Inclusion Depth for Contour Ensembles

IEEE Transactions on Visualization and Computer Graphics, vol. 30, pp. 6560–6571

Abstract

Ensembles of contours arise in various applications like simulation, computer-aided design, and semantic segmentation. Uncovering ensemble patterns and analyzing individual members is a challenging task that suffers from clutter. Ensemble statistical summarization can alleviate this issue by permitting analyzing ensembles' distributional components like the mean and median, confidence intervals, and outliers. Contour boxplots, powered by Contour Band Depth (CBD), are a popular non-parametric ensemble summarization method that benefits from CBD's generality, robustness, and theoretical properties. In this work, we introduce Inclusion Depth (ID), a new notion of contour depth with three defining characteristics. First, ID is a generalization of functional Half-Region Depth, which offers several theoretical guarantees. Second, ID relies on a simple principle: the inside/outside relationships between contours. This facilitates implementing ID and understanding its results. Third, the computational complexity of ID scales quadratically in the number of members of the ensemble, improving CBD's cubic complexity. This also in practice speeds up the computation enabling the use of ID for exploring large contour ensembles or in contexts requiring multiple depth evaluations like clustering. In a series of experiments on synthetic data and case studies with meteorological and segmentation data, we evaluate ID's performance and demonstrate its capabilities for the visual analysis of contour ensembles.

Authors 6

1. Nicolas F. Chaves-de-Plaza

   Holland Particle Therapy Center · Delft University of Technology

   Affiliation as printed

   TU Delft, Delft, CD, The Netherlands

   HollandPTC, Netherlands

2. Prerak Mody Aachen Department of Radiology, Leiden University Medical Center

   Leiden University Medical Center · Holland Particle Therapy Center

   Affiliation as printed

   Department of Radiology, Leiden University Medical Center, Leiden, ZA, The Netherlands

   HollandPTC, Netherlands

3. Marius Staring Aachen Department of Radiology Department of Radiation Oncology Department of Radiology, Leiden University Medical Center

   Leiden University Medical Center

   Affiliation as printed

   Department of Radiology, Department of Radiation Oncology, Leiden University Medical Center, Leiden, ZA, The Netherlands

   Department of Radiology, Leiden University Medical Center, Netherlands

4. René van Egmond

   Delft University of Technology

   Affiliation as printed

   TU Delft, Delft, CD, The Netherlands

5. Anna Vilanova

   Eindhoven University of Technology

   Affiliation as printed

   TU Eindhoven, Eindhoven, AZ, The Netherlands

6. Klaus Hildebrandt

   Delft University of Technology

   Affiliation as printed

   TU Delft, Delft, CD, The Netherlands