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Symmetry in Scalar Fields

People involved: Dilip Thomas and Talha Bin Masood

A scalar field is a real valued function defined on a domain of interest. Scalar field data may be generated from experimental and computational methods in different disciplines and are used to represent physical quantities measured over a domain of interest. Study of symmetric or repeating patterns in scalar fields is important in scientific data analysis because it gives deep insights into the properties of the underlying phenomenon. The top row of Figure 1 shows three examples of symmetry in scalar fields. We study the problem of detecting symmetric or repeating patterns in scalar fields and its applications to feature exploration and visualization of scalar fields. The bottom row of Figure 1 shows the symmetric regions extracted using the solutions we propose.

Figure 1. Symmetry is ubiquitous in scientific data sets as seen in the volume rendered images of (top left) the temperature distribution in a Taylor-Green vortex flow simulation, (top center) a cryo-electron microscopy image of a virus, and (top right) a CT scan image of a pair of knees. While human beings are endowed with special cognitive ability that makes symmetry detection easy, automatic detection of symmetry is a challenging task for computers. We propose algorithms to identify symmetry and show different applications that exploit symmetry information to improve visualization and exploration of scalar field data. Symmetry-aware transfer function highlights the symmetric regions detected by (bottom left) identifying similar subtrees of the contour tree, (bottom center) comparing distances between extrema using extremum graph, and (bottom right) clustering contours in a high dimensional shape descriptor space.

Motivation and Overview of Results

Symmetry Detection by Identifying Similar Subtrees in Contour Trees

Symmetry Detection by Distance Comparison using Extremum Graphs

Symmetry Detection by Clustering Contours

Symmetry Detection by Clustering Point Pairs



  1. Dilip Mathew Thomas and Vijay Natarajan.
    Symmetry in scalar field topology.
  2. Dilip Mathew Thomas and Vijay Natarajan.
    Detecting symmetry in scalar fields using augmented extremum graphs.
  3. Dilip Mathew Thomas and Vijay Natarajan.
    Multiscale symmetry detection in scalar fields by clustering contours.
  4. Talha Bin Masood, Dilip Mathew Thomas and Vijay Natarajan.
    Scalar field visualization via extraction of symmetric structures.


  1. SymmteryViewer
    A web-server for symmetry computation within EMDB data.


Contact dilthoms [at] gmail [dot] com for the software.