Medial axis

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A ellipse (red), its evolute (blue), and its medial axis (green). The symmetry set, a super-set of the medial axis is the green and yellow curves. One bi-tangent circle is shown.

The medial axis is a method for representing the shape of objects by finding the topological skeleton, a set of curves which roughly run along the middle of an object.

In 2D, the medial axis of a plane curve S is the locus of the centers of circles that are tangent to curve S in two or more points, where all such circles are contained in S. (It follows that the medial axis itself is contained in S.)

The medial axis is a subset of the symmetry set, which is defined similarly, except that it also includes circles not contained in S. (Hence, the symmetry set of S generally extends to infinity, similar to the Voronoi diagram of a point set.)

The medial axis generalizes to k-dimensional hypersurfaces by replacing 2D circles with k-dimension hyperspheres. 2D medial axis is useful for character and object recognition, while 3D medial axis has applications in surface reconstruction for physical models.

If S is given by a unit speed parametrisation , and is the unit tangent vector at each point. Then there will be a bitangent circle with center c and radius r if

For most curves, the symmetry set will form a one dimensional curve and can contain cusp. The symmetry set has end points corresponding to the vertices of S.

[edit] See also

• Voronoi diagram - which can be regarded as a discrete form of the medial axis.

[edit] References

• From the Infinitely Large to the Infinitely Small: Applications of Medial Symmetry Representations of Shape Frederic F. Leymarie1 and Benjamin B. Kimia2 [1]

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