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

where φ may be chosen to beφ(x) =11 + |∇I(x)| 2that is small on sharp discontinuities of I (∇I is the gradient of I w.r.tox). Since real images are noisy, the function φ in practice would be moreinfluenced by the noise than by the image features; for this reason, usuallythe function φ is actually defined byφ(x) =11 + |∇G ⋆ I(x)| 2where G is a smoothing kernel, such as the Gaussian.2.3 Geodesic active contour method2.3.1 The “geodesic active contour” paradigmThe general procedure for geodesic active contours goes as follows.1. Choose an appropriate geometric energy functional, E.2. Compute the directional derivative 3DE(c; h) =d dt E(c + th) ∣∣∣∣t=0where c is a curve and h is an arbitrary perturbation of c.3. Manipulate DE(c; h) into the form∫h(s) · v(s) ds .c4. Consider v to be the “gradient”, the direction which increases E fastest.5. Evolve c = c(t, θ) by the differential equation ∂ t c = −v; this is called thegradient descent flow.2.3.2 Example: geodesic active contour edge modelWe propose an explicit computation starting from the classical active contourmodel.1. Start from an energy that is minimal along image contours:∫E(c) = φ(c(s)) dscwhere φ is defined to extract relevant features; for examples as discussedafter eqn. (2.2).3 The directional derivative is the basis for the definition 3.13 of the Gâteaux differential,that will be then discussed in Section 3.7 and 4.2.13