Recent improvements in methods for acquiring and generating 3D shape data over the last few decades have motivated the need for ever complex tools to analyze and edit such information. As a result, in the last few years, numerous methods have been introduced to improve the ways in which computers understand shape. These methods vary in the type of data they measure, ranging from low-level, local geometry information to high-level semantic information. Symmetry has long been known as an important cue for non-local human recognition of shape, and as such has been a key component in Vision and Graphics applications. A second important feature of symmetry is that it is prevalent in real-world models. Pipe and gear models contain rotational symmetries, buildings and square objects contain 4-point symmetry, and nearly every natural or man-made object will contain some reflectional symmetry. This prevalence is important because this means that applications relying on symmetry information will work on a large range of models. In this thesis, we introduce the notion of a Symmetry Transform, a measure of "the amount of" or "the degree of" symmetry present in a 3D shape, under some class of transformations. This can be the set of all plane-reflectional symmetries, the set of all point reflections or some arbitrary set of transformations. We concentrate on point and reflection symmetry transforms, providing some theoretical reasoning for the transforms. We show efficient means for computing them, discuss storing them, and analyze some of their properties, including noise resistance, continuous variation under deformation, and stability with missing data. While computing the transform of an entire model can be useful for a general understanding of shape, strong symmetries (such as the center of an ellipse or the main reflection planes of a rectangle), are easily identified by humans and represent an important subset of the transform. In this thesis we provide a method for efficiently extracting these "principal symmetries", and discuss some of their properties. Finally, the symmetry transform provides mid-level information about the nature of a 3D object that can be utilized in a wide range of applications. We explore how symmetry may be used to improve alignment, matching, viewpoint selection, remeshing, and segmentation.

@phdthesis{Podolak:2008:TST,
   author = "Joshua Podolak",
   title = "The Symmetry Transform and its Applications",
   school = "Princeton University",
   year = "2008",
   month = sep
}