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Multi-field Visualization: Trait design and trait-induced merge trees

Danhua Lei, Jochen Jankowai, Petar Hristov, Hamish Carr, Leif Denby, Talha Bin Masood, Ingrid Hotz

TL;DR

This work advances multi-field visualization by integrating feature level sets (FLS) with topology-based analysis through trait-induced merge trees (TIMTs). TIMTs generalize merge trees to multi-variate and tensor data by analyzing the trait-induced distance field $h_T=d_T\circ f$, where $d_T(a)=\min_{t\in T} d_{\mathbb{A}}(a,t)$, enabling automatic feature extraction and persistent querying. It introduces Cartesian traits for intuitive, high-dimensional trait design and atom-traits via dictionary learning to auto-suggest simple, data-driven traits, both feeding into TIMT-based feature selection. The framework is implemented in Inviwo and demonstrated across five diverse case studies, illustrating robust, interactive, topology-guided exploration of complex multi-field and tensor data, with stability guarantees and practical insights for dictionary interpretability and feature extraction.

Abstract

Feature level sets (FLS) have shown significant potential in the analysis of multi-field data by using traits defined in attribute space to specify features in the domain. In this work, we address key challenges in the practical use of FLS: trait design and feature selection for rendering. To simplify trait design, we propose a Cartesian decomposition of traits into simpler components, making the process more intuitive and computationally efficient. Additionally, we utilize dictionary learning results to automatically suggest point traits. To enhance feature selection, we introduce trait-induced merge trees (TIMTs), a generalization of merge trees for feature level sets, aimed at topologically analyzing tensor fields or general multi-variate data. The leaves in the TIMT represent areas in the input data that are closest to the defined trait, thereby most closely resembling the defined feature. This merge tree provides a hierarchy of features, enabling the querying of the most relevant and persistent features. Our method includes various query techniques for the tree, allowing the highlighting of different aspects. We demonstrate the cross-application capabilities of this approach through five case studies from different domains.

Multi-field Visualization: Trait design and trait-induced merge trees

TL;DR

This work advances multi-field visualization by integrating feature level sets (FLS) with topology-based analysis through trait-induced merge trees (TIMTs). TIMTs generalize merge trees to multi-variate and tensor data by analyzing the trait-induced distance field , where , enabling automatic feature extraction and persistent querying. It introduces Cartesian traits for intuitive, high-dimensional trait design and atom-traits via dictionary learning to auto-suggest simple, data-driven traits, both feeding into TIMT-based feature selection. The framework is implemented in Inviwo and demonstrated across five diverse case studies, illustrating robust, interactive, topology-guided exploration of complex multi-field and tensor data, with stability guarantees and practical insights for dictionary interpretability and feature extraction.

Abstract

Feature level sets (FLS) have shown significant potential in the analysis of multi-field data by using traits defined in attribute space to specify features in the domain. In this work, we address key challenges in the practical use of FLS: trait design and feature selection for rendering. To simplify trait design, we propose a Cartesian decomposition of traits into simpler components, making the process more intuitive and computationally efficient. Additionally, we utilize dictionary learning results to automatically suggest point traits. To enhance feature selection, we introduce trait-induced merge trees (TIMTs), a generalization of merge trees for feature level sets, aimed at topologically analyzing tensor fields or general multi-variate data. The leaves in the TIMT represent areas in the input data that are closest to the defined trait, thereby most closely resembling the defined feature. This merge tree provides a hierarchy of features, enabling the querying of the most relevant and persistent features. Our method includes various query techniques for the tree, allowing the highlighting of different aspects. We demonstrate the cross-application capabilities of this approach through five case studies from different domains.
Paper Structure (31 sections, 5 equations, 18 figures)

This paper contains 31 sections, 5 equations, 18 figures.

Table of Contents

  1. Introduction
  2. Related work
  3. Data, traits, and merge trees
  4. Multi-variate data
  5. Feature level sets
  6. Merge trees
  7. Merge tree simplification
  8. Persistence
  9. Hypervolume
  10. Trait design
  11. Interactive trait specification
  12. Combinations using Cartesian traits
  13. Atom-traits using Dictionary learning
  14. Trait-induced merge trees
  15. Visualization and interaction
  16. ...and 16 more sections

Figures (18)

  • Figure 1: Pipeline. The first step is to design a trait representing the parameter settings of interest (Sec \ref{['Trait_design']}). For every vertex in attribute space, the distance resp similarity to the trait is calculated, which is then pulled back to the domain (Sec \ref{['FLS']}). The resulting distance field serves as input for the computation of the trait-induced merge tree (TIMT) (Sec \ref{['sec:TIMT']}) that then can be queried (Sec \ref{['sec:query_methods']}). Finally, the user can interact with the resulting domain segmentation via a legend or a slice of the data.
  • Figure 2: Simplification metrics: \ref{['subfig:metric_persistence']} The branch decomposition tree highlights the persistence of paired critical points $y_i$ and $x_i$, defined as the difference in their function values, $h' = g(y_i) - g(x_i)$. \ref{['subfig:metric_hypervolume']} Hypervolume is calculated by accumulating the volume associated with an arc of the tree multiplied by its height. While persistence relates only to the data range, hypervolume considers also the spatial embedding of the data.
  • Figure 3: Methods for manually specifying traits: (a) selecting point traits by choosing a glyph from a rendered tensor slice, (b) defining traits as lines or areas in parallel coordinates, and (c) drawing polygons within a bi-variate subspace.
  • Figure 4: Query methods. (a) original tree. Users may choose between segmentation of the tree based on (b) branch decomposition, (c) leaf nodes, (d) sub-trees, and (e) crown features.
  • Figure 5: Using TIMTs to explore two dictionaries $D_A$ (6 atoms) and $D_B$ (30 atoms) for the phantom data set. Images (a) and (b) show the volume renderings of selected atoms as point traits, highlighting segments with the highest similarity in color. Light green parts with low similarity values are rendered to provide context. (c) Atom 1 mainly represents the background, which is least similar to the trait shown. Image (d) demonstrates that the selected atom cannot be directly associated with one direction.
  • ...and 13 more figures

Theorems & Definitions (1)

  • Definition