In image processing, ridge detection is the attempt, via software, to locate ridges in an image, defined as curves whose points are local maxima of the function, akin to geographical ridges. 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. Correspondingly, the notion of valleys for a function can be defined by replacing the condition of a local maximum with the condition of a local minimum. The union of ridge sets and valley sets, together with a related set of points called the connector set, form a connected set of curves that partition, intersect, or meet at the critical points of the function. This union of sets together is called the function's relative critical set. Ridge sets, valley sets, and relative critical sets represent important geometric information intrinsic to a function. In a way, they provide a compact representation of important features of the function, but the extent to which they can be used to determine global features of the function is an open question. 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. Ridge-related representations in terms of watersheds have been used for image segmentation. 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. In this respect, ridges and valleys can be seen as a complement to natural interest points or local extremal points. 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 typical applications, ridge and valley descriptors are often used for detecting roads in aerial images and for detecting blood vessels in retinal images or three-dimensional magnetic resonance images.
Differential geometric definition of ridges and valleys at a fixed scale in a two-dimensional image Let f ( x , y ) {\displaystyle f(x,y)} denote a two-dimensional function, and let L {\displaystyle L} be the scale-space representation of f ( x , y ) {\displaystyle f(x,y)} obtained by convolving f ( x , y ) {\displaystyle f(x,y)} with a Gaussian function
g ( x , y , t ) = 1 2 π t e − ( x 2 + y 2 ) / 2 t {\displaystyle g(x,y,t)={\frac {1}{2\pi t}}e^{-(x^{2}+y^{2})/2t}} . Furthermore, let L p p {\displaystyle L_{pp}} and L q q {\displaystyle L_{qq}} denote the eigenvalues of the Hessian matrix
H = [ L x x L x y L x y L y y ] {\displaystyle H={\begin{bmatrix}L_{xx}&L_{xy}\\L_{xy}&L_{yy}\end{bmatrix}}}
of the scale-space representation L {\displaystyle L} with a coordinate transformation (a rotation) applied to local directional derivative operators,
∂ p = sin β ∂ x − cos β ∂ y , ∂ q = cos β ∂ x + sin β ∂ y {\displaystyle \partial _{p}=\sin \beta \partial _{x}-\cos \beta \partial _{y},\partial _{q}=\cos \beta \partial _{x}+\sin \beta \partial _{y}}
… excerpt ends here. Continue reading the full article.
