Abstract
This paper introduces a procedure for the calculation of the vertex positions in Marching-Cubes-like surface reconstruction methods, when the surface to reconstruct is characterised by a discrete indicator function. Linear or higher order methods for the vertex interpolation problem require a smooth input function. Therefore, the interpolation methodology to convert a discontinuous indicator function into a triangulated surface is non-trivial. Analytical formulations for this specific vertex interpolation problem have been derived for the 2D case by Manson et al. [Eurographics (2011) 30, 2] and the straightforward application of their method to a 3D case gives satisfactory visual results. A rigorous extension to 3D, however, requires a least-squares problem to be solved for the discrete values of a symmetric neighbourhood. It thus relies on an extra layer of information, and comes at a significantly higher cost. This paper proposes a novel vertex interpolation method which yields second-order-accurate reconstructed surfaces in the general 3D case, without altering the locality of the method. The associated errors are analysed and comparisons are made with linear vertex interpolation and the analytical formulations of Manson et al. [Eurographics (2011) 30, 2].
Highlights
IN computer graphics [1], [2] and computational physics [3], it is common to represent a closed subset V of the d-dimensional space with a discontinuous indicator function x : Rd ! R defined as xðxÞ 1⁄41 if x 2 V; 0 if x 2= V: (1)If the domain encompassing V is discretised into an ensemble M of small cuboids, to each cell K is associated the local fraction of volume occupied by K \ V
This paper proposes a method to accurately position the vertices of a surface reconstructed from discrete indicator function (DIF) values in a general 3D case
Prior to iteratively positioning the vertices, a linear approximation of the surface that complies with the local volume fraction is computed in each interfacial cell1 of M, using the analytical solutions of Scardovelli and Zaleski [19]
Summary
IN computer graphics [1], [2] and computational physics [3], it is common to represent a closed subset V of the d-dimensional space with a discontinuous indicator function x : Rd ! R defined as. This discrete field g, illustrated, is referred to as the volume fraction or the discrete indicator function (DIF) field. Both denominations are used in this paper. This paper proposes a method to accurately position the vertices of a surface reconstructed from DIF values in a general 3D case. It relies on the computation of local volume-fraction-compliant piecewise-linear surface approximations, based on which the positions of the reconstructed surface vertices are iteratively corrected. The proposed method conserves the locality of the interpolation procedure and does not require to solve for a least-squares problem
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