Iso-surface extraction in nD applied to tracking feature curves across scale

A method is presented to extract space curves, defined by differential invariants, at increasing scales. The curves are considered as the intersection of two iso-surfaces in 3D, so their moving paths or orbits can be explicitly obtained in scale space as the intersection of two isosurface in 4D. This method is based on a novel algorithm to search for iso-surfaces and their intersections in nD. The algorithm is a significant extension of the 3D Marching Lines algorithm with new orientation and implementation considerations. As a result of these considerations, the reconstructed iso-surfaces and their intersections can be proved to have good topological properties; moreover, the implementation is quite straightforward. Specifically, a 4D extension has been implemented to follow spatial curves efficiently via scale. The algorithm automatically finds the connection order of singularities, so tracking remains reliable even if scale is not densely sampled. As an example, the development of parabolic and crest curves across scale is visualized.