[00802] Numerical Algorithms for the Eikonal Equation and Its Applications
Session Date & Time :
00802 (1/3) : 2D (Aug.22, 15:30-17:10)
00802 (2/3) : 2E (Aug.22, 17:40-19:20)
00802 (3/3) : 3C (Aug.23, 13:20-15:00)
Type : Proposal of Minisymposium
Abstract : The minisymposium focuses on the recent state-of-art for the eikonal equation in view of the mathematical theories, diverse applications such as image processing, seismic wave travel time in layered media, 3D shape reconstruction, 3D printing, optimal control, homogenization, mean field games, and distance on a non-convex domain with polyhedral meshes, and their numerical algorithms; semi-discretization method, finite volume method, mimetic discretization method, using the Hopf-Lax formula, Jet marching method, variational methods, neural network approaches, etc. We also include a variant of the eikonal partial differential equation induced by Randers metric and a high order accurate efficient eikonal solvers on surfaces.
Laurent Cohen (University Paris Dauphine, PSL Research University, CNRS)
Jean-Marie Mirebeau (ENS Paris-Saclay, CNRS, University Paris-Saclay)
Miguel Dumett (Computational Science Research Center of San Diego State University)
Jooyoung Hahn (Slovak University of Technology in Bratislava)
Italo Capuzzo-Dolcetta (Sapienza University of Rome)
Myungjoo Kang (Seoul National University)
Silvia Tozza (University of Bologna)
Byungjoon Lee (Catholic University of Korea)
Ron Kimmel (Technion - Israel Institute of Technology)
Pierre-Alain Fayolle (University of Aizu)
Samuel Potter (The Courant Institute, New York University)
Shigetoshi Yazaki (Meiji University)
Talks in Minisymposium :
[02323] Solving the eikonal equation utilizing mimetic methods
Author(s) :
Miguel Dumett (San Diego State University)
Abstract : The Eikonal equation with Soner boundary conditions is solved utilizing high-order mimetic differences. Mimetic difference methods construct PDE discrete analogs. These preserve vector calculus identities as well as its integral theorems while keeping constant order of accuracy over the whole grid, including the boundary. The proposed algorithm leverages on the Fast Marching method and resembles a quasi-Newton iterative scheme.
[03439] Eikonal methods applied to image segmentation
Author(s) :
Laurent D. Cohen (CEREMADE, Universite Paris dauphine, PSL, CNRS)
Abstract : Minimal paths have been used for long as an interactive tool to find contours or segment tubular and tree structures, like vessels in medical images. These minimal paths correspond to minimal geodesics according to some relevant metric defined on the image domain. Finding a geodesic distance and geodesic paths can be solved by the Eikonal equation using the fast and efficient Fast Marching method. We will present various applications to image segmentation.
[03445] Regularized eikonal equation on polyhedral meshes
Author(s) :
Jooyoung Hahn (Slovak University of Technology in Bratislava)
Karol Mikula (Slovak University of Technology in Bratislava)
Peter Frolkovič (Slovak University of Technology in Bratislava)
Abstract : A cell-centered finite volume method (FVM) is discussed to compute a distance from objects on a 3D computational domain discretized by polyhedral cells. Time-relaxed and Laplacian-regularized eikonal equations are compared numerically and we show how the Soner boundary condition is straightforwardly combined in a conventional FVM code. The Laplacian regularization is more practical for large numbers of cells or distant regions of interest. Additionally, the algorithms can be implemented with parallel computing using domain decomposition.
[03535] Variational methods for distance function approximation and applications
Author(s) :
Pierre-Alain Fayolle (University of Aizu)
Alexander Belyaev (Heriot-Watt University)
Abstract : The distance function (signed or unsigned) to the boundary (curve or surface) of a given geometric domain is a useful tool in geometry processing, shape and solid modeling, as well as other related domains. It has various applications in several fields including surface reconstruction from scattered data, meshing, shape interrogation, computational fluid dynamics (turbulence modeling), computational mechanics, and robot path planning. In this talk, I will focus on several recent variational methods for computing the distance function or its approximation. Efficient numerical algorithms corresponding to these methods will also be discussed.
[03578] An Artificial Neural Network Approach for Re-distancing Implicit Surfaces
Author(s) :
Yesom Park (Seoul National University)
Chang hoon Song (Seoul National University)
Jooyoung Hahn (Slovak University of Technology in Bratislava)
Myungjoo Kang (Seoul National University)
Abstract : Following the success of machine learning tasks, the use of neural networks for solving PDEs has begun to show promising results. In this talk, we introduce a deep-learning-based method for recovering the signed distance function (SDF) from an implicit level set function representation of the hypersurface. By exploiting one of the main advantages of neural network approaches which is flexibility in network design and optimization objectives, our developments have two-fold: First, in order to increase the expressiveness of the network, we propose an augmented network that parameterizes the SDF together with the gradient of SDF as an auxiliary output while keeping the number of parameters. Second, we introduce a new objective that exploits a more global property and regularizes the singularity of the SDF by harnessing the geometric properties of the SDF. Numerical experiments on a diverse range of interfaces on two and three-dimensional domains validate the effectiveness and accuracy of the proposed method.
[03777] Approximate viscosity solutions of hamilton-Jacobi equations: a review
Author(s) :
Italo Capuzzo Dolcetta Italo (Sapienza Università di Roma)
Abstract : Semi-discretization methods to approximate viscosity solutions of Hamilton-Jacobi equations arising in different applied contexts such as optimal control, shape from shading, homogenization, and mean field games
[03845] Jet marching on unstructured meshes: algorithms and applications
Author(s) :
Samuel F Potter (Courant Institute of Mathematical Sciences)
Abstract : The jet marching method (JMM) solves the eikonal equation by marching its jet using semi-Lagrangian updates (i.e., local raytracing). Marching jets enables a compact Hermite interpolation-based scheme,
enabling the use of paraxial raytracing for marching the amplitude of the associated geometric optic wave. We use the JMM to solve high-frequency wave problems on unstructured meshes, with applications motivated by computational room acoustics and light transport in miniwasp ommatidia.
[03987] Learning to Measure Distances: High Order Accurate Efficient Eikonal Solvers on Surfaces
Author(s) :
Ron Kimmel (Technion - Israel Institute of Technologyn )
Abstract : The intimate relation between Eikonal equations and distance maps would be our starting point. When introducing a numerical solver, the balance between accuracy and complexity is at the core of computer science and a measure of quality of our solution. We will present a high accuracy deep learning method for approximating geodesic distances on surfaces at linear computational complexity. For training an accurate local solver a bootstrapping mechanism is employed.
[04630] Comparison study of image-segmentation techniques by a curvature-driven flow of planes curves
Author(s) :
Shigetoshi Yazaki (Meiji University)
Abstract : In this talk, we deal with image segmentation by a curvature-driven flow of curves in the plane.
We focus on images in the plane. Several methods to image segmentation are discussed.
For instance, the minimum radius method, L2-gradient flow method, stepwise method, etc.
Then, all methods are compared in the qualitative computational study.
[04713] Casualty and anisotropy in the design of eikonal solvers
Author(s) :
Jean-Marie Mirebeau (Centre Borelli, CNRS, ENS Paris-Saclay)
Abstract : The eikonal equation characterizes the arrival time of a front, propagating at a speed which is locally dictated by the front position and normal direction, and which crucially is always positive.
The causality property is the discrete counterpart of the monotonic progression of the front, and is at the foundation of efficient numerical solvers of the eikonal equation such as the fast marching method. I will describe an eikonal solver enjoying this property and applying to the tilted transversely anisotropy encountered in some geological media, as well as recent efforts on the formalization of the causality property, and its application to models which advect a state.
[04731] Eikonal equation in 3D shape reconstruction and 3D printing
Author(s) :
Silvia Tozza (Dept. of Mathematics, University of Bologna)
Abstract : The topic of this talk is related to Hamilton-Jacobi equations and their numerical resolution in the context of Image Processing. More in details, we will deal with the 3D reconstruction of the shape of an object (the resolution of the so-called Shape-from-Shading problem via a differential approach) and some problems related to 3D printing of the reconstructed object. In particular cases, the Hamilton-Jacobi equation describing these problems reduces to the eikonal equation.
[04972] The redistancing problem using Hopf-Lax formula and its applications
Author(s) :
Byungjoon Lee (The Catholic University of Korea)
Abstract : The redistancing, or reinitialization, problem is an important subject when one considers the signed distance function to the interface. There have been many researches on redistancing a given function to a signed distance function, based on numerical methods to solve the following eikonal equation:
$$\frac{\partial\phi}{\partial t}\left(x,t\right)+\lVert\nabla_x\phi(x,t)\rVert_2=0,\ \left(x,t\right)\in
\mathbb{R}^n\times\left(0,\infty\right)$$
In this talk, we review the novel redistancing technique based on Hopf-Lax formula equipped with the split Bregman approach and discuss on its applications.
