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.

1 Introduction

Edge detection has been a fundamental and core problem in computer vision since its inception [1], as a part of the segmentation-then-recognition paradigm where the image was to be first parsed into segments and then subjected to recognition. Under the contour-based version of this paradigm, edges need to be extracted first, linked into contours, and contours are closed to form regions. Edge detectors have evolved significantly from initial derivative operators, from the Perwitt and Sobel operators to Marr and Hildreth's zero-crossings of Laplacian of Gaussians, to the Canny edge detector, SUSAN and a host of others. Aside from a brief period where the elusiveness of segmentation led to the predominance of feature-based approaches, which bypass segmentation, hundreds of approaches to edge detection have been developed. While the majority of these approaches aim at improving the performance of edge detectors [2], or deal with specialized situations such as highly noisy images [3], they generally pay relatively less attention to finding the orientation or curvature of an edge, with few exceptions.