2-D surfaces in 3-D images can be represented as graphs, using differential geometric methods. Why is this interesting, and how could these graphs be calculated? The talk will address (not necessarily answer) these questions and supply details about the graph calculation, using geometric differential methods. In this context, the graphs are built from the set of umbilics, ridge lines and "purple flyovers".
The following concepts are used to calculate and describe the graphs.
Principal direction: Direction in which the curvature (as a function
of the angle) is extremal. Generically, exactly two such directions
The "blue" principal direction: Direction in which the curvature (as a function of the angle) is maximal.
The "red" principal direction: Direction in which the curvature (as a function of the angle) is minimal.
Principal curvature: Local curvature in the principal direction.
Umbilic: A point at which the local curvature of the surface is independent of the direction, i.e. no principal directions exist.
Ridge point: A point where the curvature - as a function of the arc length - is locally extremal along one of the principal directions.
Ridge line: A collection of neighbouring ridge points.
Purple flyover: A "double" ridge point, i.e. the curvature is extremal along both principal directions. Thus, purple flyovers represent intersections between ridge lines.
Supervisors: Adjunkt Ole Fogh Olsen, ITU and Professor Peter Johansen, DIKU
External examiner: Professor Peter Giblin, University of Liverpool.