Ridge Detection - Relations Between Edge Detection and Ridge Detection

Relations Between Edge Detection and Ridge Detection

The purpose of ridge detection is usually to capture the major axis of symmetry of an elongated object, whereas the purpose of edge detection is usually to capture the boundary of the object. However, some literature on edge detection erroneously includes the notion of ridges into the concept of edges, which confuses the situation.

In terms of definitions, there is a close connection between edge detectors and ridge detectors. With the formulation of non-maximum as given by Canny, it holds that edges are defined as the points where the gradient magnitude assumes a local maximum in the gradient direction. Following a differential geometric way of expressing this definition, we can in the above-mentioned -coordinate system state that the gradient magnitude of the scale-space representation, which is equal to the first-order directional derivative in the -direction, should have its first order directional derivative in the -direction equal to zero

while the second-order directional derivative in the -direction of should be negative, i.e.,

.

Written out as an explicit expression in terms of local partial derivatives, ..., this edge definition can be expressed as the zero-crossing curves of the differential invariant

that satisfy a sign-condition on the following differential invariant

(see the article on edge detection for more information). Notably, the edges obtained in this way are the ridges of the gradient magnitude.