Data visualization has become indispensable for efficient interpretation of large-scale data generated across diverse scientific domains, such as biomedical imaging and climate studies. Many critical decisions directly rely on the quality of data visualizations. Inaccuracies in visualizations cannot be averted due to uncertainties inherent in underlying data and non-linear transformations of data caused by the stages of visualization pipeline. The uncertainty in the final visualizations can adversely impact the decision-making process. The accurate quantification of uncertainties in data visualizations has, therefore, been recognized as the top research challenge for minimizing risks associated with scientific decisions.

In our work, we statistically quantify positional variations in features of uncertain data for two applications. First, we study the interaction of the marching cubes algorithm with uncertain data for probabilistic quantification of positional variations in level-set extractions. Second, we study spatial variability in objects of known geometry arising from their finite-resolution imaging. Specifically, we perform our second study on electrodes of fixed geometry used for deep brain stimulation (DBS) surgery. Our uncertainty visualizations for level-set extraction and DBS electrode localization confirm the significance of incorporating statistical error analysis into computational models for visualization applications.

**Quantification and Visualization of Uncertainty in Level-Set Extraction for Uncertain Data**

Level-set visualizations is one of the key visualization techniques for gaining insight into complex datasets, e.g. medical datasets. In this project, we address the challenge of extracting level sets using the marching cubes/squares algorithm in *uncertain* scalar field data. A closed-form statistical approach was derived for the probabilistic computation of topology and geometry steps of the marching cubes/squares. The uncertainty in topology and geometry is interpreted through the visualization of the probabilities of decisions.

**Fig: **The visualization of level-set extraction for the tangle function. The leftmost image visualizes level sets for the groundtruth volume. The volume is then mixed with noise. The level-set extraction with non-parametric noise assumption is better (more robust to the outliers) than the parametric noise assumption. Red, blue, and green indicate regions of relatively high, moderate, and low spatial uncertainty, respectively.

**Visualization of Positional Uncertainty in Deep Brain Stimulation (DBS) Electrodes**

In this research application, I proposed a statistical method for quantifying and visualizing positional uncertainty in DBS electrodes for a finite-resolution brain imaging (MRI/CT).

**Fig:** The sample visualization of positional uncertainty in DBS contacts/electrodes. The four colors indicate the four contacts of Medtronic 3387. The contacts lie within the inner surface with 10% confidence and they lie within the outer surface with 40% confidence.