The deep structure of an image is investigated, and a Multi-Scale Singularity Tree (MSST) is constructed based on the pair-wise annihilations of critical points. This report contains two main contributions. Firstly, we describe a fast, simple, and robust method of extracting feature lines from data sets of up to four dimen- sions, which we apply in order to extract critical paths from scale-spaces. Secondly, we investigate the extracting of MSSTs using either support regions or extrema partitions. Given an image, both methods produce a binary tree that mathematically represents the topological structures of the image. The software described in this report is available through the EU-project, Deep Structure, Singularities, and Computer Vision.