Computational geometry and topology are areas which have much potential for the analysis of arbitrarily high-dimensional data sets. In order to apply geometric or topological methods one must first generate a representative point cloud data set from the original data source, or at least a metric or distance function, which defines a distance between the elements of a given data set. Consequently, the first question is: How to get point cloud data sets? Or more precise: What is the optimal way of generating such data sets? The solution to these questions is not trivial. If a natural image is taken as an example, we are concerned more with the content, with the shape of the relevant data represented by this image than its mere matrix of pixels. Once a point cloud has been generated from a data source, it can be used as input for the application of graph theory and computational topology. In this paper we first describe the case for natural point clouds, i.e. where the data already are represented by points; we then provide some fundamentals of medical images, particularly dermoscopy, confocal laser scanning microscopy, and total-body photography; we describe the use of graph theoretic concepts for image analysis, give some medical background on skin cancer and concentrate on the challenges when dealing with lesion images. We discuss some relevant algorithms, including the watershed algorithm, region splitting (graph cuts), region merging (minimum spanning tree) and finally describe some open problems and future challenges.
Holzinger, A., Malle, B., Bloice, M., Wiltgen, M., Ferri, M., Stanganelli, I., & Hofmann-Wellenhof, R. (2014). On the generation of point cloud data sets: Step one in the knowledge discovery process. Lecture Notes in Computer Science (Including Subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), 8401, 57–80. https://doi.org/10.1007/978-3-662-43968-5_4