TY - JOUR

T1 - Simplicial isosurfacing in arbitrary dimension and codimension

AU - Min, Chohong

N1 - Funding Information:
Research supported by ONR Grants N00014-02-1-0720, UCLA PY-2029 and NSF Grant DMS-0074735.

PY - 2003/9/1

Y1 - 2003/9/1

N2 - The level set method has been successfully used for moving interface problems. The final step of the method is to construct and visualize the isosurface of a discrete function φ : {0,...,N}n → ℝm. There have existed many practical isosurfacing algorithms when n = 3, m = 1 or n = 2, m = 1. Recently we have begun to see the development of isosurfacing algorithms for higher dimensions and codimensions. This paper introduces a unified theory and an efficient isosurfacing algorithm that works in arbitrary number of dimensions and codimensions. The isosurface Γ of a discrete function φ is defined as the isosurface of its simplicial interpolant ̂f : [0,N]n → ℝm. With this simplicial definition, Γ is geometrically a piecewise intersection of a simplex and m hyperplanes. Γ is constructed as the union of simplices. The construction costs O(Nn) with a uniform grid and O(Nn-m log(N)) with a dyadic grid in numerical space and time. When n = m + 1 or m + 2, Γ is projected down into ℝ3 and can be visualized. For surface visualizations, a simple formula is presented calculating the normal vector field of the projection of Γ into R3, which gives light shadings.

AB - The level set method has been successfully used for moving interface problems. The final step of the method is to construct and visualize the isosurface of a discrete function φ : {0,...,N}n → ℝm. There have existed many practical isosurfacing algorithms when n = 3, m = 1 or n = 2, m = 1. Recently we have begun to see the development of isosurfacing algorithms for higher dimensions and codimensions. This paper introduces a unified theory and an efficient isosurfacing algorithm that works in arbitrary number of dimensions and codimensions. The isosurface Γ of a discrete function φ is defined as the isosurface of its simplicial interpolant ̂f : [0,N]n → ℝm. With this simplicial definition, Γ is geometrically a piecewise intersection of a simplex and m hyperplanes. Γ is constructed as the union of simplices. The construction costs O(Nn) with a uniform grid and O(Nn-m log(N)) with a dyadic grid in numerical space and time. When n = m + 1 or m + 2, Γ is projected down into ℝ3 and can be visualized. For surface visualizations, a simple formula is presented calculating the normal vector field of the projection of Γ into R3, which gives light shadings.

UR - http://www.scopus.com/inward/record.url?scp=0041355385&partnerID=8YFLogxK

U2 - 10.1016/S0021-9991(03)00275-4

DO - 10.1016/S0021-9991(03)00275-4

M3 - Article

AN - SCOPUS:0041355385

VL - 190

SP - 295

EP - 310

JO - Journal of Computational Physics

JF - Journal of Computational Physics

SN - 0021-9991

IS - 1

ER -