Definition of Ridges and Valleys in N Dimensions
In its broadest sense, the notion of ridge generalizes the idea of a local maximum of a real-valued function. A point in the domain of a function is a local maximum of the function if there is a distance with the property that if is within units of, then . It is well known that critical points, of which local maxima are just one type, are isolated points in a function's domain in all but the most unusual situations (i.e., the nongeneric cases).
Consider relaxing the condition that for in an entire neighborhood of slightly to require only that this hold on an dimensional subset. 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. This means that the set of ridge points will form a 1-dimensional locus, or a ridge curve. Notice that the above can be modified to generalize the idea to local minima and result in what might call 1-dimensional valley curves.
This following ridge definition follows the book by Eberly and can be seen as a generalization of some of the abovementioned ridge definitions. Let be open an open set, and be smooth. Let . Let be the gradient of at, and let be the Hessian matrix of at . Let be the ordered eigenvalues of and let be a unit eigenvector in the eigenspace for . (For this, one should assume that all the eigenvalues are distinct.)
The point is a point on the 1-dimensional ridge of if the following conditions hold:
- , and
- for .
This makes precise the concept that restricted to this particular -dimensional subspace has a local maxima at .
This definition naturally generalizes to the k-dimensional ridge as follows: the point is a point on the k-dimensional ridge of if the following conditions hold:
- , and
- for .
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 and Miller. Their properties in one-parameter families was established by Keller.
Read more about this topic: Ridge Detection
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