## Specialeforsvar

# A graph description of 2-D surfaces in 3-D images by differential geometric properties

### Bo Brandt

### Tuesday, January 15 2002, 16:00 in Lille Auditorium, DIKU

# Abstract

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
exist:

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.