Topology for grayscale images
Topological notions for binary images
We suppose that you are familiar with the topological notions for
binary images. If it is not the case, please refer, for example, to
"Digital topology: introduction and survey",
by T.Y Kong and A. Rosenfeld,
Comp. Vision, Graphics and Image Proc., 48, pp. 357-393, 1989.
In particular, the notion of simple point is essential for our
purpose. Intuitively, a point belonging to an object X is a
simple point if it may be removed from X without changing
neither the number of connected components of X,
nor the number of connected components of 
.
In order to have a correspondance between the topology of X and the
topology of , we have to consider in the square grid
two different adjacency relations: is we use the 4-adjacency for X
then we must use the 8-adjacency for , and vice-versa.
Topological notions for grayscale images
A grayscale image may be modelized by a function F from Z2 to Z, where F(x) represents the gray level of the pixel x. A grayscale image F may also be seen as a topographic relief, where F(x) stands for the elevation of the point x. With this point of view, the montains and crest lines of the relief correspond to the light areas, whereas the basins and the valleys correspond to the dark areas.
In order to extend the binary notions to a grayscale image F, we consider
the different horizontal cross-sections of the corresponding relief:
the level k cross-section of F is the set
, that is, the set of points
having an elevation greater than k. We can now introduce the notion
of destructible point, which extends the notion of simple point:
A point x is destructible for F if x is simple for the cross-section Fk , with k = F(x). It may be easily seen that one can lower the altitude of a destructible point by one without changing the topology (in the binary sense) of every cross-section.
We say that a grayscale image G is lower-homotopic to a grayscale image F if G may be derived from F by lowering destructible points.
Lower (and Upper) Homotopic Kernel
We say that a grayscale image K is a lower-homotopic kernel of a grayscale image F if K is lower-homotopic to F and if there is no destructible point for K.
Take a look at the images below. We can see that, for a real image, there is a lot of minima, each of these regions being composed of only few points. The lower-homotopic kernel transformation expands as much as possible the minima, but preserves the number of minima and also significant grayscale imformation.
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The dual notions of constructible point and upper-homotopic kernel may be defined in a similar way.
Topological Numbers and Algorithms
Following the same approach, we have defined the topological numbers for points of a grayscale image, which allow to locally characterize destructible points. Thanks to these topological numbers, efficient algorithms may be built to compute homotopic kernels. The topological numbers also lead to a classification of the local topological properties of the points in grayscale images (destructible point, constructible point, saddle point, peak, sink, ...).
Leveling Kernel
We will now introduce the leveling kernel, which may be viewed as a "filtered" lower-homotopic kernel. A point is a peak if it has only strictly lower neighbors. A leveling kernel of an image F is obtained by repeating the following procedure until stability: select a destructible point or a peak point, then lower it.
Despite the appearance, this kernel is very different from the homotopic kernel: all the points belonging to upper regions have been flattened down. This characteristic of leveling kernels will be used for the design of regularization operators.
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G. Bertrand, J. C. Everat and M. Couprie, "Image segmentation through operators based upon topology", Journal of Electronic Imaging, Vol. 6, N. 4, 395-405, 1997.
