Numerical solution of parabolic equations related to level set formulation of mean curvature flow

Variational Numerical Methods for Solving Nonlinear Diffusion Equations Arising in Image Processing

“…This idea was used in [3], where model (2) has been suggested for computational image and shape analysis. From the practical point of view, applying just the level set equation (i.e., g ≡ 1) to initial image yields the intrinsic silhouette smoothing [14]. On the other hand, Eq.…”

“…It also includes a discussion of the computational efficiency of iterative solvers used in our semiimplicit scheme (44). Since numerical experiments based on the finite element method and finite volume methods in 2D and 3D have been presented elsewhere [14,16,17,23,26,27,29], here we give some 2D examples of the usage of the complementary volume method. In our computations we have chosen g(s) = 1/(1 + K s 2 ) with a constant K > 0, f ≡ 0 while the convolution is realized using (29) with σ less than τ .…”

Stability and consistency of the semi-implicit co-volume scheme for regularized mean curvature flow equation in level set formulation

“…∇u |∇u| = 0, (1.1) as well as its nontrivial generalizations, is used in the applications as the motion of interfaces (free boundaries) in thermomechanics (solidification, crystal growth) and computational fluid dynamics (free surface flows, multi-phase flows of immiscible fluids, thin films), the smoothing and segmentation of images and the surface reconstructions in the image processing, computer vision and computer graphics (see e.g. [32,29,2,1,6,19,31,15,17]), and in many further situations related to the motion of implicit curves or surfaces. On the other hand, the convergence of numerical schemes to unique viscosity solution [9,14,7] of equation (1.1) is often an open problem, it is an exception to find an analysis of convergence of the methods used for solving the curvature driven flows in the level set formulation.…”

“…Integrating equation (1.1) in the co-volume gives the weak (integral) formulation of the problem from which the computational scheme naturally follows. One of our main motivations to solve the curvature driven level set equation and its generalizations comes from image processing applications [15,16,17,23,24,8]. The co-volume scheme has been applied to smoothing and segmentation of 2D and 3D medical images in [24,8] and is based on the original semi-implicit method studied in [16].…”

Evolution of Plane Curves Driven by a Nonlinear Function of Curvature and Anisotropy

“…There are also other direct methods based on solution of a porous medium-like equation for curvature of a curve [31,32], a crystalline curvature approximation [22,23,44], special finite difference schemes [28,29], and a method based on erosion of polygons in the affine invariant scale case [34]. By contrast to the direct approach, level set methods are based on introducing an auxiliary function whose zero level sets represent an evolving family of planar curves undergoing the geometric equation (1.1) (see, e.g., [36,41,42,43,26]). The other indirect method is based on the phase-field formulations (see, e.g., [14,35,20,13]).…”