Planar Curve Evolution Ron Kimmel Computer Science Department Technion-Israel Institute of Technology Geometric Image Processing Lab

Planar Curves qC(p)={x(p),y(p)}, p [0,1] y x C(0) C(0.1) C(0.2) C(0.4) C(0.7) C(0.95) C(0.9) C(0.8) p C =tangent

Arc-length and Curvature s(p)= | |dp C

Invariant arclength should be 1.Re-parameterization invariant 2.Invariant under the group of transformations Geometric measure Transform

Euclidean arclength qLength is preserved, thus,

Curvature flow qEuclidean geometric heat equation flow Euclidean transform

Curvature flow qTakes any simple curve into a circular point in finite time proportional to the area inside the curve qEmbedding is preserved (embedded curves keep their order along the evolution). Gage-Hamilton Grayson Given any simple planar curve First becomes convex Vanish at a Circular point

Important property qTangential components do not affect the geometry of an evolving curve

Reminder: Equi-affine arclength qArea is preserved, thus re-parameterization invariance

Affine heat equation qSpecial (equi-)affine heat flow Sapiro Given any simple planar curve First becomes convex Vanish at an elliptical point flow Affine transform

Constant flow qOffset curves qLevel sets of distance map qEqual-height contours of the distance transform qEnvelope of all disks of equal radius centered along the curve (Huygens principle)

Constant flow qOffset curves Change in topology Shock Cusp

Area inside C qArea is defined via

So far we defined qConstant flow qCurvature flow qEqui-affine flow We would like to explore evolution properties of measures like curvature, length, and area

For Length Area Curvature

Constant flow ( ) Length Area Curvature The curve vanishes at Riccati eq. Singularity (`shock’) at

Curvature flow ( ) Length Area Curvature The curve vanishes at

Equi-Affine flow ( ) Length Area Curvature

Geodesic active contours Goldenberg, Kimmel, Rivlin, Rudzsky, IEEE T-IP 2001

Tracking in color movies Goldenberg, Kimmel, Rivlin, Rudzsky, IEEE T-IP 2001

From curve to surface evolution qIt’s a bit more than invariant measures…

Surface qA surface, qFor example, in 3D qNormal qArea element qTotal area

Surface evolution qTangential velocity has no influence on the geometry qMean curvature flow, area minimizing

Segmentation in 3D Change in topology Caselles,Kimmel, Sapiro, Sbert, IEEE T-PAMI 97

Conclusions qConstant flow, geometric heat equations uEuclidean uEqui-affine uOther data dependent flows qSurface evolution