https://math.berkeley.edu/~sethian/Papers/sethian.osher.88.pdf

Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations

A novel scheme for the detection of object boundaries is presented. The technique is based on active contours deforming according to intrinsic geometric measures of the image. The evolving contours naturally split and merge, allowing the simultaneous detection of several objects and both interior and exterior boundaries. The proposed approach is based on the relation between active contours and the computation of geodesics or minimal distance curves. The minimal distance curve lays in a Riemannian space whose metric as defined by the image content. This geodesic approach for object segmentation allows to connect classical "snakes" based on energy minimization and geometric active contours based on the theory of curve evolution. Previous models of geometric active contours are improved as showed by a number of examples. Formal results concerning existence, uniqueness, stability, and correctness of the evolution are presented as well. >

/pdf/geodesic-active-contours-4wqlqxa13k.pdf

Geodesic active contours

A novel scheme for the detection of object boundaries is presented. The technique is based on active contours evolving in time according to intrinsic geometric measures of the image. The evolving contours naturally split and merge, allowing the simultaneous detection of several objects and both interior and exterior boundaries. The proposed approach is based on the relation between active contours and the computation of geodesics or minimal distance curves. The minimal distance curve lays in a Riemannian space whose metric is defined by the image content. This geodesic approach for object segmentation allows to connect classical “snakes” based on energy minimization and geometric active contours based on the theory of curve evolution. Previous models of geometric active contours are improved, allowing stable boundary detection when their gradients suffer from large variations, including gaps. Formal results concerning existence, uniqueness, stability, and correctness of the evolution are presented as well. The scheme was implemented using an efficient algorithm for curve evolution. Experimental results of applying the scheme to real images including objects with holes and medical data imagery demonstrate its power. The results may be extended to 3D object segmentation as well.

Geodesic Active Contours

1. Basic convexity 2. Boundary structure 3. Minkowski addition 4. Curvature measure and quermass integrals 5. Mixed volumes 6. Inequalities for mixed volumes 7. Selected applications Appendix.

/pdf/convex-bodies-the-brunn-minkowski-theory-3raynvod45.pdf

Convex bodies : the Brunn-Minkowski theory

Level Set Methods and Fast Marching Methods: Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision, and Materials Science (2nd edition)

Soient M et M' des varietes de Riemann et F:M→M' une application reguliere. Si M est une courbe convexe plongee dans le plan R 2 , l'equation de la chaleur contracte M a un point

/pdf/the-heat-equation-shrinking-convex-plane-curves-wr5hy2pxpu.pdf

The heat equation shrinking convex plane curves

Curve shortening makes convex curves circular

An isoperimetric inequality with applications to curve shortening

In (0.1) X :S × [0, ω) → IR is the position vector of a family of closed convex plane curves, kN is the curvature vector, with k being the curvature and N the inward pointing normal given by N = −(cos θ, sin θ). The weight function γ(θ) = γ(N) is a function of the normal vector to the curve at each point but does not depend on position in the plane. Equation (0.1) has two significant interpretations. It can be seen as the generalization of the “curve shortening” problem ([Ga8]) to Minkowski geometry or as a simplified model of the motion of the interface of a metal crystal as it melts ([AnGu],[Ta1] and [Ga8]). The proof illustrates most of the techniques that have been used recently in understanding geometric evolution equations as described in [Ha3]. It is not hard to show that the self-similar solutions correspond to positive, 2π periodic solutions of the equation

/pdf/evolving-plane-curves-by-curvature-in-relative-geometries-1fw0pkl0il.pdf

Evolving plane curves by curvature in relative geometries

This is an expository paper describing the recent progress in the study of the curve shortening equation 

$${X_{{t\,}}} = \,kN $$

(0.1)

Here X is an immersed curve in ℝ2, k the geodesic curvature and N the unit normal vector. We review the work of Gage on isoperimetric inequalities, the work of Gage and Hamilton on the associated heat equation and the work of Epstein and Weinstein on the stable manifold theorem for immersed curves. Finally we include a new proof of the Bonnesen inequality and a proof that highly symmetric immersed curves flow under (0.1) to points.

Michael E. Gage

Papers

The heat equation shrinking convex plane curves

Curve shortening makes convex curves circular

An isoperimetric inequality with applications to curve shortening

Evolving plane curves by curvature in relative geometries

The Curve Shortening Flow