Probabilistic Marching Cubes

Abstract

In this paper we revisit the computation and visualization of equivalents to isocontours in uncertain scalar fields. We model uncertainty by discrete random fields and, in contrast to previous methods, also take arbitrary spatial correlations into account. Starting with joint distributions of the random variables associated to the sample locations, we compute level crossing probabilities for cells of the sample grid. This corresponds to computing the probabilities that the well-known symmetry-reduced marching cubes cases occur in random field realizations. For Gaussian random fields, only marginal density functions that correspond to the vertices of the considered cell need to be integrated. We compute the integrals for each cell in the sample grid using a Monte Carlo method. The probabilistic ansatz does not suffer from degenerate cases that usually require case distinctions and solutions of ill-conditioned problems. Applications in 2D and 3D, both to synthetic and real data from ensemble simulations in climate research, illustrate the influence of spatial correlations on the spatial distribution of uncertain isocontours.

Application of Uncertainty Modeling Frameworks to Uncertain Isosurface Extraction

“…We propose an uncertainty-based marching cubes (UMC) template that encompasses the essential concepts from the deterministic and probabilistic marching cubes algorithm [18] and is flexible enough to be adopted for various uncertainty modeling frameworks. Similar to the conventional marching cubes algorithm, the UMC template (algorithm) proceeds through the uncertain scalar field, and at each cell it quantifies how much the available information at the corners supports the incidence of a cell crossing.…”

“…(10) and (8) in Dempster-Shafer theory. -or in special cases, such as in the probabilistic case [18], directly from the available information at the corners.…”

“…Without loss of generality, we illustrate the idea for a 1D example (similar to the example presented in [18]) while the concepts extend to higher dimensions.…”

“…The edge-crossing UMeasure values can then be computed as discussed earlier. For instance, in probabilistic modeling [18], probability measure is used to compute the U-Measure(C) values as…”

“…In order to quantify and visualize the uncertainty in isosurfaces extracted from uncertain scalar fields, parametric probabilistic models have been used to approximate Level-Crossing Probabilities (LCP) [15,16]. The concept of levelcrossing probabilities has been deployed to extend the conventional marching cubes algorithm, the predominant isosurface visualization scheme, for probabilistic modeling of uncertainty in scalar fields [17,18].…”