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Within the cubical-voxel 3D binary image context, we design an algorithm for computing the isotopic model of an image, called (6<\/jats:bold>,26<\/jats:bold>)-Homological Region Adjacency Tree ((6<\/jats:bold>,26<\/jats:bold>)-Hom-Tree<\/jats:italic>). This algorithm is based on a flexible graph scaffolding at the inter-voxel level called Homological Spanning Forest model (HSF).Hom-Trees<\/jats:italic>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-Poincar\u00e9 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$$I_D$$<\/jats:tex-math>I<\/mml:mi>D<\/mml:mi><\/mml:msub><\/mml:math><\/jats:alternatives><\/jats:inline-formula>a 3D binary well-composed image (whereD<\/jats:italic>is the set of black voxels), an almost fully parallel algorithm for constructing theHom-Tree<\/jats:italic>via HSF computation is implemented and tested here. If$$I_D$$<\/jats:tex-math>I<\/mml:mi>D<\/mml:mi><\/mml:msub><\/mml:math><\/jats:alternatives><\/jats:inline-formula>has$$m_1{\\times } m_2{\\times } m_3$$<\/jats:tex-math>m<\/mml:mi>1<\/mml:mn><\/mml:msub>\u00d7<\/mml:mo>m<\/mml:mi>2<\/mml:mn><\/mml:msub>\u00d7<\/mml:mo>m<\/mml:mi>3<\/mml:mn><\/mml:msub><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>voxels, the time complexity order of the reproducible algorithm is near$$O(\\log (m_1{+}m_2{+}m_3))$$<\/jats:tex-math>O<\/mml:mi>(<\/mml:mo>log<\/mml:mo>(<\/mml:mo>m<\/mml:mi>1<\/mml:mn><\/mml:msub>+<\/mml:mo>m<\/mml:mi>2<\/mml:mn><\/mml:msub>+<\/mml:mo>m<\/mml:mi>3<\/mml:mn><\/mml:msub>)<\/mml:mo><\/mml:mrow>)<\/mml:mo><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>, under the assumption that a processing element is available for each cubical voxel. Strategies for using the compressed information of theHom-Tree<\/jats:italic>representation to distinguish two topologically different images having the same homological information (Betti numbers) are discussed here. The topological discriminatory power of theHom-Tree<\/jats:italic>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<\/jats:italic>images.<\/jats:p>","DOI":"10.1007\/s10472-023-09913-7","type":"journal-article","created":{"date-parts":[[2024,1,29]],"date-time":"2024-01-29T10:02:34Z","timestamp":1706522554000},"page":"77-113","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Parallel homological calculus for 3D binary digital images"],"prefix":"10.1007","volume":"92","author":[{"given":"Fernando","family":"D\u00edaz-del-R\u00edo","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Helena","family":"Molina-Abril","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6853-0505","authenticated-orcid":false,"given":"Pedro","family":"Real","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Darian","family":"Onchis","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Sergio","family":"Blanco-Trejo","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2024,1,29]]},"reference":[{"key":"9913_CR1","doi-asserted-by":"publisher","first-page":"52","DOI":"10.3389\/frai.2021.681108","volume":"4","author":"F Hensel","year":"2021","unstructured":"Hensel, F., Moor, M., Rieck, B.: A survey of topological machine learning methods. 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