We propose an algorithm for minimizing the total variation of an image, and provide a proof of convergence. We show applications to image denoising, zooming, and the computation of the mean curvature motion of interfaces.

The level set method was devised by Osher and Sethian in [64] as a simple and versatile method for computing and analyzing the motion of an interface Γ in two or three dimensions. Γ bounds a (possibly multiply connected) region Ω. The goal is to compute and analyze the subsequent motion of Γ under a velocity field �v. This velocity can depend on position, time, the geometry of the interface and the external physics. The interface is captured for later time as the zero level set of a smooth (at least Lipschitz continuous) function ϕ(�x,t), i.e., Γ(t)={�x|ϕ(�x,t)=0}. ϕ is positive inside Ω, negative outside Ω andiszeroonΓ(t). Topological merging and breaking are well defined and easily performed. In this review article we discuss recent variants and extensions, including the motion of curves in three dimensions, the Dynamic Surface Extension method, fast methods for steady state problems, diffusion generated motion and the variational level set approach. We also give a user’s guide to the level set dictionary and technology, couple the method to a wide variety of problems involving external physics, such as compressible and incompressible (possibly reacting) flow, Stefan problems, kinetic crystal growth, epitaxial growth of thin films,

In this paper, we propose a new numerical method for improving the mass conservation properties of the level set method when the interface is passively advected in a flow field. Our method uses Lagrangian marker particles to rebuild the level set in regions which are under-resolved. This is often the case for flows undergoing stretching and tearing. The overall method maintains a smooth geometrical description of the interface and the implementation simplicity characteristic of the level set method. Our method compares favorably with volume of fluid methods in the conservation of mass and purely Lagrangian schemes for interface resolution. The method is presented in three spatial dimensions.

An optimal algorithm for the reconstruction of a surface from its shading image is presented. The algorithm solves the 3D reconstruction from a single shading image problem. The shading image is treated as a penalty function and the height of the reconstructed surface is a weighted distance. A consistent numerical scheme based on Sethian’s fast marching method is used to compute the reconstructed surface. The surface is a viscosity solution of an Eikonal equation for the vertical light source case. For the oblique light source case, the reconstructed surface is the viscosity solution to a different partial differential equation. A modification of the fast marching method yields a numerically consistent, computationally optimal, and practically fast algorithm for the classical shape from shading problem. Next, the fast marching method coupled with a back tracking via gradient descent along the reconstructed surface is shown to solve the path planning problem in robot navigation.

Consider the eikonal equation, = 1. If the initial condition is u = 0 on a manifold, then the solution u is the distance to the manifold. We present a new algorithm for solving this problem. More precisely, we present an algorithm for computing the closest point transform to an explicitly described manifold on a rectilinear grid in low dimensional spaces. The closest point transform finds the closest point on a manifold and the Euclidean distance to a manifold for all the points in a grid (or the grid points within a specified distance of the manifold). We consider manifolds composed of simple geometric shapes, such as, a set of points, piecewise linear curves or triangle meshes. The algorithm computes the closest point on and distance to the manifold by solving the eikonal equation = 1 by the method of characteristics. The method of characteristics is implemented efficiently with the aid of computational geometry and polygon/polyhedron scan conversion. Thus the method is named the characteristic/scan conversion algorithm. The computed distance is accurate to within machine precision. The computational complexity of the algorithm is linear in both the number of grid points and the complexity of the manifold. Thus it has optimal computational complexity. The algorithm is easily adapted to shared-memory and distributed-memory concurrent algorithms. Many query problems...

In this paper, we present an e#cient semi-Lagrangian based particle level set method for the accurate capturing of interfaces. This method retains the robust topological properties of the level set method without the adverse e#ects of numerical dissipation. Both the level set method and the particle level set method typically use high order accurate numerical discretizations in time and space, e.g. TVD RungeKutta and HJ-WENO schemes. We demonstrate that these computationally expensive schemes are not required. Instead, fast, low order accurate numerical schemes su#ce. That is, the addition of particles to the level set method not only removes the di#culties associated with numerical di#usion, but also alleviates the need for computationally expensive high order accurate schemes. We use an e#cient, first order accurate semi-Lagrangian advection scheme coupled with a first order accurate fast marching method to evolve the level set function. To accurately track the underlying flow characteristics, the particles are evolved with a second order accurate method. Since we avoid complex high order accurate numerical methods, extending the algorithm to arbitrary data structures becomes more feasible, and we show preliminary results obtained with an octree-based adaptive mesh.

A distance field is a representation where, at each point within the field, we know the distance from that point to the closest point on any object within the domain. In addition to distance, other properties may be derived from the distance field, such as the direction to the surface, and when the distance field is signed, we may also determine if the point is internal or external to objects within the domain. The distance field has been found to be a useful construction within the areas of computer vision, physics, and computer graphics. This paper serves as an exposition of methods for the production of distance fields, and a review of alternative representations and applications of distance fields. In the course of this paper, we present various methods from all three of the above areas, and we answer pertinent questions such as How accurate are these methods compared to each other? How simple are they to implement?, and What is the complexity and runtime of such methods?

We propose a numerical method for modeling multimaterial flows where the domain is decomposed into separate Eulerian and Lagrangian subdomains. That is, the equations are written in Eulerian form in one subdomain and Lagrangian form in the other subdomain. This is of interest, for example, when considering the effect of underwater explosions on the hull of a ship or the impact of a low speed projectile on a soft explosive target. On one hand, high speed fluid flows are traditionally modeled by applying shock capturing schemes to the compressible Euler equations avoiding problems associated with tangling of a Lagrangian mesh. On the other hand, solid dynamics calculations are traditionally carried out using Lagrangian numerical methods avoiding problems associated with numerical smearing in Eulerian calculations.

Since the seminal work of [92] on coupling the level set method of [69] to the equations for two-phase incompressible flow, there has been a great deal of interest in this area. That work demonstrated the most powerful aspects of the level set method, i.e. automatic handling of topological changes such as merging and pinching, as well as robust geometric information such as normals and curvature. Interestingly, this work also demonstrated the largest weakness of the level set method, i.e. mass or information loss characteristic of most Eulerian capturing techniques. In fact, [92] introduced a partial differential equation for battling this weakness, without which their work would not have been possible. In this paper, we discuss both historical and most recent works focused on improving the computational accuracy of the level set method focusing in part on applications related to incompressible flow due to both its popularity and stringent accuracy requirements. Thus, we discuss higher order accurate numerical methods such as Hamilton-Jacobi WENO [46], methods for maintaining a signed distance function, hybrid methods such as the particle level set method [27] and the coupled level set volume of fluid method [91], and adaptive gridding techniques such as the octree approach to free surface flows proposed in [56].

We present a novel method for the incorporation of shape information into active contour models, and apply it to the extraction of line networks (e.g. road, water) from satellite imagery. The method is based on a new class of contour energies. These energies are quadratic on the space of one-chains in the image, as opposed to classical energies, which are linear. They can be expressed as double integrals on the contour, and thus incorporate non-trivial interactions between different contour points. The new energies describe families of contours that share complex geometric properties, without making reference to any particular shape. Networks fall into such a family, and to model them we make a particular choice of quadratic energy whose minima are reticulated. To optimize the energies, we use a level set approach. The forces derived from the new energies are non-local however, thus necessitating an extension of standard level set methods. Promising experimental results are obtained using real images.