The medial Axis

The purpose in constructing the medial surface or medial axis (skeleton, deformation retract) of a three-dimensional object is to obtain an reduced representation of the object that is easier to analyze. Skeletonization of a digitized object is performed by careful erosion of the object's voxels, layer by layer, while preserving the object's topological and geometrical properties (essentially, a voxel is eroded (removed) only if its removal does not induce a local change in topology, e.g. breaking the object in two parts, creating a hole or cavity). Conceptually the medial erosion of a 3D object can be viewed as first generating a medial surface and, if the object has no embedded cavities, eroding this surface further to a union of digitized curves referred to as the medial axis (MA). (In practice MA algorithms generate the medial axis with no intermediate generation of a medial surface.)

The MA was introduced in 1967 by H. Blum to study biological shapes. Application of the MA to statistically reconstructed porous media was introduced by Thovert et al., 1993. Application of the MA to studies of XCMT images of real porous media was introduced by Spanne et al., 1994.

Sensitivity of the medial axis to surface noise


Fig. 1.   The medial axis of a 2D slice of a basalt image. Due to the rough grain (black) - pore (white) surface, the resulting MA has a large number of (dead end) branch-leaf paths.


Medial Axis Terminology in 3DMA-Rock

A medial axis path is a digitized version of a curve. It consists of a mutually linked set of MA path voxels, each of which is connected to exactly two distinct MA voxels. The sole exception is that a voxel at an end of the path may only connect to one other MA voxel, i.e. a "dead end".

A branch cluster voxel is a voxel connected to three or more distinct MA voxels. A branch cluster (node) is a connected set of branch cluster voxels. A cluster is the digitized version of a vertex where medial axis paths meet.

Fig. 2 shows a small portion of a medial axis for some 3D object. MA path voxels are in red, branch cluster voxels are blue.

In 3DMA-Rock, paths are classified as:

Only branch-branch and branch-leaf paths are shown in Fig. 2.

Fig. 2.   An example of the medial axis for a 3D object (object not shown). Branch clusters are shown in blue, and medial axis paths are in red. Note that while the branch clusters shown might consist of more than one voxel, they are topologically equivalent to a point.

A surface remnant is a generic name for a branch cluster that is not topologically a point. (The appellation "surface remant" is somewhat of a misnomer, as the reasons behind the appearance of such a branch cluster have more to do with voxel resolution.) Surface remnant branch clusters are illustrated, red voxels, in Fig. 3.

Surface remnant reduction refers to an algorithm in 3DMA-Rock which attempts to trim voxels of a surface remnant to reduce it to the topology of a point. This is not always possible, though a smaller sized remnant is usually achieved. Reduction is demonstrated in Fig. 3.

Fig. 3.   A 2D surface remnant is shown (red) before (left) and after (right) reduction.


Branch cluster merging

Merging of close branch clusters is done with identification of pore bodies in mind. Clusters that intuitively belong to the same pore body are merged into one super cluster. That reduces the running time for throat finding algorithms by avoiding searches on paths connecting branch clusters belonging to the same pore.

Identifying the clusters that should be merged is not trivial. Most commonly we use the criterion depicted in Fig. 4.

   Fig. 4.   One criterion for cluster merging. Clusters are merged if

N < max{B1,B2} + 2

where N is the number of voxels on the medial axis path and B1 and B2 are the burn numbers (distance to the closest grain voxel), respectively, of the two branch clusters.

This page provides only a quick reference to some basic MA terms. The reader is referred to the 3DMA General Users Manual, Section 7.3 for more details.


References

J.F. Thovert, J. Salles, and P.M. Adler, Computerized characterization of the geometry of real porous-media - their discretization, analysis and interpretation J. Microscopy-Oxford., 170, (1993) 65-79.

P. Spanne, J.F. Thovert, C.J. Jacquin, W.B. Lindquist, K.W. Jones and P.M. Adler, Synchrotron computed microtomography of porous media: topology and transports. Phys. Rev. Lett., 73, (1994) 2001-2004.