## 1 Background

We briefly discuss the vertex-based classification [AthawaleSakhaeeEntezari2016] and probabilistic marching cubes [PothkowWeberHege2011] frameworks, which are the building blocks of our proposed uncertainty quantification framework.

### 1.1 Vertex-Based Classification

The vertex-based classification predicts the most probable MS or MC topology case for each cell of a Cartesian grid, where uncertain data are assumed to be sampled from independent probability distributions. In Fig. 1, we denoted the fixed data at cell vertices with a variable

. When data have uncertainty, we denote the data at cell vertices by a random variable

. Letbe the probability distribution at each vertex estimated from sample data. For the isovalue

, let , i.e., the probability of a cell vertex (x,y) attaining a positive vertex sign. Similarly, let . In a vertex-based classification, the vertex is predicted as positive if ; otherwise, it is predicted as negative. The predicted signs, therefore, indicate the single most probable MS or MC topology case for each cell of an uncertain scalar field.### 1.2 Probabilistic Marching Cubes

The probabilistic marching cubes estimates the level-crossing probability for each cell of a Cartesian grid, where uncertain data are assumed to be sampled from multivariate Gaussians. For a 2D version of the probabilistic marching cubes, i.e., the probabilistic marching squares, let denote a random variable representing uncertain data at the 2D cell vertices. The random variable is assumed to have

a multivariate Gaussian distribution with the sample mean

and sample covariance matrix . The samples are then drawn from the distribution . If the level-set with isovalue crosses a cell for number of samples, then the level-crossing probability for the cell is estimated as . The same approach is extendable to 3D. Note that the higher value of a sample count provides a more reliable estimation of the level-crossing probability.## 2 Methods

We now describe our framework for the uncertainty quantification and visualization of the level-set topology cases. For simplicity, we limit our descriptions to the 2D MS algorithm, but they are directly applicable to the 3D MC algorithm.

### 2.1 Computing Topology Case Probability Distribution

In the uncertainty quantification step, we characterize the uncertainty of the MS topology cases by computing their probability distribution. For the independent random field assumption, we leverage the vertex-based classification framework (Sec. 1.1) to compute the topology case probability distribution. First, we compute and for each cell vertex. We then compute the probability for each of the MS topology cases per cell. For example, the probability of , , and being positive and being negative is equal to the product because of the independence assumption.

For the correlated random field assumption, we leverage the probabilistic marching cubes framework (Sec. 1.2) to compute the topology case probability distribution. First, we draw samples from a multivariate Gaussian distribution . Next, we empirically compute the histogram of topology cases depending upon the topology case observed for each of the samples. Note that the Monte Carlo sampling for the correlated noise assumption results in approximate and expensive computations unlike the closed-form and fast computations for the independent noise models.

### 2.2 Topology Case Count Visualization

In the topology case count technique, we first compute the topology case probability distribution per grid cell, as described in Sec 2.1. We then derive the topology count field in which we count the number of topology cases per cell that have a probability greater than the user-specified lower probability threshold , where . Setting a lower threshold provides users the flexibility to study uncertainty among topology cases with relatively high probability. Let

denote a discrete random variable representing the

MS topology cases for a 2D cell , and denote the topology case probability distribution for the cell . Mathematically, the topology count for cell is equal to , where is the indicator function. Finally, we visualize the topology count field via colormapping. Note that for our experiments in Sec. 3, we set .### 2.3 Entropy-Based Uncertainty Visualization

In the entropy-based technique, we first compute the topology case probability distribution per grid cell, as described in Sec 2.1. We then compute the Shannon entropy [Shannon1948] E(q) of the topology case probability distribution for each cell as: . Finally, we visualize the entropy field via colormapping. The entropy of a topology case probability distribution quantifies the level of randomness of the topology cases within each cell, which may not be captured by the topology case count technique. Low entropy implies the relatively more deterministic nature of the topology cases, whereas high entropy implies the relatively more uncertain nature of the topology cases. Our entropy-based approach is inspired by the similar entropy-based approach proposed in [Athawale:2020:AJPW] for the visualization of uncertainty in Morse complexes. Our entropy-based framework can be used to identify the isovalues that exhibit relatively high or low topological uncertainty. Specifically, we visualize a boxplot of the entropy of grid cells with nonzero entropy/uncertainty for different isovalues to compare the topological uncertainty distribution of isovalues.

## 3 Results

We demonstrate the effectiveness of our topology count and entropy-based level-set uncertainty visualizations in Fig. 2 via a synthetic experiment. The level-set for the Ackley dataset [Ackley1987] is visualized in Fig. 2a. We mix the dataset with uniform-distributed noise samples to generate an ensemble. Figs. 2b-f visualize the results for the ensemble using multiple uncertainty visualization techniques for the independent uniform noise assumption, in which the mean and width of a distribution per vertex are estimated from the ensemble.

Fig. 2b visualizes a spaghetti plot [potter2009] of level-sets, in which the orange and pink boxes enclose the positions of the relatively high and low spatial variability of the level-sets, respectively. Fig. 2c visualizes the most probable level-set extracted using the vertex-based classification (Sec. 1.1) with colormapping based on the Ilerp uncertainty [AthawaleEntezari2013]

. The relatively high Ilerp variance (mapped to red) is oberved inside the orange box in Fig.

2c. Fig. 2d visualizes a result for the probabilistic marching squares [PothkowWeberHege2011] (Sec. 1.2), in which the level-crossing probabilities are colormapped.Figs. 2e-f visualize the results of our topology case count and entropy-based techniques, respectively (Sec. 2). The yellow regions in Fig. 2e indicate the cells that have possible MS topology cases with nonzero probability of occurrence, i.e., high topological uncertainty, across the ensemble. In Fig. 2f, the high entropy mapped to yellow implies a relatively high level of randomness of level-set topology. Our proposed uncertainty visualizations clearly highlight the positions of relatively high topological uncertainty (yellow regions), which are not easily observed by visualizing the level-crossing probability (Fig. 2d). Fig. 3 visualizes entropy boxplots for the isovalues sampled in the range for the Ackley ensemble. In Fig. 3, the isovalues near exhibit relatively higher median entropy (orange segments) than the isovalues near .

In Fig. 4, we demonstrate the comparison of independent and correlated noise models and an application of interactive probability distribution queries [PotterKirbyXiu2012]. Specifically, we analyze the uncertainty in level-sets for the velocity magnitude fields derived from the wind ensemble dataset [Vitart2017] with ensemble members. Figs. 4a-c visualize the spaghetti plot, Ilerp uncertainty, and level-crossing probabilities similar to the visualizations in Figs. 2b-d for the Ackley ensemble.

Figs. 4d-e visualize the topology case counts for the independent and multivariate Gaussian noise models, respectively. The positions of relatively high topological variations are clearly highlighted (yellow regions) in Fig. 4d, which are not easily observed in Fig. 4c. The number of possible topological cases per cell with nonzero probability of occurrence is reduced or the topology becomes more deterministic for the multivariate Gaussian assumption in Fig. 4e. For the multivariate Gaussian assumption, we used a sample count for Monte Carlo sampling since increasing a sample count beyond did not visually alter the results significantly.

Fig. 4f and Fig. 4h visualize the entropy of the topological distributions for the independent and multivariate Gaussian noise assumptions, respectively. We investigate the probability distributions of the MS topology cases, interactively [PotterKirbyXiu2012], at the pixels marked with white circles in Fig. 4f and Fig. 4h and visualize them in Fig. 4g and Fig. 4i, respectively. In Fig. 4g, the high entropy of distributions is evidenced by the relatively high probability of four topology configurations , , , and , where denotes a positive and denotes a negative vertex. In contrast, the topology becomes more deterministic in Fig. 4i with relatively high probability for the topology cases , , and .

In Fig. 5, we apply our proposed uncertainty visualizations to a 3D hixel data [ThompsonLevineBennett2011] assuming the independent Gaussian-distributed uncertainty. The hixel technique produces a reduced representation of the original data by partitioning these data into blocks and summarizing each block with a probability distribution. In our example, we derived the hixel data from the stag beetle dataset [dataset-stagbeetle] with resolution . For the hixel-based representation, we partitioned the dataset into blocks of size and summarized each block with a Gaussian distribution. The hixel dataset, therefore, has a resolution of

, where each block stores the mean and standard deviation of a Gaussian distribution.

The hixel-based reduced representation comes at the cost of increased uncertainty in the data and, hence, the level-set positions. Fig. 5a visualizes the level-set extracted from the original high-resolution stag beetle dataset at isovalue . Fig. 5b visualizes the most probable isosurface extracted using the vertex-based classification [AthawaleEntezari2013] for the hixel dataset. The relatively high sensitivity of the beetle leg topology to noise results in the breaking of the beetle leg in Fig. 5b. Fig. 5c visualizes the result for the probabilistic marching cubes, in which the positions of high and low topological uncertainty are not clearly separated. Our proposed uncertainty visualizations in Fig. 5d-f clearly detect the relatively high sensitivity of the beetle leg topology to noise, as visualized with the red regions. In Fig. 5f, we overlay the most probable level-set (gray) with the level-set extracted from the entropy volume (red) for the entropy isovalue . Thus, the red isosurface encloses the positions that have relatively high topological uncertainty.

## 4 Conclusion and Future Work

In this paper, we study uncertainty arising in the topology cases of the MS and MC algorithms for level-set visualizations when the data uncertainty is modeled with independent and correlated noise distributions. Specifically, we propose the topology case count and entropy-based techniques for uncertainty quantification and visualization of the topology cases. We demonstrate the effectiveness of our proposed uncertainty visualizations by comparing the results with previously proposed spaghetti plots [potter2009], probabilistic marching cubes [PothkowWeberHege2011], and Ilerp uncertainty visualizations [AthawaleEntezari2013].

Our proposed uncertainty quantification framework has a few limitations. First, we assume no correlation among the 16 MS (or 256 MC) topology cases that are distinguished based on the cell vertex signs. A few topology cases, however, are rotated or flipped versions of other MS topology cases. Considering such correlations for uncertainty quantification could be interesting future work. Further, a specific combination of cell vertex signs may correspond to multiple possible topologies within a grid cell [Nielson:1991:TAD, Nielson:2003:OMC], which we plan to take into account for uncertainty quantification in the future. Currently, we limit interactive probability queries [PotterKirbyXiu2012] to explore the MS topological uncertainty. Applying such a framework to the MC topology cases, however, is nontrivial and impractical. We would like to study the MC topology cases further. In our study, we restrict the topology uncertainty analysis for each cell of a scalar grid similar to the MS and MC algorithms. We would like to expand this analysis to take into account correlations with a local neighborhood.