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Abstract

Crease surfaces are two-dimensional manifolds along which a scalar field
assumes a local maximum (ridge) or a local minimum (valley) in a constrained
space. Unlike isosurfaces, they are able to capture extremal structures in the
data. Creases have a long tradition in image processing and computer vision,
and have recently become a popular tool for visualization. When extracting
crease surfaces, degeneracies of the Hessian (i.e., lines along which two
eigenvalues are equal) have so far been ignored. We show that these loci,
however, have two important consequences for the topology of crease surfaces:
First, creases are bounded not only by a side constraint on eigenvalue sign,
but also by Hessian degeneracies. Second, crease surfaces are not, in general,
orientable. We describe an efficient algorithm for the extraction of crease
surfaces which takes these insights into account and demonstrate that it
produces more accurate results than previous approaches. Finally, we show that
diffusion tensor magnetic resonance imaging (DT-MRI) stream surfaces, which
were previously used for the analysis of planar regions in diffusion tensor MRI
data, are mathematically ill-defined. As an example application of our method,
creases in a measure of planarity are presented as a viable substitute.