Eli Lilly and Company, Indianapolis, IN, 46225, USA.
Department of Mathematics, William & Mary, Williamsburg, VA, 23185, USA.
Sci Rep. 2022 Apr 19;12(1):6427. doi: 10.1038/s41598-022-09464-7.
The field of mathematical morphology offers well-studied techniques for image processing and is applicable for studies ranging from materials science to ecological pattern formation. In this work, we view morphological operations through the lens of persistent homology, a tool at the heart of the field of topological data analysis. We demonstrate that morphological operations naturally form a multiparameter filtration and that persistent homology can then be used to extract information about both topology and geometry in the images as well as to automate methods for optimizing the study and rendering of structure in images. For illustration, we develop an automated approach that utilizes this framework to denoise binary, grayscale, and color images with salt and pepper and larger spatial scale noise. We measure our example unsupervised denoising approach to state-of-the-art supervised, deep learning methods to show that our results are comparable.
数学形态学领域提供了经过充分研究的图像处理技术，适用于从材料科学到生态模式形成等各种研究。在这项工作中，我们通过拓扑数据分析领域核心工具——持久同调的视角来看待形态操作。我们证明了形态操作自然地形成了一个多参数过滤，然后可以使用持久同调来提取图像中的拓扑和几何信息，以及自动化方法来优化图像中结构的研究和呈现。为了说明这一点，我们开发了一种自动化方法，利用该框架对椒盐噪声和更大空间尺度噪声的二进制、灰度和彩色图像进行去噪。我们将我们的无监督去噪方法与最先进的有监督、深度学习方法进行了比较，以表明我们的结果是可比的。
