BOAT-SAIL VORONOI DIAGRAM AND ITS APPLICATION

2009 ◽  
Vol 19 (05) ◽  
pp. 425-440 ◽  
Author(s):  
TETSUSHI NISHIDA ◽  
KOKICHI SUGIHARA
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Boundary Value Problem ◽  
Voronoi Diagram ◽  
Water Surface ◽  
Eikonal Equation ◽  
Fast Marching Method ◽  
Fast Marching ◽  
Marching Method ◽  
Generalized Voronoi Diagram

A new generalized Voronoi diagram, called a boat-sail Voronoi diagram, is defined on the basis of the time necessary for a boat to reach on water surface with flow. A new concept called a boat-sail distance is introduced on the surface of water with flow, and it is used to define a generalized Voronoi diagram, in such a way that the water surface is partitioned into regions belonging to the nearest harbors with respect to this distance. The problem of computing this Voronoi diagram is reduced to a boundary value problem of a partial differential equation, and a numerical method for solving this problem is constructed. The method is a modification of a so-called fast marching method originally proposed for the eikonal equation. Computational experiments show the efficiency and the stableness of the proposal method. We also apply our equation to the shortest path problem and the simulation of the forest fire.

scholarly journals Tsunami wave propagation by voronoi diagram

2018 ◽  
Vol 7 (3) ◽  
pp. 1233
Author(s):  
V Yuvaraj ◽  
S Rajasekaran ◽  
D Nagarajan
Keyword(s):  
Wave Propagation ◽  
Voronoi Diagram ◽  
Tsunami Wave ◽  
Fast Marching Method ◽  
Coastal Regions ◽  
Fast Marching ◽  
Tsunami Waves ◽  
The Finite Difference Method ◽  
Run Up ◽  
Marching Method

Cellular automata is the model applied in very complicated situations and complex problems. It involves the Introduction of voronoi diagram in tsunami wave propagation with the help of a fast-marching method to find the spread of the tsunami waves in the coastal regions. In this study we have modelled and predicted the tsunami wave propagation using the finite difference method. This analytical method gives the horizontal and vertical layers of the wave run up and enables the calculation of reaching time.  

On: “3-D traveltime compuation using the fast marching method” (J. A. Sethian and A. M. Popovici, GEOPHYSICS, 64, 516–523).

Geophysics ◽  
2000 ◽  
Vol 65 (2) ◽  
pp. 682-682
Author(s):  
Fuhao Qin
Keyword(s):  
Finite Difference ◽  
Eikonal Equation ◽  
Fast Marching Method ◽  
Fast Marching ◽  
First Arrival ◽  
Finite Difference Solution ◽  
Marching Method ◽  
Difference Solution

The Sethian and Popovici paper “3-D traveltime computation using the fast marching method” that appeared in Geophysics, Vol. 64, 516–523, discussed a method to solve the eikonal equation for first arrival traveltimes which was called the “fast marching” method. The method, as the authors demonstrated, is very fast and stable. However, their method is very similar to the method discussed by F. Qin et al. (1992), entitled “Finite difference solution of the eikonal equation along expanding wavefronts,” Geophysics, Vol. 57, 478–487. F. Qin et al. first proposed the “expanding wavefront” method for solving eikonal equation in the 60th Ann. Internat. Mtg. of the SEG in 1990.

Mobile Robot Path Planning using Voronoi Diagram and Fast Marching

2015 ◽  
pp. 92-108 ◽  
Author(s):  
S. Garrido ◽  
L. Moreno
Keyword(s):  
Path Planning ◽  
Mobile Robot ◽  
Voronoi Diagram ◽  
High Speed ◽  
Fast Marching Method ◽  
Robot Path Planning ◽  
Fast Marching ◽  
Path Planner ◽  
Marching Method ◽  
Robot Path

This chapter presents a new sensor-based path planner, which gives a fast local or global motion plan capable to incorporate new obstacles data. Within the first step, the safest areas in the environment are extracted by means of a Voronoi Diagram. Within the second step, the fast marching method is applied to the Voronoi extracted areas so as to get the trail. This strategy combines map-based and sensor-based designing operations to supply a reliable motion plan, whereas it operates at the frequency of the sensor. The most interesting characteristics are high speed and reliability, as the map dimensions are reduced to a virtually one-dimensional map and this map represents the safest areas within the environment.

3D Facial Shape Reconstruction from a Single Frontal Image

2014 ◽  
Vol 989-994 ◽  
pp. 3544-3547
Author(s):  
Qian Ma
Keyword(s):  
Eikonal Equation ◽  
Shape From Shading ◽  
Shape Reconstruction ◽  
Fast Marching Method ◽  
Improved Method ◽  
Fast Marching ◽  
Facial Shape ◽  
Shape Invariant ◽  
Marching Method

In this paper, we propose an improved method for reconstruct 3D facial shape from a single frontal image. We use improved fast marching method to solve the Eikonal equation which can obtained from the method of shape from shading (SFS). In order to overcome the concave-convex ambiguity problems inherent to SFS, we find out the concave region and recover concave into the convex keeping the relative shape invariant to reconstruct the accuracy facial shape.

PyKonal: A Python Package for Solving the Eikonal Equation in Spherical and Cartesian Coordinates Using the Fast Marching Method

2020 ◽  
Vol 91 (4) ◽  
pp. 2378-2389
Author(s):  
Malcolm C. A. White ◽  
Hongjian Fang ◽  
Nori Nakata ◽  
Yehuda Ben-Zion
Keyword(s):  
Open Source ◽  
Eikonal Equation ◽  
Heterogeneous Media ◽  
Fast Marching Method ◽  
Fast Marching ◽  
Cartesian Coordinates ◽  
Source Codes ◽  
High Level ◽  
Marching Method ◽  
Python Package

Abstract This article introduces PyKonal: a new open-source Python package for computing travel times and tracing ray paths in 2D or 3D heterogeneous media using the fast marching method for solving the eikonal equation in spherical and Cartesian coordinates. Compiled with the Cython compiler framework, PyKonal offers a Python application program interface (API) with execution speeds comparable to C or Fortran codes. Designed to be accurate, stable, fast, general, extensible, and easy to use, PyKonal offers low- and high-level API functions for full control and convenience, respectively. A scale-independent implementation allows problems to be solved at micro, local, regional, and global scales, and precision can be improved over existing open-source codes by combining different coordinate systems. The resulting code makes state-of-the-art computational capabilities accessible to novice programmers and is efficient enough for modern research problems in seismology.

scholarly journals Fast Marching Method Aplication for Forward Modelling of Seismic Wave Propagation

2018 ◽  
Vol 16 (3) ◽  
pp. 1
Author(s):  
Wahyu Srigutomo ◽  
Ghany Hanifan Muslim
Keyword(s):  
Wave Propagation ◽  
Viscosity Solution ◽  
Seismic Wave ◽  
Eikonal Equation ◽  
Heterogeneous Media ◽  
Seismic Wave Propagation ◽  
Time Travel ◽  
Fast Marching Method ◽  
Fast Marching ◽  
Marching Method

One of the classical problem in seismology is to determine time travel and ray path of seismic wave betweentwo points at a given heterogeneous media. This problem is expressed by eikonal equation and can be seen as a propagation of a wavefront and interface evolution. One of methods to solve this problem is Fast Marching Method abbreviated as FMM. This method is used to produce entropy-satisfying viscosity solution of eikonal equation. FMM combines viscosity solution of Hamilton-Jacobi equation and Huygen's Principle that centered on min-heap data structure to determine the minimum value at every loop. In this study, FMM is applied to determine time travel and raypath of seismic wave. FMM also is used to determine the location of wavesource using simple algorithm. From our forward modeling schemes, it is found that FMM is an accurate, robust, and effcient method to simulate seismic wave propagation. For further study, FMM also can be used to be a part of passive seismic inverse scheme to locate hypocenter location.

3-D traveltime computation using the fast marching method

Geophysics ◽  
1999 ◽  
Vol 64 (2) ◽  
pp. 516-523 ◽  
Author(s):  
James A. Sethian ◽  
A. Mihai Popovici
Keyword(s):  
Point Source ◽  
Eikonal Equation ◽  
Initial Conditions ◽  
Plane Waves ◽  
Three Dimensions ◽  
Computational Domain ◽  
Fast Marching Method ◽  
Fast Marching ◽  
Grid Points ◽  
Marching Method

We present a fast algorithm for solving the eikonal equation in three dimensions, based on the fast marching method. The algorithm is of the order O(N log N), where N is the total number of grid points in the computational domain. The algorithm can be used in any orthogonal coordinate system and globally constructs the solution to the eikonal equation for each point in the coordinate domain. The method is unconditionally stable and constructs solutions consistent with the exact solution for arbitrarily large gradient jumps in velocity. In addition, the method resolves any overturning propagation wavefronts. We begin with the mathematical foundation for solving the eikonal equation using the fast marching method and follow with the numerical details. We then show examples of traveltime propagation through the SEG/EAGE salt model using point‐source and plane‐wave initial conditions and analyze the error in constant velocity media. The algorithm allows for any shape of the initial wavefront. While a point source is the most commonly used initial condition, initial plane waves can be used for controlled illumination or for downward continuation of the traveltime field from one depth to another or from a topographic depth surface to another. The algorithm presented here is designed for computing first‐arrival traveltimes. Nonetheless, since it exploits the fast marching method for solving the eikonal equation, we believe it is the fastest of all possible consistent schemes to compute first arrivals.

A Third Order Accurate Fast Marching Method for the Eikonal Equation in Two Dimensions

2011 ◽  
Vol 33 (5) ◽  
pp. 2402-2420 ◽  
Author(s):  
Shahnawaz Ahmed ◽  
Stanley Bak ◽  
Joyce McLaughlin ◽  
Daniel Renzi
Keyword(s):  
Eikonal Equation ◽  
Two Dimensions ◽  
Fast Marching Method ◽  
Fast Marching ◽  
Third Order ◽  
Marching Method
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