For a function of N variables, its ridges are a set of curves whose points are local maxima in N − 1 dimensions.
In this respect, the notion of ridge points extends the concept of a local maximum.
The primary motivation for the creation of ridge detection and valley detection procedures has come from image analysis and computer vision and is to capture the interior of elongated objects in the image domain.
There have also been attempts to capture the shapes of objects by graph-based representations that reflect ridges, valleys and critical points in the image domain.
Such representations may, however, be highly noise sensitive if computed at a single scale only.
Because scale-space theoretic computations involve convolution with the Gaussian (smoothing) kernel, it has been hoped that use of multi-scale ridges, valleys and critical points in the context of scale space theory should allow for more a robust representation of objects (or shapes) in the image.
With appropriately defined concepts, ridges and valleys in the intensity landscape (or in some other representation derived from the intensity landscape) may form a scale invariant skeleton for organizing spatial constraints on local appearance, with a number of qualitative similarities to the way the Blum's medial axis transform provides a shape skeleton for binary images.
in the transformed coordinate system is zero if we choose Then, a formal differential geometric definition of the ridges of
direction parallel to the image gradient where it can be shown that this ridge and valley definition can instead be equivalently[4] written as where and the sign of
Experiments show that the scale parameter of the Gaussian pre-smoothing kernel must be carefully tuned to the width of the ridge structure in the image domain, in order for the ridge detector to produce a connected curve reflecting the underlying image structures.
In the literature, a number of different approaches have been proposed based on this idea.
Then, for a two-dimensional image, a scale-space ridge is the set of points that satisfy where
-normalized derivatives is essential here, since it allows the ridge and valley detector algorithms to be calibrated properly.
By requiring that for a one-dimensional Gaussian ridge embedded in two (or three dimensions) the detection scale should be equal to the width of the ridge structure when measured in units of length (a requirement of a match between the size of the detection filter and the image structure it responds to), it follows that one should choose
is a general purpose ridge strength measure with many applications such as blood vessel detection and road extraction.
has been used in applications such as fingerprint enhancement,[6] real-time hand tracking and gesture recognition[7] as well as for modelling local image statistics for detecting and tracking humans in images and video.
[8] There are also other closely related ridge definitions that make use of normalized derivatives with the implicit assumption of
The notion of ridges and valleys in digital images was introduced by Haralick in 1983[10] and by Crowley concerning difference of Gaussians pyramids in 1984.
-normalized derivatives and scale-space ridges defined from local maximization of the appropriately normalized main principal curvature of the Hessian matrix (or other measures of ridge strength) over space and over scale.
[22] A review of vessel extraction techniques has been presented by Kirbas and Quek.
[23] In its broadest sense, the notion of ridge generalizes the idea of a local maximum of a real-valued function.
Presumably this relaxation allows the set of points which satisfy the criteria, which we will call the ridge, to have a single degree of freedom, at least in the generic case.
Notice that the above can be modified to generalize the idea to local minima and result in what might call 1-dimensional valley curves.
if the following conditions hold: In many ways, these definitions naturally generalize that of a local maximum of a function.
Properties of maximal convexity ridges are put on a solid mathematical footing by Damon[1] and Miller.
[25] The following definition can be traced to Fritsch[26] who was interested in extracting geometric information about figures in two dimensional greyscale images.
is a point on the maximal scale ridge if and only if The purpose of ridge detection is usually to capture the major axis of symmetry of an elongated object,[citation needed] whereas the purpose of edge detection is usually to capture the boundary of the object.
-coordinate system state that the gradient magnitude of the scale-space representation, which is equal to the first-order directional derivative in the
should be negative, i.e., Written out as an explicit expression in terms of local partial derivatives
Notably, the edges obtained in this way are the ridges of the gradient magnitude.