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#### Mathematical description

Given a set of real values , we define the median filter of level r, for a r [0, 1], by

medr() = inf , suchthat#{ , } = r#

where # is the cardinal operator. Note that for r = 0, the median of level r coresponds to the inf operator, and for r = 1, it coresponds to the sup operator.

Now, consider a set of shapes, for example the set of the ellipses of same area, and an original image u0. This program will compute the sequences of images ui defined by

ui+1(x) = meddeginfmeddegsup{ui(y + x), y B}, B ,

or, when using the -a option,
 ui+1(x) = meddeginfmeddegsup{ui(y + x), y B}, B + meddegsupmeddeginf{ui(y + x), y B}, B /2.

In a mathematical viewpoint, when the area of the elements of tends to zero, this scheme (with option -a set) computes a discretisation of an equation of type

= | Du| G(Du/| Du|, curv(u)), qquadu(t = 0) = u0

For example, it has been proved that when is the set of segments with same lenght (file seg_mask), and centered into 0, we have G(Du/| Du|, curv(u)) = curv(u).

And, in the same way, when is the sets of the ellipses, or rectangles with same areas, also centered into 0, G(Du/| Du|, curv(u)) = curv(u)13.

At last, notice that when contains only one element then with deginf = degsup = 0 we have the erosion, with deginf = degsup = 1 we have the dilation, and with deginf = degsup = 0.5 we have the median filter.

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mw 2004-05-05