Smoothed Local Symmetries and Their Implementation

Abstract

We introduce a novel representation of two-dimensional shape that we call smoothed local symmetries (SLS). Smoothed local symmetries represent both the bounding contour of a shape fragment and the region that it occupies. In this paper we develop the main features of the SLS representation and describe an implemented algorithm that computes it. The performance of the algorithm is illustrated for a set of tools. We conclude by sketching a method for determining the articulation of a shape into subshapes.

Area-Based Medial Axis of Planar Curves

“…But λ = 1 means C(s) and C(t(s)) coincide, so for us the interesting solution is λ = −1. This means that X is the mid-point of the segment (this result has also been obtained in Moisan (1998) and Brady and Asada (1984)): The area function d has a stationary point at (s, X), i.e., ∂d/∂s = 0, if and only if X is the midpoint of the segment from C(s) to C(t(s)).…”

“…The medial axis may be used to detect symmetries, a topic that has been the subject of extensive research in the computer vision community (e.g., Brady and Asada, 1984;Brooks, 1981;Cham and Cipolla, 1995;Ponce, 1989;van Gool et al, 1985;van Gool et al, 1996). In particular, affine skeletons may be used to detect skew symmetries: numerical experiments show that if the skeleton contains a straight branch, a portion of the curve has skew symmetry with respect to this line.…”

Optimal Contour Closure by Superpixel Grouping

“…Lowe's [10] early work on perceptual grouping was one of the first to develop a computational model for parallelism, collinearity, and proximity. Many computational models exist for symmetry-based grouping, including Brady and Asada [11], Cham and Cipolla [12], Saint-Marc et al [13], Ylä-Jääski and Ade [14], and more recently, Stahl and Wang [15]. One significant challenge faced by these systems is the complexity of pairwise contour grouping to detect symmetry-related contour pairs.…”

Multiresolution Shape Descriptions and their Applications in Medical Imaging

“…Several methods use this approach. (Brady and Asada, 1984). Arcs of tangent circles are used to define Process Inferred Axes (Leyton, 1986).…”

“…Several researchers have focused on boundary curvature because it reflects the bending of the object, an essential aspect of shape (Brady and Asada, 1984). Extreme points of curvature (local maxima and minima) on the boundary can be used to characterize shapes.…”