Parallel Homological Calculus for 3D Binary Digital Images

Topological representations of binary digital images usually take into consideration different adjacency types between colors. Within the cubical-voxel 3D binary image context, we design an algorithm for computing the isotopic model of an image, called ( 6 , 26 )-Homological Region Adjacency Tree (( 6 , 26 )- Hom-Tree ). This algorithm is based on a flexible graph scaffolding at the inter-voxel level called Homological Spanning Forest model (HSF). Hom-Trees are edge-weighted trees in which each node is a maximally connected set of constant-value voxels, which is interpreted as a subtree of the HSF. This representation integrates and relates the homological information (connected components, tunnels and cavities) of the maximally connected regions of constant color using 6-adjacency and 26-adjacency for black and white voxels, respectively (the criteria most commonly used for 3D images). The Euler-Poincaré numbers (which may as well be computed by counting the number of cells of each dimension on a cubical complex) and the connected component labeling of the foreground and background of a given image can also be straightforwardly computed from its Hom-Trees. Being I D a 3D binary well-composed image (where D is the set of black voxels), an almost fully parallel algorithm for constructing the Hom-Tree via HSF computation is implemented and tested here. If I D has m 1 × m 2 × m 3 voxels, the time complexity order of the reproducible algorithm is near O ( log ( m 1 + m 2 + m 3 ) ) , under the assumption that a processing element is available for each cubical voxel. Strategies for using the compressed information of the Hom-Tree representation to distinguish two topologically different images having the same homological information (Betti numbers) are discussed here. The topological discriminatory power of the Hom-Tree and the low time complexity order of the proposed implementation guarantee its usability within machine learning methods for the classification and comparison of natural 3 D images.