要
持久性同源性是一种抽象数据中辨别结构的代数方法，依赖于构造一系列被称为过滤的嵌套拓扑空间。两参数持续的同源性允许分析通过两个参数过滤的数据，但需要分叉 - 一系列拓扑空间序列同时由两个参数索引。要将两参数持久性应用于数字图像，我们首先必须考虑通过数字图像构建的分叉，而数字图像几乎没有被研究。我们介绍了灰度数字图像数据的价值偏移分叉。我们提出了有效的算法，用于计算相对于出租车距离的该分叉，并相对于欧几里得距离进行近似。我们分析了算法的运行时复杂性，在样本图像上演示了结果，并将从真实图像获得的分叉与从随机噪声获得的分叉进行了对比。
Persistent homology, an algebraic method for discerning structure in abstract data, relies on the construction of a sequence of nested topological spaces known as a filtration. Two-parameter persistent homology allows the analysis of data simultaneously filtered by two parameters, but requires a bifiltration -- a sequence of topological spaces simultaneously indexed by two parameters. To apply two-parameter persistence to digital images, we first must consider bifiltrations constructed from digital images, which have scarcely been studied. We introduce the value-offset bifiltration for grayscale digital image data. We present efficient algorithms for computing this bifiltration with respect to the taxicab distance and for approximating it with respect to the Euclidean distance. We analyze the runtime complexity of our algorithms, demonstrate the results on sample images, and contrast the bifiltrations obtained from real images with those obtained from random noise.
