International Journal for Uncertainty Quantification, 3 (2): 101–117 (2013) APPROXIMATE LEVEL-CROSSING PROBABILITIES FOR INTERACTIVE VISUALIZATION OF UNCERTAIN ISOCONTOURS Kai Pöthkow, * Christoph Petz, & Hans-Christian Hege Zuse Institute Berlin, Takustrasse 7, 14195 Berlin, Germany Original Manuscript Submitted: 09/02/2011; Final Draft Received: 07/12/2012 A major method for quantitative visualization of a scalar field is depiction of its isocontours. If the scalar field is afflicted with uncertainties, uncertain counterparts to isocontours have to be extracted and depicted. We consider the case where the input data is modeled as a discretized Gaussian field with spatial correlations. For this situation we want to compute level-crossing probabilities that are associated to grid cells. To avoid the high computational cost of Monte Carlo integrations and direction dependencies of raycasting methods, we formulate two approximations for these probabilities that can be utilized during rendering by looking up precomputed univariate and bivariate distribution functions. The first method, called maximum edge crossing probability, considers only pairwise correlations at a time. The second method, called linked-pairs method, considers joint and conditional probabilities between vertices along paths of a spanning tree over the n vertices of the grid cell; with each possible tree an n-dimensional approximate distribution is associated; the choice of the distribution is guided by minimizing its Bhattacharyya distance to the original distribution. We perform a quantitative and qualitative evaluation of the approximation errors on synthetic data and show the utility of both approximations on the example of climate simulation data. KEY WORDS: scalar fields, spatial uncertainty, level-crossing probabilities, approximation, visualization 1. INTRODUCTION Data acquired by measurements or simulations are always affected by uncertainty. Important sources of uncertainty include the measurement process, parameter selection, discretization and quantization of continuous quantities, as well as numerical simulations with finite precision. Modeling and visualization techniques taking care of uncertainties therefore are of interest for a variety of applications. In this paper we revisit the extraction of uncertain counterparts to isocontours. In previous work we have proposed a method to compute level-crossing probabilities for cells in Gaussian fields defined on grids, considering the spatial correlation structure of the input data [1]. A disadvantage of this approach is the high computational cost of the Monte Carlo (MC) integration. For the specific case of Gaussian fields with exponential correlation functions, Pfaffelmoser et al. [2] presented a raycasting approach that computes first-crossing probabilities along rays using lookup tables for fast evaluation. The results of this method depend on the viewing direction. Our aim is to improve the computation of local cellwise level-crossing probabilities that arise from arbitrary spatial correlations and that are independent of the viewing direction, i.e., are objective, in the sense that they are independent of the observer. Since the input data are usually given on some grid, it is a natural choice to consider grid cells and to compute cell-related probabilities. The numerical computation of high-dimensional integrals in general is expensive, both with deterministic and MC methods. There are two ways to deal with this problem: Either utilize specific properties of the problem to facilitate the computation, or find good and fast approximations of the integrals. Here we consider the latter approach: We * Correspond to Kai P ¨ othkow, E-mail: poethkow@zib.de, URL: http://www.zib.de/visual 2152–5080/13/$35.00 c  2013 by Begell House, Inc. 101