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Surface Evolution Under Curvature Flows
In many areas of computer vision, such as multiscale analysis and shape description, an image or surface is smoothed by a nonlinear parabolic partial differential equation to eliminate noise and to reveal the large global features. An ideal flow, or smoothing process, should not create new features. In this paper we describe in detail the effect of a number of flows on surfaces on the parabolic curves, the ridge curves, and umbilic points. In particular we look at the mean curvature flow and the two principal curvature flows. Our calculations show that two principal curvature flows never create parabolic and ridge curves of the same type as the flow, but no flow is found capable of simultaneously smoothing out all features. In fact, we find that the principal curvature flows in some cases create a highly degenerate type of umbilic. We illustrate the effect of these flows by an example of a 3-D face evolving under principal curvature flows.Mathematic
Geometry of isophote curves
In this paper, we consider the intensity surface of a 2D image, we study the
evolution of the symmetry sets (and medial axes) of 1-parameter families of
iso-intensity curves. This extends the investigation done on 1-parameter
families of smooth plane curves (Bruce and Giblin, Giblin and Kimia, etc.) to
the general case when the family of curves includes a singular member, as will
happen if the curves are obtained by taking plane sections of a smooth surface,
at the moment when the plane becomes tangent to the surface.
Looking at those surface sections as isophote curves, of the pixel values of
an image embedded in the real plane, this allows us to propose to combine
object representation using a skeleton or symmetry set representation and the
appearance modelling by representing image information as a collection of
medial representations for the level-sets of an image.Comment: 15 pages, 7 figure
Curve diffusion and straightening flows on parallel lines
In this paper, we study families of immersed curves
with free boundary supported
on parallel lines
evolving by the curve diffusion flow and the curve straightening flow. The
evolving curves are orthogonal to the boundary and satisfy a no-flux condition.
We give estimates and monotonicity on the normalised oscillation of curvature,
yielding global results for the flows.Comment: 35 pages, 3 figure
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