Skeleton Extraction from Binary Images Kalman Palagyi University of Szeged, Hungary

The generic model of a modular machine vision system

Feature extraction

Shape representation to describe the boundary that surrounds an object; to describe the region that is occupied by an object.

Skeleton result of the Medial Axis Transform: object points having at least two nearest boundary points; praire-fire analogy: the boundary is set on fire and skeleton is formed by the loci where the fire fronts meet and quench each other; the locus of the centers of all the maximal inscribed hyper-spheres.

Nearest boundary points and inscribed hyper-spheres

Skeleton of a 3D solid box The skeleton in 3D generally contains surface patches (2D segments).

Properties: It represents the general form of an object, the topological structure of an object, and local object symmetries. It is invariant to translation, rotation, and (uniform) scale change. It is thin.

Uniqueness The same skeleton may belong to different elongated objects.

Stability

Representing local object symmetries and the topological structure

Skeletonization techniques distance transform, Voronoi diagram, and thinning.

Distance transform Input: Binary array A containing feature elements (1’s) and non-feature elements (0’s). Output: Non-binary array B containing the distance to the nearest feature element.

Example: distance map (non-binary image) input (binary image)

M.C. Escher: Reptiles

Distance transform using city-block (or 4) distance

Distance transform using chess-board (or 8) distance

Chamfer distance transform in linear time (G. Borgefors, 1984)

forward scan backward scan

Chamfer masks in 2D

Chamfer masks in 3D

original binary image initialization forward scan backward scan

Skeletonization based on distance transform

Positions marked boldface numbers belong to the skeleton.

Voronoi diagram

Incremental construction

Delauney triangulation/tessalation

Voronoi & Delauney

Duality

Skeletal elements of a Voronoi diagram

A 3D example original Voronoi diagram regularization M. Näf (ETH, Zürich)

‘Thinning’ before after

Thinning It is an iterative object reduction technique in a topology preserving way.

Topology preservation in 2D (a counter example) 

Hole It is a new concept in 3D  ”A topologist is a man who does not know the difference between a coffee cup and a doughnut.”

Shape preservation

End-points in 3D thinning original medial surface topological kernel medial lines

Types of voxels in 3D medial lines

A 2D thinning algorithm using 8 subiterations

A 3D thinning algorithm using 6 subiterations

Blood vessel (infra-renal aortic aneurysms)

Airway (trachealstenosis)

Calculating cross sectional profiles and estimating diameter

Colon (cadaveric phantom)

Airway (intrathoracic airway tree)

Example Centerlines Segmented tree Labeled tree Formal tree

Requirements Geometrical: The skeleton must be in the middle of the original object and must be invariant to translation, rotation, and scale change. Topological: The skeleton must retain the topology of the original object.

Comparison