Computational Complexity of Geometric Symmetry Detection in Graphs

Constructing a visually informative drawing of an abstract graph is a problem of considerable practical importance, and has recently been the focus of much investigation. Displaying symmetry has emerged as one of the foremost criteria for achieving good drawings. Linear-time algorithms are already known for the detection and display of symmetry in trees, outerplanar graphs, and embedded planar graphs. The central results of this paper show that for general graphs, however, detecting the presence of even a single axial or rotational symmetry is NP-complete. A number of related results are also established, including the #P-completeness of counting the axial or rotational symmetries of a graph

A Framework for Symmetric Part Detection in Cluttered Scenes

The role of symmetry in computer vision has waxed and waned in importance during the evolution of the field from its earliest days. At first figuring prominently in support of bottom-up indexing, it fell out of favor as shape gave way to appearance and recognition gave way to detection. With a strong prior in the form of a target object, the role of the weaker priors offered by perceptual grouping was greatly diminished. However, as the field returns to the problem of recognition from a large database, the bottom-up recovery of the parts that make up the objects in a cluttered scene is critical for their recognition. The medial axis community has long exploited the ubiquitous regularity of symmetry as a basis for the decomposition of a closed contour into medial parts. However, today's recognition systems are faced with cluttered scenes, and the assumption that a closed contour exists, i.e. that figure-ground segmentation has been solved, renders much of the medial axis community's work inapplicable. In this article, we review a computational framework, previously reported in Lee et al. (2013), Levinshtein et al. (2009, 2013), that bridges the representation power of the medial axis and the need to recover and group an object's parts in a cluttered scene. Our framework is rooted in the idea that a maximally inscribed disc, the building block of a medial axis, can be modeled as a compact superpixel in the image. We evaluate the method on images of cluttered scenes.Comment: 10 pages, 8 figure