Estimation and Visualization of Isosurface Uncertainty from Linear and High-Order Interpolation Methods

TL;DR

This paper addresses the problem of quantifying and visualizing isosurface uncertainty in MC-based visualization, focusing on errors arising from interpolation methods rather than data noise. It introduces a closed-form edge-crossing error estimator derived from polynomial interpolation and a hidden feature detection/reconstruction approach that uses cubic interpolation on subdivided cells. An integrated visualization tool enables interactive exploration, local querying, and cross-method comparison between linear, cubic, and WENO interpolations. The results on synthetic and real-world datasets demonstrate where linear interpolation fails, how higher-order methods can improve accuracy, and how hidden features can be recovered to produce more reliable isosurfaces.

Abstract

Isosurface visualization is fundamental for exploring and analyzing 3D volumetric data. Marching cubes (MC) algorithms with linear interpolation are commonly used for isosurface extraction and visualization. Although linear interpolation is easy to implement, it has limitations when the underlying data is complex and high-order, which is the case for most real-world data. Linear interpolation can output vertices at the wrong location. Its inability to deal with sharp features and features smaller than grid cells can lead to an incorrect isosurface with holes and broken pieces. Despite these limitations, isosurface visualizations typically do not include insight into the spatial location and the magnitude of these errors. We utilize high-order interpolation methods with MC algorithms and interactive visualization to highlight these uncertainties. Our visualization tool helps identify the regions of high interpolation errors. It also allows users to query local areas for details and compare the differences between isosurfaces from different interpolation methods. In addition, we employ high-order methods to identify and reconstruct possible features that linear methods cannot detect. We showcase how our visualization tool helps explore and understand the extracted isosurface errors through synthetic and real-world data.

Figures (10)

• Figure 1: Comparison between measured and approximate error. The first row corresponds to the measured error and the second to the approximated error. Each column from left to right corresponds the Tangle ( \ref{['subfig:tangle_512x512x512_32x32x32']} and \ref{['subfig:tangle_Approx_32x32x32']} with $k=0.1$), Torus ( \ref{['subfig:torus_512x512x512_64x64x64']} and \ref{['subfig:torus_Approx_64x64x64']} with $k=0.0$), Marschner and Lobb (\ref{['subfig:marschnerlobb_512x512x512_32x32x32']} and \ref{['subfig:marschnerlobb_Approx_32x32x32']} with $k=0.5$), and Teardrop (\ref{['subfig:teardrop_512x512x512_32x32x32']} and \ref{['subfig:teardrop_Approx_32x32x32']} with $k=-0.001$) and Tubey (\ref{['subfig:tubey_512x512x512_32x32x32']} and \ref{['subfig:tubey_Approx_32x32x32']} with $k=0.0$) examples. Our approximated errors show similar patterns to the measured errors. In most cases, it slightly overestimates the errors.
• Figure 2: Comparison of measured (in green), our estimated (in blue), and standard approach for error approximation (light blue). The columns from left to right show errors for the Tangle (\ref{['subfig:tangle_rms_error']}), Torus (\ref{['subfig:torus_rms_error']}), Marschner and Lobb (\ref{['subfig:marschnerlobb_rms_error']}), Teardrop (\ref{['subfig:teardrop_rms_error']}), and Tubey (\ref{['subfig:tubery_rms_error']}). Our error estimation in \ref{['eq:approx_linear_error']} is much closer to the measured error compared to the standard approach in \ref{['eq:error_bound_approx']}.
• Figure 3: Edge-crossing uncertainty. The underlying function and the linear interpolation are shown in black and orange, respectively. The black line segment with the positive and negative nodes is the edge and the blue line indicates the target isovalue. The red double arrow indicates the approximation error. The isovalue is indicated by the blue horizontal line.
• Figure 4: Hidden feature recovery in 1D and 2D.\ref{['subfig:hidden_feature_1D']} shows an example of a hidden feature between $i$ and $i+1$ that can be detected by noting that $U[i-1, i+1]*U[i+1, i+2] < 0$, meaning the slopes surrounding the hidden features have a different sign. Our method subdivides the cell at the orange dotted line to recover the hidden feature. The isovalue is indicated by the blue horizontal line. \ref{['subfig:hidden_feature_2D']} shows a 2D example with hidden features that can be recovered by refining the cell. The orange curves are the isocontour inside the cell. MC will miss the contour because all four corners have the same sign. Our method identifies the hidden feature and subdivides the cell at the dotted black lines.
• Figure 5: Crack patching. \ref{['subfig:crack']} shows an isosurface crack caused by the refined cell. The cell on the left is divided according to our algorithm, while the cell on the right is from the original Marching Cubes. The extracted vertices on the interface of two cells are mismatched. In \ref{['subfig:crack_free']} the crack is fixed by (1) matching the boundaries of the blue triangle in \ref{['subfig:crack']} to the boundary of the blue line in \ref{['subfig:crack']}, (2) connecting the edges of the new polygon to the center of the triangle in \ref{['subfig:crack']} to form the crack-free triangulated patch..