Depth for Multi‐Modal Contour Ensembles

Computer Graphics Forum, vol. 43

Abstract

Abstract The contour depth methodology enables non‐parametric summarization of contour ensembles by extracting their representatives, confidence bands, and outliers for visualization (via contour boxplots) and robust downstream procedures. We address two shortcomings of these methods. Firstly, we significantly expedite the computation and recomputation of Inclusion Depth (ID), introducing a linear‐time algorithm for epsilon ID, a variant used for handling ensembles with contours with multiple intersections. We also present the inclusion matrix, which contains the pairwise inclusion relationships between contours, and leverage it to accelerate the recomputation of ID. Secondly, extending beyond the single distribution assumption, we present the Relative Depth (ReD), a generalization of contour depth for ensembles with multiple modes. Building upon the linear‐time eID, we introduce CDclust, a clustering algorithm that untangles ensemble modes of variation by optimizing ReD. Synthetic and real datasets from medical image segmentation and meteorological forecasting showcase the speed advantages, illustrate the use case of progressive depth computation and enable non‐parametric multimodal analysis. To promote research and adoption, we offer the contour‐depth Python package.

Authors 7

1. Nicolas F. Chaves-de-Plaza corresponding

   Holland Particle Therapy Center · Delft University of Technology

   Affiliation as printed

   HollandPTC Netherlands

   TU Delft Netherlands

   HollandPTC, Netherlands

   TU Delft, Netherlands

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

   HollandPTC Netherlands

   Department of Radiology, Leiden University Medical Center, Netherlands

   HollandPTC, Netherlands

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

   Leiden University Medical Center

   Affiliation as printed

   Department of Radiation Oncology Leiden University Medical Center Netherlands

   Department of Radiology Leiden University Medical Center Netherlands

   Department of Radiation Oncology, Leiden University Medical Center, Netherlands

   Department of Radiology, Leiden University Medical Center, Netherlands

4. René van Egmond

   Delft University of Technology

   Affiliation as printed

   TU Delft Netherlands

   TU Delft, Netherlands

5. Elmar Eisemann

   Delft University of Technology

   Affiliation as printed

   TU Delft Netherlands

   TU Delft, Netherlands

6. Anna Vilanova

   Eindhoven University of Technology

   Affiliation as printed

   TU Eindhoven Netherlands

   TU Eindhoven, Netherlands

7. Klaus Hildebrandt

   Delft University of Technology

   Affiliation as printed

   TU Delft Netherlands

   TU Delft, Netherlands