An improved level-set method for tracking the interface of fluids

A fast marching algorithm for the factored eikonal equation

Feature extraction from mammographic images using fast marching methods

The association between daily as well as monthly winter precipitation and concurrent 700 hPa heights for 3 different North American precipitation regimes was examined using correlation fields and a typing of daily 700 hPa height patterns. Wet days, regardless of circulation type, were found to be associated with negative height anomalies immediately to the west or northwest of the precipitation regions, at the 'local' grid point. These small-scale height anomalies were found to rep- resent traveling short waves that were missed by the circulation typing scheme. The correlation fields between monthly precipitation and concurrent 700 hPa heights revealed a westward shift of the height minima to a location west of 100°W, to a 'remote' grid point. The nature of these remote min- ima was examined using the 10 wettest and 10 driest months. Decomposition of the height anomalies at the remote grid points into between-type and within-type contributions revealed that while some of the negative heights during wet months are due to an increase in the frequency of circulation pat- terns with below normal heights (such as troughs or weak ridges), more than two-thirds of the con- tribution comes from height depressions superimposed upon the larger-scale patterns and most of these are associated with positive vorticity. This leads to the conclusion that the correlation minima west of 100° W at the monthly time scale reflect the frequent generation of short waves during wet months over the well-known North American cyclogenetic region east of the Rocky Mountains. These results have 2 implications with respect to statistical downscaling. First, regarding techniques designed to extract information from a single grid point: the optimum grid point location may be a function of the time scale under consideration. Second, regarding techniques designed to extract information from large-scale circulation patterns: the difficulties in capturing the very important but small short waves limit their usefulness.

A fast marching approach to multidimensional extrapolation

Describes an approach to solving optimal hybrid control problems using level set methods. Level set methods are powerful techniques for generating equipotential contours with applications in the realm of fluid mechanics, computer vision, material science, robotics, and geometry. The paper specifically deals with the problem of determining an optimal control path in a hybrid system by extending the fast marching method to a hybrid setting. We formalize the hybrid problem, provide an algorithm to solve it, and give a constructive proof of the algorithm's correctness. We also solve two examples in our hybrid setup and discuss upper- and lower-bounds of numerical solutions.

A new method for mesh generation is formulated based on the level set functions, which are solutions of the standard level set evolution equation with the Cartesian coordinates as initial values (Liao et al. J Comput Phys 159 (2000), 103–122; Osher and Sethian J Comput Phys 79 (1988), 12; Sethian, Level set methods and fast marching methods, Cambridge University Press, 1999; Di et al. J Sci Comput 31 (2007), 75–98). The intersection of the level contours of the evolving functions form a new grid at each time. The velocity vector in the evolution equation is chosen according to the Geometric Conservation Law (GCL) method (Cao et al., SIAM J Sci Comput 24 (2002), 118–142.). This method has precise control over the Jacobian of transformation because of using the GCL method. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009

Technology is proliferating. Many methods are used for medical imaging. The important methods used here are fast marching and fast level set in comparison with the watershed transform. Since watershed algorithm was applied to an image has over clusters in segmentation. Both methods are applied to segment the medical images. First, fast marching method is used to extract the rough contours. Then fast level set method is utilized to finely tune the initial boundary. Moreover, Traditional fast marching method was modified by the use of watershed transform. The method is feasible in medical imaging and deserves further research. In the future, we will integrate level set method with statistical shape analysis to make it applicable to more kinds of medical images and have better robustness to noise.

This course is as an introduction to the PhysBAM simulation library developed at Stanford University and used in both academic and industrial settings, including Intel Corporation, Industrial Light & Magic, Walt Disney Animation Studios, and Pixar Animation Studios. The course contains information on the release of PhysBAM as well as information on how to obtain the source code, set up the library, and use it to run example smoke and water simulations. It also summarizes a visualization tool and a rendering tool included in the release of the library.Physically based simulation is a vital part of the special-effects toolkit. Traditionally, special effects are obtained by constructing scale or full-size models, whether the scene calls for a sinking ship or a burning building. Unfortunately, this is very expensive and at times not feasible. Physical simulation is also used for special effects in animated films, where traditional special effects methods cannot easily be applied. And it is increasingly used in the gaming community as enhanced processing power enables simulations to occur at real-time rates.In addition to the PhysBAM library, the course explains the underlying techniques that make these simulations possible, in particular level set methods such as fast marching, fast sweeping, and the particle level set method. It also addresses the important aspects of a fluid simulation, including advection, viscosity, and projection.

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 the points in a grid. We consider manifolds composed of simple geometric shapes, such as, a set of points, piecewise linear curves or triangle meshes. The algorithm solves the eikonal equation |∇u| = 1 with the method of characteristics. For many problems, the computational complexity of the algorithm is linear in both the number of grid points and the complexity of the manifold. Many query problems can be aided by using orthogonal range queries (ORQ). There are several standard data structures for performing ORQ's in 3-D, including kd-trees, octrees, and cell arrays. We develop additional data structures based on cell arrays. We study the characteristics of each data structure and compare their performance. We present a new algorithm for solving the single-source, non-negative weight, shortest-paths problem. Dijkstra's algorithm solves this problem with computational complexity O ((E + V)log V) where E is the number of edges and V is the number of vertices. The new algorithm, called Marching with a Correctness Criterion (MCC), has computational complexity O (E + RV), where R is the ratio of the largest to smallest edge weight. Sethian's Fast Marching Method (FMM) may be used to solve static Hamilton-Jacobi equations. It has computational complexity O (N log N), where N is the number of grid points. The FMM has been regarded as an optimal algorithm because it is closely related to Dijkstra's algorithm. The new shortest-paths algorithm discussed above can be used to develop an ordered, upwind, finite difference algorithm for solving static Hamilton-Jacobi equations. This algorithm requires difference schemes that difference not only in coordinate directions, but in diagonal directions as well. It has computational complexity O(RN), where R is the ratio of the largest to smallest propagation speed and N is the number of grid points.

The level-set method is used in many different applications to describe the propagation of shapes and domains. When scalar speed fields are used to encode the desired shape evolution, this leads to the classical level-set equation. We present a concise Hopf–Lax representation formula that can be used to characterise the evolved domains at arbitrary times. This result is also applicable for the case of speed fields without a fixed sign, even though the level-set equation has a non-convex Hamiltonian in these situations. The representation formula is based on the same idea that underpins the Fast-Marching Method, and it provides a strong theoretical justification for a generalised Composite Fast-Marching method. Based on our Hopf–Lax formula, we are also able to present new theoretical results. In particular, we show non-fattening of the zero level set in a measure-theoretic sense, derive a very general shapesensitivity calculus that does not require the usual regularity assumptions on the domains, prove optimal Lipschitz constants for the evolved level-set function and discuss the effect of perturbations in both the speed field and the initial geometry.

The geometric distribution of immiscible fluid phases in fractures is not readily accessible experimentally, so aperture-scale simulations of drainage and imbibition in realistic fractures can provide valuable insight. We implement a level set method for computing the location within a fracture of the interface between two fluids controlled by capillary forces. The movement of the interface in response to changes in capillary pressure is approximated as quasi-static displacement. Fluid interfaces are thus constant mean curvature surfaces, satisfying the Young Laplace equation. We apply a progressive-quasistatic (PQS) algorithm to determine when spontaneous pore-level events occur during fluid displacement. The algorithm captures reversible and irreversible behavior. We illustrate the approach in two types of rough-walled fractures. One type is a 3D crack between irregular, impermeable surfaces; the other type is a gap between irregular 2D and 3D grain packs. We focus on the disconnected (defending) fluid volumes and (advancing) fluid main pathway as the geometric properties of the fracture are varied, notably aperture and the number of contact points between the upper and lower fracture surfaces. The curvature of the fluid-fluid interface in the plane of the fracture is often ignored by invasion percolation simulation techniques, yet it is known to influence strongly the fluid cluster properties. Our simulations establish the exact position and shape of the interface in realistic fracture geometries, from which fluid volumes, contact areas and interface curvatures can be obtained. This establishes a new, mechanistic basis for predicting relative permeabilities in fractures and for evaluating transfer functions in dual-porosity flow models.

Many meshes in computer animation practice are meant to approximate solid objects, but the provided triangular geometry is often unoriented, non-manifold or contains self-intersections, causing inside/outside of objects to be mathematically ill-defined. We describe a robust and efficient automatic approach to define and compute a signed distance field for arbitrary triangular geometry. Starting with arbitrary (non-manifold) triangular geometry, we first define and extract an offset manifold surface using an unsigned distance field. We then automatically remove any interior surface components. Finally, we exploit the manifoldness of the offset surface to quickly detect interior distance field grid points. We prove that exterior grid points can reuse a shifted original unsigned distance field, whereas for interior cells, we compute the signed field from the offset surface geometry. We demonstrate improved performance both using exact distance fields computed using an octree, and approximate distance fields computed using fast marching. We analyze the time and memory costs for complex meshes that include self-intersections and non-manifold geometry. We demonstrate the effectiveness of our algorithm by using the signed distance field for collision detection and generation of tetrahedral meshes for physically based simulation.

The injection of an air bubble in a liquid at rest is an interface flow problem where surface tension and its modeling at solid boundaries is a key factor. It is the subject of this study. Numerical simulations have been performed to study 3D axi-symmetrical bubble growth from an orifice through a horizontal wall. The gas inflow velocity used was sufficiently small to ensure that the bubble growth is quasi-static so that surface tension and buoyancy forces are dominant. The wall was considered non-wettable to avoid spreading of the interface along the wall. The Navier-Stokes equations were solved with two different interface capturing methods based on Volume of Fluid (VOF) and Level Set (LS) as well as coupled CVOFLS. In the VOF method the bubble interface was tracked using either an algebraic solver which results in some diffusion of the interface (compressive scheme implemented in OpenFOAM), or it was determined using a geometric reconstruction scheme (Geo-Reconstruct Scheme from Fluent). The TransAT code was used for the LS model which captures the interface using signed distance function.The bubble volume and center of gravity have been investigated during the growth using the three solvers and numerical results have been assessed against experimental data. These results have shown that reconstructing the interface using the LS method gives good agreement with the experiments. In VOF (compressive scheme), the bubble detaches at earlier times resulting in a smaller detachment volume. The coupled CVOFLS-GeoReconstruct was found to be more computationally expensive than the VOF-GeoReconstruct and to present bubble oscillation during the growth.

Rapid and accurate evaluation of gamma radiation dose in dynamic virtual environments is crucial for radiation protection and safety assessment. The point kernel method has been widely used for rapid dose calculation due to its high efficiency. However, the 'point kernel calculations (PKCs) count' is defined as the product of the point kernels number and the counting grids number. Consequently, when either the number of point kernels generated from the discretised radiation source or the number of counting grids in the space becomes large, the time consumption increases significantly. This issue is further exacerbated when the geometric model is no longer a regular geometry describable by mathematical formulas but rather an arbitrary 3D model in a virtual environment, where the cost of ray tracing grows substantially. To address this challenge, an efficient computational approach is proposed and a high-performance point kernel code, PLSPK, is developed in this work. The main features of PLSPK include using triangular mesh models as geometric representations to handle arbitrary shapes, and optimising the ray-tracing process through the application of the maximised-parallelism bounding volume hierarchy technology, enabling rapid selection of target triangles for ray tracing in complex scenes. Furthermore, by leveraging graphics processing unit parallel computation, the code significantly accelerates large-scale PKC in gamma radiation fields, allowing PLSPK to meet the demand for continuous rapid updates of radiation field in dynamic environments. Compared with serial implementations, the acceleration achieved by PLSPK becomes more pronounced as the PKC count (PKC count) increases, when the PKC count reaches 1E + 06, the speedup is approximately 2000×, with small impact from scene complexity. Accuracy validation against existing point kernel codes and the Monte Carlo particle transport program MCNP demonstrates that PLSPK provides reliable dose estimations. These features make PLSPK a reliable tool for high-efficiency gamma radiation field calculation in complex dynamic virtual environments.

Current computer games are being set in increasingly more complex and dynamic virtual environments. Massively multiplayer online games, for example, are played in persistent virtual worlds, which evolve and change as players create and personalize their own virtual property. In contrast, technologies for controlling the behavior of nonplayer characters that populate virtual game worlds are frequently limited to preprogrammed rules. Characters using fixed rule-sets lack the ability to adapt in time with their environment. Motivated reinforcement learning offers an alternative to character design that can achieve nonplayer characters that both evolve and adapt in dynamic environments. This article presents and evaluates two computational models of motivation for use in nonplayer characters in persistent computer game worlds. These models represent motivation as an ongoing search for novelty, interest, and competence. Two metrics are introduced to evaluate the adaptability of characters controlled by motivated reinforcement learning agents using different models of motivation. These metrics characterize the behavior of nonplayer characters in terms of the variety and complexity of learned behaviors. An empirical evaluation of characters in simulated game scenarios shows that characters motivated by the search for competence are more adaptable in dynamic environments than those motivated by interest and novelty alone.