Center median

The median axis is a method making it possible to represent the shape of an object by finding its topological skeleton, i.e. a whole of curves which “runs” along the “medium” of the object.

In 2D, the median axis of a curve S is the place of the centers of the circles which are tangent with S in 2 points or plusin, and which is contained in S (thus, the median axis is contained in S ).

The median axis is a subset of the Symmetrical unit , which is defined in a similar way, except that it includes circles not contained in S (thus, the symmetrical unit of S extends generally ad infinitum, in the same way as the Diagramme of Voronoï of a number of points).

The median axis can spread with hypersurfaces of dimension K by replacing the circles 2D by Hypersphère S of dimension K . The median axis in 2D is useful for the character recognition and objects, whereas that in 3D is used for the rebuilding of surface intervening in the physical models.

If S is given by a unit parameterization $\ gamma: \ mathbf {R} \ to \ mathbf {R} ^2$ , and $\ underline {T} (T) = {D \ gamma \ over dt}$ indicates the tangent vector in each point. Then there exists a bitangential circle with a center C and a ray R if

$(C \ gamma (S))\ cdot \ underline {T} (S) = (C \ gamma (T))\ cdot \ underline {T} (T) =0,$
$|C \ gamma (S)|=|C \ gamma (T)|=r. \,$

For the majority of the curves, the symmetrical unit will form a unidimensional curve and will be able to contain singularities. The symmetrical unit have final points which correspond to the flat of S .

See too

Diagram of Voronoï, which can be seen like the discrete shape of the median axis.

References

Broad From the Infinitely to the Infinitely Small: Applications off Medial Symmetry Representations off Shape Frederic F. Leymarie1 and Benjamin B. Kimia2

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