A ridge detector can be expressed in a conceptually similar way
as follows:
Introduce at any image point a local (p, q)-system aligned to
the principal curvature directions such that the mixed second-order
derivative is zero, i.e., .
Then, we can define a bright (dark) ridge point as a point
for which the intensity assumes a local maximum in the
main principal curvature direction.
In terms of the (p, q)-coordinates, this definition can be written
depending on whether the p- or the q-direction
corresponds to the maximum absolute value of the
principal curvatures.
At points where the gradient does not vanish,
this condition can equivalently be expressed
as follows in the (u, v)-system
and in terms of Cartesian partial derivatives
Figure 5(b) shows the result of applying
this ridge detector to an image of an arm.
As can be seen, the types of ridge curves that are obtained are
strongly strongly scale dependent.
At very fine scales, the ridge detector responds mainly to
noise and spurious fine-scale textures.
Then, the fingers give rise to ridge curves at scale level t = 16.0,
and the arm as a whole is extracted as a long ridge curve at t = 256.0.
Notably, these ridge descriptors are much more sensitive to the
choice of scale levels than the edge features in
figure 5(a).
In particular, no single scale level is appropriate for describing
the dominant ridge structures in this image.