Medial axis

In geometry, the medial axis of a region is a set of points that lie in a sort of geometric center of the area. It was proposed in the 1960s by Harry Blum to describe biological shapes. Since the medial axis has found a variety of applications in many different areas, from the formation of galaxies, via path planning for robot or the recognition of characters up to the representation of molecular structures.

Medial axis in the plane

Blum defined the medial axis of a region in the plane as the set of centers of maximal circles. A circle is a maximum in when he is fully in, and there is no other district in which contains. It follows immediately that the points of the medial axis must also lie in the interior of. It is observed that maximum circles touch the edge of the area tangentially, ie the tangent direction of the circle right at the contact point with the tangent direction of the edge match ( if it is defined - for polygons as this is not in the corners of the case). In general, the maximum circles touch the edge at two points, but there are also situations with one, or infinitely many points of contact. The contact points are also known as base points.

Assigns to each point of the medial axis of the radius of the circle corresponding to the maximum, to obtain an image. This so-called radius function thus assigns to each point of the medial axis to its distance to the edge. Medial axis and radius function is referred to as Medial axis transformation, since it is possible with them, the original area to reconstruct.

An alternative definition of medial axis arises from the observation that there is more than one shortest path to the edge there for a point of the MA in general - these routes are the routes to the base points. Thus, the medial axis can also be defined as the set of points within the area of which there is no clearly defined edge of the shortest path to the amount. That is, there are at least two directions in which one is the fastest from a point on the medial axis of the rim.

Wherein said metric is the distance of the points and the distance of the point to the set. Usually one uses the Euclidean distance for the former, the latter more for the minimum of the distances to points of the set:

The medial axis depends only on the choice of a metric - the first definition contains this implicitly: a circle with center and radius is defined as the set of all points that have a distance, the edge of the circle are the points with distance.

Special situations

Assigns the edge of the area on corners touched the medial axis at these points the edge. At all other places, this is not the case.

Higher dimensions

The above definition of the medial axis can be canonically extended to higher dimensions, it is, as I said, depends only on the existence of a metric on the space in which lies the area. Instead of maximum circles maximum - dimensional ( hyper-) spheres are considered in the -dimensional space then.

Calculation

In the 2D case, the medial axis can be an area approximated by computing the Voronoi diagram of a sample of the edge. The Voronoi node then approximate the medial axis, the accuracy of the approximation of the sampling density and fineness of the structures depends on the boundary.