Ridge Detection - Computation of Variable Scale Ridges From Two-dimensional Images

Computation of Variable Scale Ridges From Two-dimensional Images

A main problem with the fixed scale ridge definition presented above is that it can be very sensitive to the choice of the scale level. 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. To handle this problem in the absence of prior information, the notion of scale-space ridges has been introduced, which treats the scale parameter as an inherent property of the ridge definition and allows the scale levels to vary along a scale-space ridge. Moreover, the concept of a scale-space ridge also allows the scale parameter to be automatically tuned to the width of the ridge structures in the image domain, in fact as a consequence of a well-stated definition. In the literature, a number of different approaches have been proposed based on this idea.

Let denote a measure of ridge strength (to be specified below). Then, for a two-dimensional image, a scale-space ridge is the set of points that satisfy

where is the scale parameter in the scale-space representation. Similarly, a scale-space valley is the set of points that satisfy

An immediate consequence of this definition is that for a two-dimensional image the concept of scale-space ridges sweeps out a set of one-dimensional curves in the three-dimensional scale-space, where the scale parameter is allowed to vary along the scale-space ridge (or the scale-space valley). The ridge descriptor in the image domain will then be a projection of this three-dimensional curve into the two-dimensional image plane, where the attribute scale information at every ridge point can be used as a natural estimate of the width of the ridge structure in the image domain in a neighbourhood of that point.

In the literature, various measures of ridge strength have been proposed. When Lindeberg (1996, 1998) coined the term scale-space ridge, he considered three measures of ridge strength:

• The main principal curvature
  expressed in terms of -normalized derivatives with

  .

• The square of the -normalized square eigenvalue difference

• The square of the -normalized eigenvalue difference

The notion of -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 . Out of these three measures of ridge strength, the first entity is a general purpose ridge strength measure with many applications such as blood vessel detection and road extraction. Nevertheless, the entity has been used in applications such as fingerprint enhancement, real-time hand tracking and gesture recognition as well as for modelling local image statistics for detecting and tracking humans in images and video.

There are also other closely related ridge definitions that make use of normalized derivatives with the implicit assumption of . Develop these approaches in further detail. When detecting ridges with, however, the detection scale will be twice as large as for, resulting in more shape distortions and a lower ability to capture ridges and valleys with nearby interfering image structures in the image domain.