The Island Hierachy

The hexagonal island hierachy

The two-dimensional CSC operates on a hexagonal island hierarchy. Hexagonal islands of level 0 consist of seven pixels (one center and its 6 neighbor pixels). Two hexagonal islands of level 0 overlap each other in one common sub pixel, that is to say each pixel is either a island center or it is enclosed in two islands of level 0.

hexagone

On the next hierarchy level we could consider the islands of level 0 as macro pixels. Therefore we could also combine seven islands of level 0 in one new island of level 1. Two islands of level 1 overlap each other in one common subisland (island of level 0). Islands of level n+1 could now be formed in the same manner. Seven islands of level n form a new island of leveln+1. From one hierarchy level to the next, the number of islands is reduced by factor 4. On the top most hierarchy level exists only one island that cover the entire image.

 

The three-dimensional island hierachy

On the basis of the hexagonal island hierarchy we could formulate the following features of a three-dimensional island hierarchy:

  • The structure of each island is the same.

  • Two islands of one level overlaps each other in one common sub island or in one common sub pixel.

  • Each island resp. each pixel is either an island center or belongs to exactly two islands of the next hierarchy level.

  • The structure of each hierarchy level is the same.

 Three-dimensional island hierarchies, that holding the features above, could not achieved trivial from the hexagonal island hierarchy.

 

spheres-layer0spheres-layer1spheres-layer2
(a) Layer 0(b) Layer 1(c) Layer 2
Figure 2: Three dimensional sphere raster. The spheres are organized in a dense package. White Spheres represent the center spheres of level 0 islands. Black Spheres are not a neighbor of any center. Spheres that are neighbors of a center sphere are either blue (layer 0 and 2) or red (layer 1).

 

 
 
3D Island 1 3D Island 2
(a) (b)
 Figure 3: 3D-Island of level 0. (a) Center + 12 Neighbors (b) Center + 14 Neighbors
 

In our three-dimensional island hierarchy the islands of level 0 consists of 1 center voxel and its 14 neighbor voxels. On the next hierarchy level 15 islands of level 0 are combined to a island of level 1. Two three-dimensional islands overlap each other in one commmon sub island. From one hierarchy level to the next, the number of islands is reduced by factor 8.