Abstract
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 \(m_1{\times } m_2{\times } m_3\) 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 3D images.
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Acknowledgements
This work has been partially supported by the research projects Par-Hot (PID2019-110455GB-I00) and CIUCAP-HSF (US-1381077) funded by MCIN/AEI/10.13039/501100011033 and Feder Funds.
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The first three authors are supported by project PID2019-110455GB-I00 (AEI/FEDER,UE) and project US-1381077 (JJAA/FEDER, UE).
Appendices
Appendix
1.1 A parallel generation of Hom-Tree. HSF construction via crack transports
For each critical MrSF 0-crack, we calculate in parallel the movements along (1, 2) and (0, 1) trees. Firstly, there are six possible paths leaving from each critical 0-cell and moving through the (1, 2)-tree. One of them is selected for every critical MrSF 0-crack, and the (1, 2)-trees (tracking frontier digital surfaces), fall onto the corresponding 2-cells (black arches in Fig. 11). From these 2-cells, it is resolved what critical 1-crack has a primal vector to them.
Then, every 1-cell has two incident 0-cells (red arrows in Fig. 11) , and the two back movements for these 0-cells are computed in parallel in order to determine if the couple of (0, 1)-cells can be eliminated, and so the transport is carried out. The movements of these 0-cells through the (0, 1)-tree yield three different cases (processed in an independent and parallel manner for each couple of (0, 1)-cells). These cases are: a) the 1-cell can be deleted (a crack transport is carried out) with the initial critical 0-cell, because only one of its two (0, 1)-paths fell into the initial 0-cell; b) the 1-cell is the representative of a tunnel (it is a true critical 1-cell), since these two (0,1) paths fell into the same 0-cell; c) nothing can be determined, because the two (0, 1)-paths fell into two different 0-cells (in this case, the 1-cell remains the same).
Note that moving along trees guarantees that the uniqueness in the selection of the different couples of (0, 1)-cells is ensured, and therefore there is no need for synchronization primitives. In this sense, perfect parallelism is achieved in these six paths, thus, their time order can be considered as O(1). Summing up, for each critical MrSF 0-crack, six set of paths must be computed (all the paths of each set in parallel). Likewise, for the pairing of couples of critical MrSF 1-crack, a similar parallel procedure can be performed, preserving the time order as O(1). The number of incident cells is inferior in this ulterior pairing. More concretely, each critical MrSF 1-cell has only two possible paths that go through (2, 3)-trees. These trees would fall into 3-cells that have associated (through primal vectors) with a possible critical 2-cell. Each of these 2-cells has up to four 1-cells in their borders; thus, four back movements along (1, 2)-trees must be then calculated. A comparison of the 1-cells where the back movements fall will result in these cases: 1) the 2-cell may be the representative of a cavity (a true critical HSF 2-cell), since at least two of them fell into the initial critical MrSF 1-cell; 2) the 2-cell can be paired with the initial critical MrSF 1-cell, because only one of the back travel fell into this last; 3) nothing can be determined, because the two previous cases did not occur.
B Further experimental results
Figure 12 shows examples of deformed objects generated with the automatic homotopy deformation (thinning-thickening) software, based on the notion of a simple 3D point. Up to 500 26-simple points have been randomly added/removed 100 times for each of these cases: Three concentric spheres, one torus with two spheres inside, and one sphere with two spheres inside. These examples were used to corroborate that homotopically (in fact, isotopically) equivalent nesting return the same Hom-Tree.
In Fig. 13 an HSF of a foreground sphere with two external handles is completely drawn to distinguish that the borders of its two critical 1-cells (marked with thickest yellow lines at \(Z=2\), \(Y=2\), \(X=2.5\) and \(X=6.5\)) fall into the foreground component. Moreover, there are two additional tunnels for the external background component (ahead of the two previous ones at \(Y=1.5\)).
Tables 2 and 3 show the mean number of critical cells for the first four parallel and last sequential transport iterations of (0, 1)-cell pairs, and for the following parallel and final sequential transport of (1, 2)-cell pairs for a set of five \(20 \times 20 \times 20\) and \(30 \times 30 \times 30\) B/W random images, resp. These images were generated using a 50/50 probability for colors, and surrounded by six thin faces of background voxels according to our working premises. Actually, a fifth iteration would transport another one or two (0, 1)-pairs more for two of the bigger images tested.
Table 4 presents a summary of the results for the images of the trabecular bone.
The Hom-Tree is able to distinguish non-isotopic patterns having the same homology. An interesting example occurs when a ring inside a sphere is contrasted against a sphere surrounded by a ring (see Figs. 14 and 15 )(Fig. 16).
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Díaz-del-Río, F., Molina-Abril, H., Real, P. et al. Parallel homological calculus for 3D binary digital images. Ann Math Artif Intell 92, 77–113 (2024). https://doi.org/10.1007/s10472-023-09913-7
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DOI: https://doi.org/10.1007/s10472-023-09913-7
Keywords
- 3D digital images
- Binary images
- Parallel computing
- Cavity
- Tunnel
- Connected component
- Homological spanning forest
- Inter-voxel
- Homological region adjacency tree