Differential Geometric Definition of Ridges and Valleys At A Fixed Scale in A Two-dimensional Image
Let denote a two-dimensional function, and let be the scale-space representation of obtained by convolving with a Gaussian function
- .
Furthermore, let and denote the eigenvalues of the Hessian matrix
of the scale-space representation . With a coordinate transformation (a rotation) applied to local directional derivative operators,
where p and q are coordinates of the rotated coordinate system.
It can be shown that the mixed derivative in the transformed coordinate system is zero if we choose
- ,.
Then, a formal differential geometric definition of the ridges of at a fixed scale can be expressed as the set of points that satisfy
Correspondingly, the valleys of at scale are the set of points
In terms of a coordinate system with the direction parallel to the image gradient
where
it can be shown that this ridge and valley definition can instead be equivalently be written as
where
and the sign of determines the polarity; for ridges and for valleys.
Read more about this topic: Ridge Detection
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