Several natural and man-made objects exhibit symmetry in different forms, both in their geometry and in the material distribution. The study of symmetry plays an important role in understanding both the structure of these objects and their physical properties. The notion of symmetry with respect to the geometry of an object or domain is well understood. In this talk, I will introduce the problem of symmetry detection in a scalar field. This refers to identifying regions within the domain of a scalar field that remain invariant under geometric transformations with respect to both the domain geometry and the scalar values. Symmetry detection in scientific data is still a nascent area of research and existing methods that detect symmetry are either not robust in the presence of noise or are computationally costly.

I will present a recently developed method that detects symmetry in, potentially noisy, 3D scalar fields without an explicit construction of the corresponding geometric transformation. The key ingredient in the algorithm is a data structure called the augmented extremum graph that captures both topological and geometric properties of the scalar field. I will also briefly discuss other methods that we have developed to identify symmetric regions and also demonstrate applications to symmetry-aware volume rendering, isosurface extraction, query-based exploration, linked selection and volume editing.