Differential Geometry in Edge Detection: Accurate Estimation of Position, Orientation and Curvature.

The vast majority of edge detection literature has aimed at improving edge recall and precision, with relatively few addressing the accuracy of edge orientation estimates which are often based on gradient. We show that first-order estimates of orientation can have significant error and this can be remedied by employing Third-Order estimates. This paper aims at estimating differential geometry attributes of an edge, namely, localization, orientation, and curvature, as well as edge topology, and develop robust numerical techniques in gray-scale and color images, applicable to a variety of popular edge detectors, such as gradient-based, gPb and SE. Second, a combinatorial model of edge grouping in a small neighborhood is developed to capture all geometrically consistent grouping called curvels, which establish: (i) edge topology in the form of potential links between an edge and other edges; (ii) an accurate curvature estimate for each possible grouping, whose performance is comparable to methods which use global and multi-scale methods; (iii) a more accurate localization of an edge. These have been evaluated using four distinct methodologies (i) traditional human annotated datasets; (ii) using coherence measure; (iii) stability analysis under visual perturbation, and (iv) utilitarian evaluation, and show meaningful improvements.