Metrics of curves in shape optimization and analysis - Andrea Carlo ...

2.4.1 Example: geometric heat flowWe first review one of the most mathematically studied gradient descent flows ofcurves. By direct computation, the Gâteaux differential of the length of a closedcurve is∫∫D len(c; h) = ∂ s h · T ds = − h · H ds (2.5)cLet C = C(t, θ) be an evolving family of curves. The geometric heat flow(also known as motion by mean curvature) is∂C∂t = Hwhere H := ∂ s ∂ s C is the mean curvature. In the H 0 inner-product, this flow isthe gradient descent for curve length. 5This flow has been studied deeply by the mathematical community, and isknown to enjoy the following properties.Properties 2.7• Embedded planar curves remain embedded.• Embedded planar curves converge to a circular point.• Comparison principle: if two embedded curves are used as initialization,and the first contains the second, then it will contain the second for alltime of evolution. This is important for level set methods.• The flow is well posed only for positive times, that is, for increasing t(similarly to the usual heat equation).For the proofs, see Gage and Hamilton [21], Grayson [23].2.4.2 Short length biasThe usual edge-based active contour energy is of the form∫E(c) = g(c(s)) dscsupposing that g is reasonably constant in the region where c is running, thenE(c) ≃ g len(c), due to the integral being performed w.r.to the arc parameter ds.So one way to reduce E is to shrink the curve. Consequently, when the image issmooth and featureless (as in medical imaging), the usual edge based energieswould often drive the curve to a point.To avoid it, an inflationary term νN (with ν > 0) was usually added tothe curve evolution, to obtain∂c= φκN − ∇φ + νN ,∂tsee [7, 27]. Unfortunately, this adds a parameter that has to be tuned to matchthe characteristics of each specific image.5 A different gradient descent flow for curve length will be discussed in Remark 10.32.c15