--- _id: '13134' abstract: - lang: eng text: We propose a characterization of discrete analytical spheres, planes and lines in the body-centered cubic (BCC) grid, both in the Cartesian and in the recently proposed alternative compact coordinate system, in which each integer triplet addresses some voxel in the grid. We define spheres and planes through double Diophantine inequalities and investigate their relevant topological features, such as functionality or the interrelation between the thickness of the objects and their connectivity and separation properties. We define lines as the intersection of planes. The number of the planes (up to six) is equal to the number of the pairs of faces of a BCC voxel that are parallel to the line. acknowledgement: The first author has been partially supported by the Ministry of Science, Technological Development and Innovation of the Republic of Serbia through the project no. 451-03-47/2023-01/200156. The fourth author is funded by the DFG Collaborative Research Center TRR 109, ‘Discretization in Geometry and Dynamics’, Austrian Science Fund (FWF), grant no. I 02979-N35. article_number: '109693' article_processing_charge: No article_type: original author: - first_name: Lidija full_name: Čomić, Lidija last_name: Čomić - first_name: Gaëlle full_name: Largeteau-Skapin, Gaëlle last_name: Largeteau-Skapin - first_name: Rita full_name: Zrour, Rita last_name: Zrour - first_name: Ranita full_name: Biswas, Ranita id: 3C2B033E-F248-11E8-B48F-1D18A9856A87 last_name: Biswas orcid: 0000-0002-5372-7890 - first_name: Eric full_name: Andres, Eric last_name: Andres citation: ama: Čomić L, Largeteau-Skapin G, Zrour R, Biswas R, Andres E. Discrete analytical objects in the body-centered cubic grid. Pattern Recognition. 2023;142(10). doi:10.1016/j.patcog.2023.109693 apa: Čomić, L., Largeteau-Skapin, G., Zrour, R., Biswas, R., & Andres, E. (2023). Discrete analytical objects in the body-centered cubic grid. Pattern Recognition. Elsevier. https://doi.org/10.1016/j.patcog.2023.109693 chicago: Čomić, Lidija, Gaëlle Largeteau-Skapin, Rita Zrour, Ranita Biswas, and Eric Andres. “Discrete Analytical Objects in the Body-Centered Cubic Grid.” Pattern Recognition. Elsevier, 2023. https://doi.org/10.1016/j.patcog.2023.109693. ieee: L. Čomić, G. Largeteau-Skapin, R. Zrour, R. Biswas, and E. Andres, “Discrete analytical objects in the body-centered cubic grid,” Pattern Recognition, vol. 142, no. 10. Elsevier, 2023. ista: Čomić L, Largeteau-Skapin G, Zrour R, Biswas R, Andres E. 2023. Discrete analytical objects in the body-centered cubic grid. Pattern Recognition. 142(10), 109693. mla: Čomić, Lidija, et al. “Discrete Analytical Objects in the Body-Centered Cubic Grid.” Pattern Recognition, vol. 142, no. 10, 109693, Elsevier, 2023, doi:10.1016/j.patcog.2023.109693. short: L. Čomić, G. Largeteau-Skapin, R. Zrour, R. Biswas, E. Andres, Pattern Recognition 142 (2023). date_created: 2023-06-18T22:00:45Z date_published: 2023-10-01T00:00:00Z date_updated: 2023-10-10T07:37:16Z day: '01' department: - _id: HeEd doi: 10.1016/j.patcog.2023.109693 external_id: isi: - '001013526000001' intvolume: ' 142' isi: 1 issue: '10' language: - iso: eng month: '10' oa_version: None project: - _id: 2561EBF4-B435-11E9-9278-68D0E5697425 call_identifier: FWF grant_number: I02979-N35 name: Persistence and stability of geometric complexes - _id: 0aa4bc98-070f-11eb-9043-e6fff9c6a316 grant_number: I4887 name: Discretization in Geometry and Dynamics publication: Pattern Recognition publication_identifier: issn: - 0031-3203 publication_status: published publisher: Elsevier quality_controlled: '1' scopus_import: '1' status: public title: Discrete analytical objects in the body-centered cubic grid type: journal_article user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 volume: 142 year: '2023' ... --- _id: '13182' abstract: - lang: eng text: "We characterize critical points of 1-dimensional maps paired in persistent homology\r\ngeometrically and this way get elementary proofs of theorems about the symmetry\r\nof persistence diagrams and the variation of such maps. In particular, we identify\r\nbranching points and endpoints of networks as the sole source of asymmetry and\r\nrelate the cycle basis in persistent homology with a version of the stable marriage\r\nproblem. Our analysis provides the foundations of fast algorithms for maintaining a\r\ncollection of sorted lists together with its persistence diagram." acknowledgement: Open access funding provided by Austrian Science Fund (FWF). This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme, grant no. 788183, from the Wittgenstein Prize, Austrian Science Fund (FWF), Grant No. Z 342-N31, and from the DFG Collaborative Research Center TRR 109, ‘Discretization in Geometry and Dynamics’, Austrian Science Fund (FWF), Grant No. I 02979-N35. The authors of this paper thank anonymous reviewers for their constructive criticism and Monika Henzinger for detailed comments on an earlier version of this paper. article_processing_charge: Yes (via OA deal) article_type: original author: - first_name: Ranita full_name: Biswas, Ranita id: 3C2B033E-F248-11E8-B48F-1D18A9856A87 last_name: Biswas orcid: 0000-0002-5372-7890 - first_name: Sebastiano full_name: Cultrera Di Montesano, Sebastiano id: 34D2A09C-F248-11E8-B48F-1D18A9856A87 last_name: Cultrera Di Montesano orcid: 0000-0001-6249-0832 - first_name: Herbert full_name: Edelsbrunner, Herbert id: 3FB178DA-F248-11E8-B48F-1D18A9856A87 last_name: Edelsbrunner orcid: 0000-0002-9823-6833 - first_name: Morteza full_name: Saghafian, Morteza id: f86f7148-b140-11ec-9577-95435b8df824 last_name: Saghafian citation: ama: Biswas R, Cultrera di Montesano S, Edelsbrunner H, Saghafian M. Geometric characterization of the persistence of 1D maps. Journal of Applied and Computational Topology. 2023. doi:10.1007/s41468-023-00126-9 apa: Biswas, R., Cultrera di Montesano, S., Edelsbrunner, H., & Saghafian, M. (2023). Geometric characterization of the persistence of 1D maps. Journal of Applied and Computational Topology. Springer Nature. https://doi.org/10.1007/s41468-023-00126-9 chicago: Biswas, Ranita, Sebastiano Cultrera di Montesano, Herbert Edelsbrunner, and Morteza Saghafian. “Geometric Characterization of the Persistence of 1D Maps.” Journal of Applied and Computational Topology. Springer Nature, 2023. https://doi.org/10.1007/s41468-023-00126-9. ieee: R. Biswas, S. Cultrera di Montesano, H. Edelsbrunner, and M. Saghafian, “Geometric characterization of the persistence of 1D maps,” Journal of Applied and Computational Topology. Springer Nature, 2023. ista: Biswas R, Cultrera di Montesano S, Edelsbrunner H, Saghafian M. 2023. Geometric characterization of the persistence of 1D maps. Journal of Applied and Computational Topology. mla: Biswas, Ranita, et al. “Geometric Characterization of the Persistence of 1D Maps.” Journal of Applied and Computational Topology, Springer Nature, 2023, doi:10.1007/s41468-023-00126-9. short: R. Biswas, S. Cultrera di Montesano, H. Edelsbrunner, M. Saghafian, Journal of Applied and Computational Topology (2023). date_created: 2023-07-02T22:00:44Z date_published: 2023-06-17T00:00:00Z date_updated: 2024-03-20T09:36:56Z day: '17' ddc: - '000' department: - _id: HeEd doi: 10.1007/s41468-023-00126-9 ec_funded: 1 file: - access_level: open_access checksum: 697249d5d1c61dea4410b9f021b70fce content_type: application/pdf creator: alisjak date_created: 2023-07-03T09:41:05Z date_updated: 2023-07-03T09:41:05Z file_id: '13185' file_name: 2023_Journal of Applied and Computational Topology_Biswas.pdf file_size: 487355 relation: main_file success: 1 file_date_updated: 2023-07-03T09:41:05Z has_accepted_license: '1' language: - iso: eng month: '06' oa: 1 oa_version: Published Version project: - _id: 266A2E9E-B435-11E9-9278-68D0E5697425 call_identifier: H2020 grant_number: '788183' name: Alpha Shape Theory Extended - _id: 0aa4bc98-070f-11eb-9043-e6fff9c6a316 grant_number: I4887 name: Discretization in Geometry and Dynamics - _id: 268116B8-B435-11E9-9278-68D0E5697425 call_identifier: FWF grant_number: Z00342 name: The Wittgenstein Prize publication: Journal of Applied and Computational Topology publication_identifier: eissn: - 2367-1734 issn: - 2367-1726 publication_status: epub_ahead publisher: Springer Nature quality_controlled: '1' related_material: record: - id: '15094' relation: dissertation_contains status: public scopus_import: '1' status: public title: Geometric characterization of the persistence of 1D maps tmp: image: /images/cc_by.png legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0) short: CC BY (4.0) type: journal_article user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 year: '2023' ... --- _id: '10773' abstract: - lang: eng text: The Voronoi tessellation in Rd is defined by locally minimizing the power distance to given weighted points. Symmetrically, the Delaunay mosaic can be defined by locally maximizing the negative power distance to other such points. We prove that the average of the two piecewise quadratic functions is piecewise linear, and that all three functions have the same critical points and values. Discretizing the two piecewise quadratic functions, we get the alpha shapes as sublevel sets of the discrete function on the Delaunay mosaic, and analogous shapes as superlevel sets of the discrete function on the Voronoi tessellation. For the same non-critical value, the corresponding shapes are disjoint, separated by a narrow channel that contains no critical points but the entire level set of the piecewise linear function. acknowledgement: Open access funding provided by the Institute of Science and Technology (IST Austria). article_processing_charge: Yes (via OA deal) article_type: original author: - first_name: Ranita full_name: Biswas, Ranita id: 3C2B033E-F248-11E8-B48F-1D18A9856A87 last_name: Biswas orcid: 0000-0002-5372-7890 - first_name: Sebastiano full_name: Cultrera Di Montesano, Sebastiano id: 34D2A09C-F248-11E8-B48F-1D18A9856A87 last_name: Cultrera Di Montesano orcid: 0000-0001-6249-0832 - first_name: Herbert full_name: Edelsbrunner, Herbert id: 3FB178DA-F248-11E8-B48F-1D18A9856A87 last_name: Edelsbrunner orcid: 0000-0002-9823-6833 - first_name: Morteza full_name: Saghafian, Morteza last_name: Saghafian citation: ama: Biswas R, Cultrera di Montesano S, Edelsbrunner H, Saghafian M. Continuous and discrete radius functions on Voronoi tessellations and Delaunay mosaics. Discrete and Computational Geometry. 2022;67:811-842. doi:10.1007/s00454-022-00371-2 apa: Biswas, R., Cultrera di Montesano, S., Edelsbrunner, H., & Saghafian, M. (2022). Continuous and discrete radius functions on Voronoi tessellations and Delaunay mosaics. Discrete and Computational Geometry. Springer Nature. https://doi.org/10.1007/s00454-022-00371-2 chicago: Biswas, Ranita, Sebastiano Cultrera di Montesano, Herbert Edelsbrunner, and Morteza Saghafian. “Continuous and Discrete Radius Functions on Voronoi Tessellations and Delaunay Mosaics.” Discrete and Computational Geometry. Springer Nature, 2022. https://doi.org/10.1007/s00454-022-00371-2. ieee: R. Biswas, S. Cultrera di Montesano, H. Edelsbrunner, and M. Saghafian, “Continuous and discrete radius functions on Voronoi tessellations and Delaunay mosaics,” Discrete and Computational Geometry, vol. 67. Springer Nature, pp. 811–842, 2022. ista: Biswas R, Cultrera di Montesano S, Edelsbrunner H, Saghafian M. 2022. Continuous and discrete radius functions on Voronoi tessellations and Delaunay mosaics. Discrete and Computational Geometry. 67, 811–842. mla: Biswas, Ranita, et al. “Continuous and Discrete Radius Functions on Voronoi Tessellations and Delaunay Mosaics.” Discrete and Computational Geometry, vol. 67, Springer Nature, 2022, pp. 811–42, doi:10.1007/s00454-022-00371-2. short: R. Biswas, S. Cultrera di Montesano, H. Edelsbrunner, M. Saghafian, Discrete and Computational Geometry 67 (2022) 811–842. date_created: 2022-02-20T23:01:34Z date_published: 2022-04-01T00:00:00Z date_updated: 2023-08-02T14:31:25Z day: '01' ddc: - '510' department: - _id: HeEd doi: 10.1007/s00454-022-00371-2 external_id: isi: - '000752175300002' file: - access_level: open_access checksum: 9383d3b70561bacee905e335dc922680 content_type: application/pdf creator: dernst date_created: 2022-08-02T06:07:55Z date_updated: 2022-08-02T06:07:55Z file_id: '11718' file_name: 2022_DiscreteCompGeometry_Biswas.pdf file_size: 2518111 relation: main_file success: 1 file_date_updated: 2022-08-02T06:07:55Z has_accepted_license: '1' intvolume: ' 67' isi: 1 language: - iso: eng month: '04' oa: 1 oa_version: Published Version page: 811-842 publication: Discrete and Computational Geometry publication_identifier: eissn: - 1432-0444 issn: - 0179-5376 publication_status: published publisher: Springer Nature quality_controlled: '1' scopus_import: '1' status: public title: Continuous and discrete radius functions on Voronoi tessellations and Delaunay mosaics tmp: image: /images/cc_by.png legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0) short: CC BY (4.0) type: journal_article user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8 volume: 67 year: '2022' ... --- _id: '11660' abstract: - lang: eng text: 'We characterize critical points of 1-dimensional maps paired in persistent homology geometrically and this way get elementary proofs of theorems about the symmetry of persistence diagrams and the variation of such maps. In particular, we identify branching points and endpoints of networks as the sole source of asymmetry and relate the cycle basis in persistent homology with a version of the stable marriage problem. Our analysis provides the foundations of fast algorithms for maintaining collections of interrelated sorted lists together with their persistence diagrams. ' acknowledgement: 'This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme, grant no. 788183, from the Wittgenstein Prize, Austrian Science Fund (FWF), grant no. Z 342-N31, and from the DFG Collaborative Research Center TRR 109, ‘Discretization in Geometry and Dynamics’, Austrian Science Fund (FWF), grant no. I 02979-N35. ' alternative_title: - LIPIcs article_processing_charge: No author: - first_name: Ranita full_name: Biswas, Ranita id: 3C2B033E-F248-11E8-B48F-1D18A9856A87 last_name: Biswas orcid: 0000-0002-5372-7890 - first_name: Sebastiano full_name: Cultrera di Montesano, Sebastiano id: 34D2A09C-F248-11E8-B48F-1D18A9856A87 last_name: Cultrera di Montesano orcid: 0000-0001-6249-0832 - first_name: Herbert full_name: Edelsbrunner, Herbert id: 3FB178DA-F248-11E8-B48F-1D18A9856A87 last_name: Edelsbrunner orcid: 0000-0002-9823-6833 - first_name: Morteza full_name: Saghafian, Morteza last_name: Saghafian citation: ama: 'Biswas R, Cultrera di Montesano S, Edelsbrunner H, Saghafian M. A window to the persistence of 1D maps. I: Geometric characterization of critical point pairs. LIPIcs.' apa: 'Biswas, R., Cultrera di Montesano, S., Edelsbrunner, H., & Saghafian, M. (n.d.). A window to the persistence of 1D maps. I: Geometric characterization of critical point pairs. LIPIcs. Schloss Dagstuhl - Leibniz-Zentrum für Informatik.' chicago: 'Biswas, Ranita, Sebastiano Cultrera di Montesano, Herbert Edelsbrunner, and Morteza Saghafian. “A Window to the Persistence of 1D Maps. I: Geometric Characterization of Critical Point Pairs.” LIPIcs. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, n.d.' ieee: 'R. Biswas, S. Cultrera di Montesano, H. Edelsbrunner, and M. Saghafian, “A window to the persistence of 1D maps. I: Geometric characterization of critical point pairs,” LIPIcs. Schloss Dagstuhl - Leibniz-Zentrum für Informatik.' ista: 'Biswas R, Cultrera di Montesano S, Edelsbrunner H, Saghafian M. A window to the persistence of 1D maps. I: Geometric characterization of critical point pairs. LIPIcs.' mla: 'Biswas, Ranita, et al. “A Window to the Persistence of 1D Maps. I: Geometric Characterization of Critical Point Pairs.” LIPIcs, Schloss Dagstuhl - Leibniz-Zentrum für Informatik.' short: R. Biswas, S. Cultrera di Montesano, H. Edelsbrunner, M. Saghafian, LIPIcs (n.d.). date_created: 2022-07-27T09:31:15Z date_published: 2022-07-25T00:00:00Z date_updated: 2024-03-20T09:36:56Z day: '25' ddc: - '510' department: - _id: GradSch - _id: HeEd ec_funded: 1 file: - access_level: open_access checksum: 95903f9d1649e8e437a967b6f2f64730 content_type: application/pdf creator: scultrer date_created: 2022-07-27T09:30:30Z date_updated: 2022-07-27T09:30:30Z file_id: '11661' file_name: window 1.pdf file_size: 564836 relation: main_file file_date_updated: 2022-07-27T09:30:30Z has_accepted_license: '1' language: - iso: eng month: '07' oa: 1 oa_version: Submitted Version project: - _id: 266A2E9E-B435-11E9-9278-68D0E5697425 call_identifier: H2020 grant_number: '788183' name: Alpha Shape Theory Extended - _id: 268116B8-B435-11E9-9278-68D0E5697425 call_identifier: FWF grant_number: Z00342 name: The Wittgenstein Prize - _id: 2561EBF4-B435-11E9-9278-68D0E5697425 call_identifier: FWF grant_number: I02979-N35 name: Persistence and stability of geometric complexes publication: LIPIcs publication_status: submitted publisher: Schloss Dagstuhl - Leibniz-Zentrum für Informatik quality_controlled: '1' related_material: record: - id: '15094' relation: dissertation_contains status: public status: public title: 'A window to the persistence of 1D maps. I: Geometric characterization of critical point pairs' tmp: image: /images/cc_by.png legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0) short: CC BY (4.0) type: journal_article user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 year: '2022' ... --- _id: '11658' abstract: - lang: eng text: The depth of a cell in an arrangement of n (non-vertical) great-spheres in Sd is the number of great-spheres that pass above the cell. We prove Euler-type relations, which imply extensions of the classic Dehn–Sommerville relations for convex polytopes to sublevel sets of the depth function, and we use the relations to extend the expressions for the number of faces of neighborly polytopes to the number of cells of levels in neighborly arrangements. acknowledgement: This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme, grant no. 788183, from the Wittgenstein Prize, Austrian Science Fund (FWF), grant no. Z 342-N31, and from the DFG Collaborative Research Center TRR 109, ‘Discretization in Geometry and Dynamics’, Austrian Science Fund (FWF), grant no. I 02979-N35. article_processing_charge: No author: - first_name: Ranita full_name: Biswas, Ranita id: 3C2B033E-F248-11E8-B48F-1D18A9856A87 last_name: Biswas orcid: 0000-0002-5372-7890 - first_name: Sebastiano full_name: Cultrera di Montesano, Sebastiano id: 34D2A09C-F248-11E8-B48F-1D18A9856A87 last_name: Cultrera di Montesano orcid: 0000-0001-6249-0832 - first_name: Herbert full_name: Edelsbrunner, Herbert id: 3FB178DA-F248-11E8-B48F-1D18A9856A87 last_name: Edelsbrunner orcid: 0000-0002-9823-6833 - first_name: Morteza full_name: Saghafian, Morteza id: f86f7148-b140-11ec-9577-95435b8df824 last_name: Saghafian citation: ama: 'Biswas R, Cultrera di Montesano S, Edelsbrunner H, Saghafian M. Depth in arrangements: Dehn–Sommerville–Euler relations with applications. Leibniz International Proceedings on Mathematics.' apa: 'Biswas, R., Cultrera di Montesano, S., Edelsbrunner, H., & Saghafian, M. (n.d.). Depth in arrangements: Dehn–Sommerville–Euler relations with applications. Leibniz International Proceedings on Mathematics. Schloss Dagstuhl - Leibniz Zentrum für Informatik.' chicago: 'Biswas, Ranita, Sebastiano Cultrera di Montesano, Herbert Edelsbrunner, and Morteza Saghafian. “Depth in Arrangements: Dehn–Sommerville–Euler Relations with Applications.” Leibniz International Proceedings on Mathematics. Schloss Dagstuhl - Leibniz Zentrum für Informatik, n.d.' ieee: 'R. Biswas, S. Cultrera di Montesano, H. Edelsbrunner, and M. Saghafian, “Depth in arrangements: Dehn–Sommerville–Euler relations with applications,” Leibniz International Proceedings on Mathematics. Schloss Dagstuhl - Leibniz Zentrum für Informatik.' ista: 'Biswas R, Cultrera di Montesano S, Edelsbrunner H, Saghafian M. Depth in arrangements: Dehn–Sommerville–Euler relations with applications. Leibniz International Proceedings on Mathematics.' mla: 'Biswas, Ranita, et al. “Depth in Arrangements: Dehn–Sommerville–Euler Relations with Applications.” Leibniz International Proceedings on Mathematics, Schloss Dagstuhl - Leibniz Zentrum für Informatik.' short: R. Biswas, S. Cultrera di Montesano, H. Edelsbrunner, M. Saghafian, Leibniz International Proceedings on Mathematics (n.d.). date_created: 2022-07-27T09:27:34Z date_published: 2022-07-27T00:00:00Z date_updated: 2024-03-20T09:36:56Z day: '27' ddc: - '510' department: - _id: GradSch - _id: HeEd ec_funded: 1 file: - access_level: open_access checksum: b2f511e8b1cae5f1892b0cdec341acac content_type: application/pdf creator: scultrer date_created: 2022-07-27T09:25:53Z date_updated: 2022-07-27T09:25:53Z file_id: '11659' file_name: D-S-E.pdf file_size: 639266 relation: main_file file_date_updated: 2022-07-27T09:25:53Z has_accepted_license: '1' language: - iso: eng month: '07' oa: 1 oa_version: Submitted Version project: - _id: 266A2E9E-B435-11E9-9278-68D0E5697425 call_identifier: H2020 grant_number: '788183' name: Alpha Shape Theory Extended - _id: 268116B8-B435-11E9-9278-68D0E5697425 call_identifier: FWF grant_number: Z00342 name: The Wittgenstein Prize - _id: 2561EBF4-B435-11E9-9278-68D0E5697425 call_identifier: FWF grant_number: I02979-N35 name: Persistence and stability of geometric complexes publication: Leibniz International Proceedings on Mathematics publication_status: submitted publisher: Schloss Dagstuhl - Leibniz Zentrum für Informatik quality_controlled: '1' related_material: record: - id: '15094' relation: dissertation_contains status: public status: public title: 'Depth in arrangements: Dehn–Sommerville–Euler relations with applications' tmp: image: /images/cc_by.png legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0) short: CC BY (4.0) type: journal_article user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 year: '2022' ... --- _id: '15090' abstract: - lang: eng text: Given a locally finite set A⊆Rd and a coloring χ:A→{0,1,…,s}, we introduce the chromatic Delaunay mosaic of χ, which is a Delaunay mosaic in Rs+d that represents how points of different colors mingle. Our main results are bounds on the size of the chromatic Delaunay mosaic, in which we assume that d and s are constants. For example, if A is finite with n=#A, and the coloring is random, then the chromatic Delaunay mosaic has O(n⌈d/2⌉) cells in expectation. In contrast, for Delone sets and Poisson point processes in Rd, the expected number of cells within a closed ball is only a constant times the number of points in this ball. Furthermore, in R2 all colorings of a dense set of n points have chromatic Delaunay mosaics of size O(n). This encourages the use of chromatic Delaunay mosaics in applications. article_number: '2212.03121' article_processing_charge: No author: - first_name: Ranita full_name: Biswas, Ranita id: 3C2B033E-F248-11E8-B48F-1D18A9856A87 last_name: Biswas orcid: 0000-0002-5372-7890 - first_name: Sebastiano full_name: Cultrera di Montesano, Sebastiano id: 34D2A09C-F248-11E8-B48F-1D18A9856A87 last_name: Cultrera di Montesano orcid: 0000-0001-6249-0832 - first_name: Ondrej full_name: Draganov, Ondrej id: 2B23F01E-F248-11E8-B48F-1D18A9856A87 last_name: Draganov - first_name: Herbert full_name: Edelsbrunner, Herbert id: 3FB178DA-F248-11E8-B48F-1D18A9856A87 last_name: Edelsbrunner orcid: 0000-0002-9823-6833 - first_name: Morteza full_name: Saghafian, Morteza id: f86f7148-b140-11ec-9577-95435b8df824 last_name: Saghafian citation: ama: Biswas R, Cultrera di Montesano S, Draganov O, Edelsbrunner H, Saghafian M. On the size of chromatic Delaunay mosaics. arXiv. apa: Biswas, R., Cultrera di Montesano, S., Draganov, O., Edelsbrunner, H., & Saghafian, M. (n.d.). On the size of chromatic Delaunay mosaics. arXiv. chicago: Biswas, Ranita, Sebastiano Cultrera di Montesano, Ondrej Draganov, Herbert Edelsbrunner, and Morteza Saghafian. “On the Size of Chromatic Delaunay Mosaics.” ArXiv, n.d. ieee: R. Biswas, S. Cultrera di Montesano, O. Draganov, H. Edelsbrunner, and M. Saghafian, “On the size of chromatic Delaunay mosaics,” arXiv. . ista: Biswas R, Cultrera di Montesano S, Draganov O, Edelsbrunner H, Saghafian M. On the size of chromatic Delaunay mosaics. arXiv, 2212.03121. mla: Biswas, Ranita, et al. “On the Size of Chromatic Delaunay Mosaics.” ArXiv, 2212.03121. short: R. Biswas, S. Cultrera di Montesano, O. Draganov, H. Edelsbrunner, M. Saghafian, ArXiv (n.d.). date_created: 2024-03-08T09:54:20Z date_published: 2022-12-06T00:00:00Z date_updated: 2024-03-20T09:36:56Z day: '06' department: - _id: HeEd ec_funded: 1 external_id: arxiv: - '2212.03121' language: - iso: eng main_file_link: - open_access: '1' url: https://arxiv.org/abs/2212.03121 month: '12' oa: 1 oa_version: Preprint project: - _id: 266A2E9E-B435-11E9-9278-68D0E5697425 call_identifier: H2020 grant_number: '788183' name: Alpha Shape Theory Extended - _id: 0aa4bc98-070f-11eb-9043-e6fff9c6a316 grant_number: I4887 name: Discretization in Geometry and Dynamics - _id: 268116B8-B435-11E9-9278-68D0E5697425 call_identifier: FWF grant_number: Z00342 name: The Wittgenstein Prize publication: arXiv publication_status: submitted related_material: record: - id: '15094' relation: dissertation_contains status: public status: public title: On the size of chromatic Delaunay mosaics tmp: image: /images/cc_by.png legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0) short: CC BY (4.0) type: preprint user_id: 8b945eb4-e2f2-11eb-945a-df72226e66a9 year: '2022' ... --- _id: '9604' abstract: - lang: eng text: Generalizing Lee’s inductive argument for counting the cells of higher order Voronoi tessellations in ℝ² to ℝ³, we get precise relations in terms of Morse theoretic quantities for piecewise constant functions on planar arrangements. Specifically, we prove that for a generic set of n ≥ 5 points in ℝ³, the number of regions in the order-k Voronoi tessellation is N_{k-1} - binom(k,2)n + n, for 1 ≤ k ≤ n-1, in which N_{k-1} is the sum of Euler characteristics of these function’s first k-1 sublevel sets. We get similar expressions for the vertices, edges, and polygons of the order-k Voronoi tessellation. alternative_title: - LIPIcs article_number: '16' article_processing_charge: No author: - first_name: Ranita full_name: Biswas, Ranita id: 3C2B033E-F248-11E8-B48F-1D18A9856A87 last_name: Biswas orcid: 0000-0002-5372-7890 - first_name: Sebastiano full_name: Cultrera di Montesano, Sebastiano id: 34D2A09C-F248-11E8-B48F-1D18A9856A87 last_name: Cultrera di Montesano orcid: 0000-0001-6249-0832 - first_name: Herbert full_name: Edelsbrunner, Herbert id: 3FB178DA-F248-11E8-B48F-1D18A9856A87 last_name: Edelsbrunner orcid: 0000-0002-9823-6833 - first_name: Morteza full_name: Saghafian, Morteza last_name: Saghafian citation: ama: 'Biswas R, Cultrera di Montesano S, Edelsbrunner H, Saghafian M. Counting cells of order-k voronoi tessellations in ℝ3 with morse theory. In: Leibniz International Proceedings in Informatics. Vol 189. Schloss Dagstuhl - Leibniz-Zentrum für Informatik; 2021. doi:10.4230/LIPIcs.SoCG.2021.16' apa: 'Biswas, R., Cultrera di Montesano, S., Edelsbrunner, H., & Saghafian, M. (2021). Counting cells of order-k voronoi tessellations in ℝ3 with morse theory. In Leibniz International Proceedings in Informatics (Vol. 189). Online: Schloss Dagstuhl - Leibniz-Zentrum für Informatik. https://doi.org/10.4230/LIPIcs.SoCG.2021.16' chicago: Biswas, Ranita, Sebastiano Cultrera di Montesano, Herbert Edelsbrunner, and Morteza Saghafian. “Counting Cells of Order-k Voronoi Tessellations in ℝ3 with Morse Theory.” In Leibniz International Proceedings in Informatics, Vol. 189. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. https://doi.org/10.4230/LIPIcs.SoCG.2021.16. ieee: R. Biswas, S. Cultrera di Montesano, H. Edelsbrunner, and M. Saghafian, “Counting cells of order-k voronoi tessellations in ℝ3 with morse theory,” in Leibniz International Proceedings in Informatics, Online, 2021, vol. 189. ista: 'Biswas R, Cultrera di Montesano S, Edelsbrunner H, Saghafian M. 2021. Counting cells of order-k voronoi tessellations in ℝ3 with morse theory. Leibniz International Proceedings in Informatics. SoCG: International Symposium on Computational Geometry, LIPIcs, vol. 189, 16.' mla: Biswas, Ranita, et al. “Counting Cells of Order-k Voronoi Tessellations in ℝ3 with Morse Theory.” Leibniz International Proceedings in Informatics, vol. 189, 16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021, doi:10.4230/LIPIcs.SoCG.2021.16. short: R. Biswas, S. Cultrera di Montesano, H. Edelsbrunner, M. Saghafian, in:, Leibniz International Proceedings in Informatics, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. conference: end_date: 2021-06-11 location: Online name: 'SoCG: International Symposium on Computational Geometry' start_date: 2021-06-07 date_created: 2021-06-27T22:01:48Z date_published: 2021-06-02T00:00:00Z date_updated: 2023-02-23T14:02:28Z day: '02' ddc: - '516' department: - _id: HeEd doi: 10.4230/LIPIcs.SoCG.2021.16 ec_funded: 1 file: - access_level: open_access checksum: 22b11a719018b22ecba2471b51f2eb40 content_type: application/pdf creator: asandaue date_created: 2021-06-28T13:11:39Z date_updated: 2021-06-28T13:11:39Z file_id: '9611' file_name: 2021_LIPIcs_Biswas.pdf file_size: 727817 relation: main_file success: 1 file_date_updated: 2021-06-28T13:11:39Z has_accepted_license: '1' intvolume: ' 189' language: - iso: eng month: '06' oa: 1 oa_version: Published Version project: - _id: 266A2E9E-B435-11E9-9278-68D0E5697425 call_identifier: H2020 grant_number: '788183' name: Alpha Shape Theory Extended - _id: 268116B8-B435-11E9-9278-68D0E5697425 call_identifier: FWF grant_number: Z00342 name: The Wittgenstein Prize - _id: 0aa4bc98-070f-11eb-9043-e6fff9c6a316 grant_number: I4887 name: Discretization in Geometry and Dynamics publication: Leibniz International Proceedings in Informatics publication_identifier: isbn: - '9783959771849' issn: - '18688969' publication_status: published publisher: Schloss Dagstuhl - Leibniz-Zentrum für Informatik quality_controlled: '1' scopus_import: '1' status: public title: Counting cells of order-k voronoi tessellations in ℝ3 with morse theory tmp: image: /images/cc_by.png legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0) short: CC BY (4.0) type: conference user_id: D865714E-FA4E-11E9-B85B-F5C5E5697425 volume: 189 year: '2021' ... --- _id: '9824' abstract: - lang: eng text: We define a new compact coordinate system in which each integer triplet addresses a voxel in the BCC grid, and we investigate some of its properties. We propose a characterization of 3D discrete analytical planes with their topological features (in the Cartesian and in the new coordinate system) such as the interrelation between the thickness of the plane and the separability constraint we aim to obtain. acknowledgement: 'This work has been partially supported by the Ministry of Education, Science and Technological Development of the Republic of Serbia through the project no. 451-03-68/2020-14/200156: “Innovative scientific and artistic research from the FTS (activity) domain” (LČ), the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme, grant no. 788183 (RB), and the DFG Collaborative Research Center TRR 109, ‘Discretization in Geometry and Dynamics’, Austrian Science Fund (FWF), grant no. I 02979-N35 (RB).' alternative_title: - LNCS article_processing_charge: No author: - first_name: Lidija full_name: Čomić, Lidija last_name: Čomić - first_name: Rita full_name: Zrour, Rita last_name: Zrour - first_name: Gaëlle full_name: Largeteau-Skapin, Gaëlle last_name: Largeteau-Skapin - first_name: Ranita full_name: Biswas, Ranita id: 3C2B033E-F248-11E8-B48F-1D18A9856A87 last_name: Biswas orcid: 0000-0002-5372-7890 - first_name: Eric full_name: Andres, Eric last_name: Andres citation: ama: 'Čomić L, Zrour R, Largeteau-Skapin G, Biswas R, Andres E. Body centered cubic grid - coordinate system and discrete analytical plane definition. In: Discrete Geometry and Mathematical Morphology. Vol 12708. Springer Nature; 2021:152-163. doi:10.1007/978-3-030-76657-3_10' apa: 'Čomić, L., Zrour, R., Largeteau-Skapin, G., Biswas, R., & Andres, E. (2021). Body centered cubic grid - coordinate system and discrete analytical plane definition. In Discrete Geometry and Mathematical Morphology (Vol. 12708, pp. 152–163). Uppsala, Sweden: Springer Nature. https://doi.org/10.1007/978-3-030-76657-3_10' chicago: Čomić, Lidija, Rita Zrour, Gaëlle Largeteau-Skapin, Ranita Biswas, and Eric Andres. “Body Centered Cubic Grid - Coordinate System and Discrete Analytical Plane Definition.” In Discrete Geometry and Mathematical Morphology, 12708:152–63. Springer Nature, 2021. https://doi.org/10.1007/978-3-030-76657-3_10. ieee: L. Čomić, R. Zrour, G. Largeteau-Skapin, R. Biswas, and E. Andres, “Body centered cubic grid - coordinate system and discrete analytical plane definition,” in Discrete Geometry and Mathematical Morphology, Uppsala, Sweden, 2021, vol. 12708, pp. 152–163. ista: 'Čomić L, Zrour R, Largeteau-Skapin G, Biswas R, Andres E. 2021. Body centered cubic grid - coordinate system and discrete analytical plane definition. Discrete Geometry and Mathematical Morphology. DGMM: International Conference on Discrete Geometry and Mathematical Morphology, LNCS, vol. 12708, 152–163.' mla: Čomić, Lidija, et al. “Body Centered Cubic Grid - Coordinate System and Discrete Analytical Plane Definition.” Discrete Geometry and Mathematical Morphology, vol. 12708, Springer Nature, 2021, pp. 152–63, doi:10.1007/978-3-030-76657-3_10. short: L. Čomić, R. Zrour, G. Largeteau-Skapin, R. Biswas, E. Andres, in:, Discrete Geometry and Mathematical Morphology, Springer Nature, 2021, pp. 152–163. conference: end_date: 2021-05-27 location: Uppsala, Sweden name: 'DGMM: International Conference on Discrete Geometry and Mathematical Morphology' start_date: 2021-05-24 date_created: 2021-08-08T22:01:29Z date_published: 2021-05-16T00:00:00Z date_updated: 2022-05-31T06:58:21Z day: '16' department: - _id: HeEd doi: 10.1007/978-3-030-76657-3_10 ec_funded: 1 intvolume: ' 12708' language: - iso: eng month: '05' oa_version: None page: 152-163 project: - _id: 266A2E9E-B435-11E9-9278-68D0E5697425 call_identifier: H2020 grant_number: '788183' name: Alpha Shape Theory Extended - _id: 2561EBF4-B435-11E9-9278-68D0E5697425 call_identifier: FWF grant_number: I02979-N35 name: Persistence and stability of geometric complexes publication: Discrete Geometry and Mathematical Morphology publication_identifier: eissn: - '16113349' isbn: - '9783030766566' issn: - '03029743' publication_status: published publisher: Springer Nature quality_controlled: '1' scopus_import: '1' status: public title: Body centered cubic grid - coordinate system and discrete analytical plane definition type: conference user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 volume: 12708 year: '2021' ... --- _id: '9249' abstract: - lang: eng text: Rhombic dodecahedron is a space filling polyhedron which represents the close packing of spheres in 3D space and the Voronoi structures of the face centered cubic (FCC) lattice. In this paper, we describe a new coordinate system where every 3-integer coordinates grid point corresponds to a rhombic dodecahedron centroid. In order to illustrate the interest of the new coordinate system, we propose the characterization of 3D digital plane with its topological features, such as the interrelation between the thickness of the digital plane and the separability constraint we aim to obtain. We also present the characterization of 3D digital lines and study it as the intersection of multiple digital planes. Characterization of 3D digital sphere with relevant topological features is proposed as well along with the 48-symmetry appearing in the new coordinate system. acknowledgement: "This work has been partially supported by the European Research Council (ERC) under\r\nthe European Union’s Horizon 2020 research and innovation programme, grant no. 788183, and the DFG Collaborative Research Center TRR 109, ‘Discretization in Geometry and Dynamics’, Austrian Science Fund (FWF), grant no. I 02979-N35. " article_processing_charge: No article_type: original author: - first_name: Ranita full_name: Biswas, Ranita id: 3C2B033E-F248-11E8-B48F-1D18A9856A87 last_name: Biswas orcid: 0000-0002-5372-7890 - first_name: Gaëlle full_name: Largeteau-Skapin, Gaëlle last_name: Largeteau-Skapin - first_name: Rita full_name: Zrour, Rita last_name: Zrour - first_name: Eric full_name: Andres, Eric last_name: Andres citation: ama: Biswas R, Largeteau-Skapin G, Zrour R, Andres E. Digital objects in rhombic dodecahedron grid. Mathematical Morphology - Theory and Applications. 2020;4(1):143-158. doi:10.1515/mathm-2020-0106 apa: Biswas, R., Largeteau-Skapin, G., Zrour, R., & Andres, E. (2020). Digital objects in rhombic dodecahedron grid. Mathematical Morphology - Theory and Applications. De Gruyter. https://doi.org/10.1515/mathm-2020-0106 chicago: Biswas, Ranita, Gaëlle Largeteau-Skapin, Rita Zrour, and Eric Andres. “Digital Objects in Rhombic Dodecahedron Grid.” Mathematical Morphology - Theory and Applications. De Gruyter, 2020. https://doi.org/10.1515/mathm-2020-0106. ieee: R. Biswas, G. Largeteau-Skapin, R. Zrour, and E. Andres, “Digital objects in rhombic dodecahedron grid,” Mathematical Morphology - Theory and Applications, vol. 4, no. 1. De Gruyter, pp. 143–158, 2020. ista: Biswas R, Largeteau-Skapin G, Zrour R, Andres E. 2020. Digital objects in rhombic dodecahedron grid. Mathematical Morphology - Theory and Applications. 4(1), 143–158. mla: Biswas, Ranita, et al. “Digital Objects in Rhombic Dodecahedron Grid.” Mathematical Morphology - Theory and Applications, vol. 4, no. 1, De Gruyter, 2020, pp. 143–58, doi:10.1515/mathm-2020-0106. short: R. Biswas, G. Largeteau-Skapin, R. Zrour, E. Andres, Mathematical Morphology - Theory and Applications 4 (2020) 143–158. date_created: 2021-03-16T08:55:19Z date_published: 2020-11-17T00:00:00Z date_updated: 2021-03-22T09:01:50Z day: '17' ddc: - '510' department: - _id: HeEd doi: 10.1515/mathm-2020-0106 ec_funded: 1 file: - access_level: open_access checksum: 4a1043fa0548a725d464017fe2483ce0 content_type: application/pdf creator: dernst date_created: 2021-03-22T08:56:37Z date_updated: 2021-03-22T08:56:37Z file_id: '9272' file_name: 2020_MathMorpholTheoryAppl_Biswas.pdf file_size: 3668725 relation: main_file success: 1 file_date_updated: 2021-03-22T08:56:37Z has_accepted_license: '1' intvolume: ' 4' issue: '1' language: - iso: eng month: '11' oa: 1 oa_version: Published Version page: 143-158 project: - _id: 266A2E9E-B435-11E9-9278-68D0E5697425 call_identifier: H2020 grant_number: '788183' name: Alpha Shape Theory Extended - _id: 2561EBF4-B435-11E9-9278-68D0E5697425 call_identifier: FWF grant_number: I02979-N35 name: Persistence and stability of geometric complexes publication: Mathematical Morphology - Theory and Applications publication_identifier: issn: - 2353-3390 publication_status: published publisher: De Gruyter quality_controlled: '1' status: public title: Digital objects in rhombic dodecahedron grid tmp: image: /images/cc_by.png legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0) short: CC BY (4.0) type: journal_article user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 volume: 4 year: '2020' ... --- _id: '6163' abstract: - lang: eng text: We propose a new non-orthogonal basis to express the 3D Euclidean space in terms of a regular grid. Every grid point, each represented by integer 3-coordinates, corresponds to rhombic dodecahedron centroid. Rhombic dodecahedron is a space filling polyhedron which represents the close packing of spheres in 3D space and the Voronoi structures of the face centered cubic (FCC) lattice. In order to illustrate the interest of the new coordinate system, we propose the characterization of 3D digital plane with its topological features, such as the interrelation between the thickness of the digital plane and the separability constraint we aim to obtain. A characterization of a 3D digital sphere with relevant topological features is proposed as well with the help of a 48 symmetry that comes with the new coordinate system. alternative_title: - LNCS article_processing_charge: No author: - first_name: Ranita full_name: Biswas, Ranita id: 3C2B033E-F248-11E8-B48F-1D18A9856A87 last_name: Biswas orcid: 0000-0002-5372-7890 - first_name: Gaëlle full_name: Largeteau-Skapin, Gaëlle last_name: Largeteau-Skapin - first_name: Rita full_name: Zrour, Rita last_name: Zrour - first_name: Eric full_name: Andres, Eric last_name: Andres citation: ama: 'Biswas R, Largeteau-Skapin G, Zrour R, Andres E. Rhombic dodecahedron grid—coordinate system and 3D digital object definitions. In: 21st IAPR International Conference on Discrete Geometry for Computer Imagery. Vol 11414. Berlin, Heidelberg: Springer Berlin Heidelberg; 2019:27-37. doi:10.1007/978-3-030-14085-4_3' apa: 'Biswas, R., Largeteau-Skapin, G., Zrour, R., & Andres, E. (2019). Rhombic dodecahedron grid—coordinate system and 3D digital object definitions. In 21st IAPR International Conference on Discrete Geometry for Computer Imagery (Vol. 11414, pp. 27–37). Berlin, Heidelberg: Springer Berlin Heidelberg. https://doi.org/10.1007/978-3-030-14085-4_3' chicago: 'Biswas, Ranita, Gaëlle Largeteau-Skapin, Rita Zrour, and Eric Andres. “Rhombic Dodecahedron Grid—Coordinate System and 3D Digital Object Definitions.” In 21st IAPR International Conference on Discrete Geometry for Computer Imagery, 11414:27–37. Berlin, Heidelberg: Springer Berlin Heidelberg, 2019. https://doi.org/10.1007/978-3-030-14085-4_3.' ieee: R. Biswas, G. Largeteau-Skapin, R. Zrour, and E. Andres, “Rhombic dodecahedron grid—coordinate system and 3D digital object definitions,” in 21st IAPR International Conference on Discrete Geometry for Computer Imagery, Marne-la-Vallée, France, 2019, vol. 11414, pp. 27–37. ista: 'Biswas R, Largeteau-Skapin G, Zrour R, Andres E. 2019. Rhombic dodecahedron grid—coordinate system and 3D digital object definitions. 21st IAPR International Conference on Discrete Geometry for Computer Imagery. DGCI: International Conference on Discrete Geometry for Computer Imagery, LNCS, vol. 11414, 27–37.' mla: Biswas, Ranita, et al. “Rhombic Dodecahedron Grid—Coordinate System and 3D Digital Object Definitions.” 21st IAPR International Conference on Discrete Geometry for Computer Imagery, vol. 11414, Springer Berlin Heidelberg, 2019, pp. 27–37, doi:10.1007/978-3-030-14085-4_3. short: R. Biswas, G. Largeteau-Skapin, R. Zrour, E. Andres, in:, 21st IAPR International Conference on Discrete Geometry for Computer Imagery, Springer Berlin Heidelberg, Berlin, Heidelberg, 2019, pp. 27–37. conference: end_date: 2019-03-28 location: Marne-la-Vallée, France name: 'DGCI: International Conference on Discrete Geometry for Computer Imagery' start_date: 2019-03-26 date_created: 2019-03-21T12:12:19Z date_published: 2019-02-23T00:00:00Z date_updated: 2022-01-27T14:25:17Z day: '23' doi: 10.1007/978-3-030-14085-4_3 extern: '1' intvolume: ' 11414' language: - iso: eng month: '02' oa_version: None page: 27-37 place: Berlin, Heidelberg publication: 21st IAPR International Conference on Discrete Geometry for Computer Imagery publication_identifier: isbn: - 978-3-6624-6446-5 - 978-3-6624-6447-2 issn: - 0302-9743 - 1611-3349 publication_status: published publisher: Springer Berlin Heidelberg quality_controlled: '1' status: public title: Rhombic dodecahedron grid—coordinate system and 3D digital object definitions type: conference user_id: 8b945eb4-e2f2-11eb-945a-df72226e66a9 volume: 11414 year: '2019' ... --- _id: '6164' abstract: - lang: eng text: In this paper, we propose an algorithm to build discrete spherical shell having integer center and real-valued inner and outer radii on the face-centered cubic (FCC) grid. We address the problem by mapping it to a 2D scenario and building the shell layer by layer on hexagonal grids with additive manufacturing in mind. The layered hexagonal grids get shifted according to need as we move from one layer to another and forms the FCC grid in 3D. However, we restrict our computation strictly to 2D in order to utilize symmetry and simplicity. alternative_title: - LNCS article_processing_charge: No author: - first_name: Girish full_name: Koshti, Girish last_name: Koshti - first_name: Ranita full_name: Biswas, Ranita id: 3C2B033E-F248-11E8-B48F-1D18A9856A87 last_name: Biswas orcid: 0000-0002-5372-7890 - first_name: Gaëlle full_name: Largeteau-Skapin, Gaëlle last_name: Largeteau-Skapin - first_name: Rita full_name: Zrour, Rita last_name: Zrour - first_name: Eric full_name: Andres, Eric last_name: Andres - first_name: Partha full_name: Bhowmick, Partha last_name: Bhowmick citation: ama: 'Koshti G, Biswas R, Largeteau-Skapin G, Zrour R, Andres E, Bhowmick P. Sphere construction on the FCC grid interpreted as layered hexagonal grids in 3D. In: 19th International Workshop. Vol 11255. Cham: Springer; 2018:82-96. doi:10.1007/978-3-030-05288-1_7' apa: 'Koshti, G., Biswas, R., Largeteau-Skapin, G., Zrour, R., Andres, E., & Bhowmick, P. (2018). Sphere construction on the FCC grid interpreted as layered hexagonal grids in 3D. In 19th International Workshop (Vol. 11255, pp. 82–96). Cham: Springer. https://doi.org/10.1007/978-3-030-05288-1_7' chicago: 'Koshti, Girish, Ranita Biswas, Gaëlle Largeteau-Skapin, Rita Zrour, Eric Andres, and Partha Bhowmick. “Sphere Construction on the FCC Grid Interpreted as Layered Hexagonal Grids in 3D.” In 19th International Workshop, 11255:82–96. Cham: Springer, 2018. https://doi.org/10.1007/978-3-030-05288-1_7.' ieee: G. Koshti, R. Biswas, G. Largeteau-Skapin, R. Zrour, E. Andres, and P. Bhowmick, “Sphere construction on the FCC grid interpreted as layered hexagonal grids in 3D,” in 19th International Workshop, Porto, Portugal, 2018, vol. 11255, pp. 82–96. ista: 'Koshti G, Biswas R, Largeteau-Skapin G, Zrour R, Andres E, Bhowmick P. 2018. Sphere construction on the FCC grid interpreted as layered hexagonal grids in 3D. 19th International Workshop. IWCIA: International Workshop on Combinatorial Image Analysis, LNCS, vol. 11255, 82–96.' mla: Koshti, Girish, et al. “Sphere Construction on the FCC Grid Interpreted as Layered Hexagonal Grids in 3D.” 19th International Workshop, vol. 11255, Springer, 2018, pp. 82–96, doi:10.1007/978-3-030-05288-1_7. short: G. Koshti, R. Biswas, G. Largeteau-Skapin, R. Zrour, E. Andres, P. Bhowmick, in:, 19th International Workshop, Springer, Cham, 2018, pp. 82–96. conference: end_date: 2018-11-24 location: Porto, Portugal name: 'IWCIA: International Workshop on Combinatorial Image Analysis' start_date: 2018-11-22 date_created: 2019-03-21T12:16:58Z date_published: 2018-11-22T00:00:00Z date_updated: 2022-01-27T15:26:39Z day: '22' doi: 10.1007/978-3-030-05288-1_7 extern: '1' intvolume: ' 11255' language: - iso: eng month: '11' oa_version: None page: 82-96 place: Cham publication: 19th International Workshop publication_identifier: eisbn: - 978-3-030-05288-1 eissn: - 1611-3349 isbn: - 978-3-030-05287-4 issn: - 0302-9743 publication_status: published publisher: Springer quality_controlled: '1' status: public title: Sphere construction on the FCC grid interpreted as layered hexagonal grids in 3D type: conference user_id: 8b945eb4-e2f2-11eb-945a-df72226e66a9 volume: 11255 year: '2018' ... --- _id: '5800' abstract: - lang: eng text: This paper presents a novel study on the functional gradation of coordinate planes in connection with the thinnest and tunnel-free (i.e., naive) discretization of sphere in the integer space. For each of the 48-symmetric quadraginta octants of naive sphere with integer radius and integer center, we show that the corresponding voxel set forms a bijection with its projected pixel set on a unique coordinate plane, which thereby serves as its functional plane. We use this fundamental property to prove several other theoretical results for naive sphere. First, the quadraginta octants form symmetry groups and subgroups with certain equivalent topological properties. Second, a naive sphere is always unique and consists of fewest voxels. Third, it is efficiently constructible from its functional-plane projection. And finally, a special class of 4-symmetric discrete 3D circles can be constructed on a naive sphere based on back projection from the functional plane. author: - first_name: Ranita full_name: Biswas, Ranita id: 3C2B033E-F248-11E8-B48F-1D18A9856A87 last_name: Biswas orcid: 0000-0002-5372-7890 - first_name: Partha full_name: Bhowmick, Partha last_name: Bhowmick citation: ama: Biswas R, Bhowmick P. On the functionality and usefulness of Quadraginta octants of naive sphere. Journal of Mathematical Imaging and Vision. 2017;59(1):69-83. doi:10.1007/s10851-017-0718-4 apa: Biswas, R., & Bhowmick, P. (2017). On the functionality and usefulness of Quadraginta octants of naive sphere. Journal of Mathematical Imaging and Vision. Springer Nature. https://doi.org/10.1007/s10851-017-0718-4 chicago: Biswas, Ranita, and Partha Bhowmick. “On the Functionality and Usefulness of Quadraginta Octants of Naive Sphere.” Journal of Mathematical Imaging and Vision. Springer Nature, 2017. https://doi.org/10.1007/s10851-017-0718-4. ieee: R. Biswas and P. Bhowmick, “On the functionality and usefulness of Quadraginta octants of naive sphere,” Journal of Mathematical Imaging and Vision, vol. 59, no. 1. Springer Nature, pp. 69–83, 2017. ista: Biswas R, Bhowmick P. 2017. On the functionality and usefulness of Quadraginta octants of naive sphere. Journal of Mathematical Imaging and Vision. 59(1), 69–83. mla: Biswas, Ranita, and Partha Bhowmick. “On the Functionality and Usefulness of Quadraginta Octants of Naive Sphere.” Journal of Mathematical Imaging and Vision, vol. 59, no. 1, Springer Nature, 2017, pp. 69–83, doi:10.1007/s10851-017-0718-4. short: R. Biswas, P. Bhowmick, Journal of Mathematical Imaging and Vision 59 (2017) 69–83. date_created: 2019-01-08T20:42:08Z date_published: 2017-09-01T00:00:00Z date_updated: 2021-01-12T08:03:34Z day: '01' doi: 10.1007/s10851-017-0718-4 extern: '1' intvolume: ' 59' issue: '1' language: - iso: eng month: '09' oa_version: None page: 69-83 publication: Journal of Mathematical Imaging and Vision publication_identifier: issn: - '09249907' publication_status: published publisher: Springer Nature quality_controlled: '1' status: public title: On the functionality and usefulness of Quadraginta octants of naive sphere type: journal_article user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 volume: 59 year: '2017' ... --- _id: '5799' abstract: - lang: eng text: We construct a polyhedral surface called a graceful surface, which provides best possible approximation to a given sphere regarding certain criteria. In digital geometry terms, the graceful surface is uniquely characterized by its minimality while guaranteeing the connectivity of certain discrete (polyhedral) curves defined on it. The notion of “gracefulness” was first proposed in Brimkov and Barneva (1999) and shown to be useful for triangular mesh discretization through graceful planes and graceful lines. In this paper we extend the considerations to a nonlinear object such as a sphere. In particular, we investigate the properties of a discrete geodesic path between two voxels and show that discrete 3D circles, circular arcs, and Mobius triangles are all constructible on a graceful sphere, with guaranteed minimum thickness and the desired connectivity in the discrete topological space. author: - first_name: Ranita full_name: Biswas, Ranita id: 3C2B033E-F248-11E8-B48F-1D18A9856A87 last_name: Biswas orcid: 0000-0002-5372-7890 - first_name: Partha full_name: Bhowmick, Partha last_name: Bhowmick - first_name: Valentin E. full_name: Brimkov, Valentin E. last_name: Brimkov citation: ama: Biswas R, Bhowmick P, Brimkov VE. On the polyhedra of graceful spheres and circular geodesics. Discrete Applied Mathematics. 2017;216:362-375. doi:10.1016/j.dam.2015.11.017 apa: Biswas, R., Bhowmick, P., & Brimkov, V. E. (2017). On the polyhedra of graceful spheres and circular geodesics. Discrete Applied Mathematics. Elsevier. https://doi.org/10.1016/j.dam.2015.11.017 chicago: Biswas, Ranita, Partha Bhowmick, and Valentin E. Brimkov. “On the Polyhedra of Graceful Spheres and Circular Geodesics.” Discrete Applied Mathematics. Elsevier, 2017. https://doi.org/10.1016/j.dam.2015.11.017. ieee: R. Biswas, P. Bhowmick, and V. E. Brimkov, “On the polyhedra of graceful spheres and circular geodesics,” Discrete Applied Mathematics, vol. 216. Elsevier, pp. 362–375, 2017. ista: Biswas R, Bhowmick P, Brimkov VE. 2017. On the polyhedra of graceful spheres and circular geodesics. Discrete Applied Mathematics. 216, 362–375. mla: Biswas, Ranita, et al. “On the Polyhedra of Graceful Spheres and Circular Geodesics.” Discrete Applied Mathematics, vol. 216, Elsevier, 2017, pp. 362–75, doi:10.1016/j.dam.2015.11.017. short: R. Biswas, P. Bhowmick, V.E. Brimkov, Discrete Applied Mathematics 216 (2017) 362–375. date_created: 2019-01-08T20:41:12Z date_published: 2017-01-10T00:00:00Z date_updated: 2021-01-12T08:03:33Z day: '10' doi: 10.1016/j.dam.2015.11.017 extern: '1' intvolume: ' 216' language: - iso: eng month: '01' oa_version: None page: 362-375 publication: Discrete Applied Mathematics publication_identifier: issn: - 0166-218X publication_status: published publisher: Elsevier quality_controlled: '1' status: public title: On the polyhedra of graceful spheres and circular geodesics type: journal_article user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 volume: 216 year: '2017' ... --- _id: '5801' abstract: - lang: eng text: Space filling circles and spheres have various applications in mathematical imaging and physical modeling. In this paper, we first show how the thinnest (i.e., 2-minimal) model of digital sphere can be augmented to a space filling model by fixing certain “simple voxels” and “filler voxels” associated with it. Based on elementary number-theoretic properties of such voxels, we design an efficient incremental algorithm for generation of these space filling spheres with successively increasing radius. The novelty of the proposed technique is established further through circular space filling on 3D digital plane. As evident from a preliminary set of experimental result, this can particularly be useful for parallel computing of 3D Voronoi diagrams in the digital space. alternative_title: - LNCS article_processing_charge: No author: - first_name: Shivam full_name: Dwivedi, Shivam last_name: Dwivedi - first_name: Aniket full_name: Gupta, Aniket last_name: Gupta - first_name: Siddhant full_name: Roy, Siddhant last_name: Roy - first_name: Ranita full_name: Biswas, Ranita id: 3C2B033E-F248-11E8-B48F-1D18A9856A87 last_name: Biswas orcid: 0000-0002-5372-7890 - first_name: Partha full_name: Bhowmick, Partha last_name: Bhowmick citation: ama: 'Dwivedi S, Gupta A, Roy S, Biswas R, Bhowmick P. Fast and Efficient Incremental Algorithms for Circular and Spherical Propagation in Integer Space. In: 20th IAPR International Conference. Vol 10502. Cham: Springer Nature; 2017:347-359. doi:10.1007/978-3-319-66272-5_28' apa: 'Dwivedi, S., Gupta, A., Roy, S., Biswas, R., & Bhowmick, P. (2017). Fast and Efficient Incremental Algorithms for Circular and Spherical Propagation in Integer Space. In 20th IAPR International Conference (Vol. 10502, pp. 347–359). Cham: Springer Nature. https://doi.org/10.1007/978-3-319-66272-5_28' chicago: 'Dwivedi, Shivam, Aniket Gupta, Siddhant Roy, Ranita Biswas, and Partha Bhowmick. “Fast and Efficient Incremental Algorithms for Circular and Spherical Propagation in Integer Space.” In 20th IAPR International Conference, 10502:347–59. Cham: Springer Nature, 2017. https://doi.org/10.1007/978-3-319-66272-5_28.' ieee: S. Dwivedi, A. Gupta, S. Roy, R. Biswas, and P. Bhowmick, “Fast and Efficient Incremental Algorithms for Circular and Spherical Propagation in Integer Space,” in 20th IAPR International Conference, Vienna, Austria, 2017, vol. 10502, pp. 347–359. ista: 'Dwivedi S, Gupta A, Roy S, Biswas R, Bhowmick P. 2017. Fast and Efficient Incremental Algorithms for Circular and Spherical Propagation in Integer Space. 20th IAPR International Conference. DGCI: International Conference on Discrete Geometry for Computer Imagery, LNCS, vol. 10502, 347–359.' mla: Dwivedi, Shivam, et al. “Fast and Efficient Incremental Algorithms for Circular and Spherical Propagation in Integer Space.” 20th IAPR International Conference, vol. 10502, Springer Nature, 2017, pp. 347–59, doi:10.1007/978-3-319-66272-5_28. short: S. Dwivedi, A. Gupta, S. Roy, R. Biswas, P. Bhowmick, in:, 20th IAPR International Conference, Springer Nature, Cham, 2017, pp. 347–359. conference: end_date: 2017-09-21 location: Vienna, Austria name: 'DGCI: International Conference on Discrete Geometry for Computer Imagery' start_date: 2017-09-19 date_created: 2019-01-08T20:42:22Z date_published: 2017-08-22T00:00:00Z date_updated: 2022-01-27T15:34:25Z day: '22' doi: 10.1007/978-3-319-66272-5_28 extern: '1' intvolume: ' 10502' language: - iso: eng month: '08' oa_version: None page: 347-359 place: Cham publication: 20th IAPR International Conference publication_identifier: eisbn: - 978-3-319-66272-5 eissn: - 1611-3349 isbn: - 978-3-319-66271-8 issn: - 0302-9743 publication_status: published publisher: Springer Nature quality_controlled: '1' status: public title: Fast and Efficient Incremental Algorithms for Circular and Spherical Propagation in Integer Space type: conference user_id: 8b945eb4-e2f2-11eb-945a-df72226e66a9 volume: 10502 year: '2017' ... --- _id: '5803' abstract: - lang: eng text: Different distance metrics produce Voronoi diagrams with different properties. It is a well-known that on the (real) 2D plane or even on any 3D plane, a Voronoi diagram (VD) based on the Euclidean distance metric produces convex Voronoi regions. In this paper, we first show that this metric produces a persistent VD on the 2D digital plane, as it comprises digitally convex Voronoi regions and hence correctly approximates the corresponding VD on the 2D real plane. Next, we show that on a 3D digital plane D, the Euclidean metric spanning over its voxel set does not guarantee a digital VD which is persistent with the real-space VD. As a solution, we introduce a novel concept of functional-plane-convexity, which is ensured by the Euclidean metric spanning over the pedal set of D. Necessary proofs and some visual result have been provided to adjudge the merit and usefulness of the proposed concept. alternative_title: - LNCS article_processing_charge: No author: - first_name: Ranita full_name: Biswas, Ranita id: 3C2B033E-F248-11E8-B48F-1D18A9856A87 last_name: Biswas orcid: 0000-0002-5372-7890 - first_name: Partha full_name: Bhowmick, Partha last_name: Bhowmick citation: ama: 'Biswas R, Bhowmick P. Construction of persistent Voronoi diagram on 3D digital plane. In: Combinatorial Image Analysis. Vol 10256. Cham: Springer Nature; 2017:93-104. doi:10.1007/978-3-319-59108-7_8' apa: 'Biswas, R., & Bhowmick, P. (2017). Construction of persistent Voronoi diagram on 3D digital plane. In Combinatorial image analysis (Vol. 10256, pp. 93–104). Cham: Springer Nature. https://doi.org/10.1007/978-3-319-59108-7_8' chicago: 'Biswas, Ranita, and Partha Bhowmick. “Construction of Persistent Voronoi Diagram on 3D Digital Plane.” In Combinatorial Image Analysis, 10256:93–104. Cham: Springer Nature, 2017. https://doi.org/10.1007/978-3-319-59108-7_8.' ieee: 'R. Biswas and P. Bhowmick, “Construction of persistent Voronoi diagram on 3D digital plane,” in Combinatorial image analysis, vol. 10256, Cham: Springer Nature, 2017, pp. 93–104.' ista: 'Biswas R, Bhowmick P. 2017.Construction of persistent Voronoi diagram on 3D digital plane. In: Combinatorial image analysis. LNCS, vol. 10256, 93–104.' mla: Biswas, Ranita, and Partha Bhowmick. “Construction of Persistent Voronoi Diagram on 3D Digital Plane.” Combinatorial Image Analysis, vol. 10256, Springer Nature, 2017, pp. 93–104, doi:10.1007/978-3-319-59108-7_8. short: R. Biswas, P. Bhowmick, in:, Combinatorial Image Analysis, Springer Nature, Cham, 2017, pp. 93–104. conference: end_date: 2017-06-21 location: Plovdiv, Bulgaria name: 'IWCIA: International Workshop on Combinatorial Image Analysis' start_date: 2017-06-19 date_created: 2019-01-08T20:42:56Z date_published: 2017-05-17T00:00:00Z date_updated: 2022-01-28T07:48:24Z day: '17' department: - _id: HeEd doi: 10.1007/978-3-319-59108-7_8 extern: '1' intvolume: ' 10256' language: - iso: eng month: '05' oa_version: None page: 93-104 place: Cham publication: Combinatorial image analysis publication_identifier: isbn: - 978-3-319-59107-0 - 978-3-319-59108-7 issn: - 0302-9743 - 1611-3349 publication_status: published publisher: Springer Nature quality_controlled: '1' status: public title: Construction of persistent Voronoi diagram on 3D digital plane type: book_chapter user_id: 8b945eb4-e2f2-11eb-945a-df72226e66a9 volume: 10256 year: '2017' ... --- _id: '5802' abstract: - lang: eng text: This papers introduces a definition of digital primitives based on focal points and weighted distances (with positive weights). The proposed definition is applicable to general dimensions and covers in its gamut various regular curves and surfaces like circles, ellipses, digital spheres and hyperspheres, ellipsoids and k-ellipsoids, Cartesian k-ovals, etc. Several interesting properties are presented for this class of digital primitives such as space partitioning, topological separation, and connectivity properties. To demonstrate further the potential of this new way of defining digital primitives, we propose, as extension, another class of digital conics defined by focus-directrix combination. alternative_title: - LNCS article_processing_charge: No author: - first_name: Eric full_name: Andres, Eric last_name: Andres - first_name: Ranita full_name: Biswas, Ranita id: 3C2B033E-F248-11E8-B48F-1D18A9856A87 last_name: Biswas orcid: 0000-0002-5372-7890 - first_name: Partha full_name: Bhowmick, Partha last_name: Bhowmick citation: ama: 'Andres E, Biswas R, Bhowmick P. Digital primitives defined by weighted focal set. In: 20th IAPR International Conference. Vol 10502. Cham: Springer Nature; 2017:388-398. doi:10.1007/978-3-319-66272-5_31' apa: 'Andres, E., Biswas, R., & Bhowmick, P. (2017). Digital primitives defined by weighted focal set. In 20th IAPR International Conference (Vol. 10502, pp. 388–398). Cham: Springer Nature. https://doi.org/10.1007/978-3-319-66272-5_31' chicago: 'Andres, Eric, Ranita Biswas, and Partha Bhowmick. “Digital Primitives Defined by Weighted Focal Set.” In 20th IAPR International Conference, 10502:388–98. Cham: Springer Nature, 2017. https://doi.org/10.1007/978-3-319-66272-5_31.' ieee: E. Andres, R. Biswas, and P. Bhowmick, “Digital primitives defined by weighted focal set,” in 20th IAPR International Conference, Vienna, Austria, 2017, vol. 10502, pp. 388–398. ista: 'Andres E, Biswas R, Bhowmick P. 2017. Digital primitives defined by weighted focal set. 20th IAPR International Conference. DGCI: International Conference on Discrete Geometry for Computer Imagery, LNCS, vol. 10502, 388–398.' mla: Andres, Eric, et al. “Digital Primitives Defined by Weighted Focal Set.” 20th IAPR International Conference, vol. 10502, Springer Nature, 2017, pp. 388–98, doi:10.1007/978-3-319-66272-5_31. short: E. Andres, R. Biswas, P. Bhowmick, in:, 20th IAPR International Conference, Springer Nature, Cham, 2017, pp. 388–398. conference: end_date: 2017-09-21 location: Vienna, Austria name: 'DGCI: International Conference on Discrete Geometry for Computer Imagery' start_date: 2017-09-19 date_created: 2019-01-08T20:42:39Z date_published: 2017-08-22T00:00:00Z date_updated: 2022-01-27T15:38:35Z day: '22' doi: 10.1007/978-3-319-66272-5_31 extern: '1' intvolume: ' 10502' language: - iso: eng month: '08' oa_version: None page: 388-398 place: Cham publication: 20th IAPR International Conference publication_identifier: eisbn: - 978-3-319-66272-5 eissn: - 1611-3349 isbn: - 978-3-319-66271-8 issn: - 0302-9743 publication_status: published publisher: Springer Nature quality_controlled: '1' status: public title: Digital primitives defined by weighted focal set type: conference user_id: 8b945eb4-e2f2-11eb-945a-df72226e66a9 volume: 10502 year: '2017' ... --- _id: '5806' abstract: - lang: eng text: Although the concept of functional plane for naive plane is studied and reported in the literature in great detail, no similar study is yet found for naive sphere. This article exposes the first study in this line, opening up further prospects of analyzing the topological properties of sphere in the discrete space. We show that each quadraginta octant Q of a naive sphere forms a bijection with its projected pixel set on a unique coordinate plane, which thereby serves as the functional plane of Q, and hence gives rise to merely mono-jumps during back projection. The other two coordinate planes serve as para-functional and dia-functional planes for Q, as the former is ‘mono-jumping’ but not bijective, whereas the latter holds neither of the two. Owing to this, the quadraginta octants form symmetry groups and subgroups with equivalent jump conditions. We also show a potential application in generating a special class of discrete 3D circles based on back projection and jump bridging by Steiner voxels. A circle in this class possesses 4-symmetry, uniqueness, and bounded distance from the underlying real sphere and real plane. alternative_title: - LNCS article_processing_charge: No author: - first_name: Ranita full_name: Biswas, Ranita id: 3C2B033E-F248-11E8-B48F-1D18A9856A87 last_name: Biswas orcid: 0000-0002-5372-7890 - first_name: Partha full_name: Bhowmick, Partha last_name: Bhowmick citation: ama: 'Biswas R, Bhowmick P. On functionality of quadraginta octants of naive sphere with application to circle drawing. In: Discrete Geometry for Computer Imagery. Vol 9647. Cham: Springer Nature; 2016:256-267. doi:10.1007/978-3-319-32360-2_20' apa: 'Biswas, R., & Bhowmick, P. (2016). On functionality of quadraginta octants of naive sphere with application to circle drawing. In Discrete Geometry for Computer Imagery (Vol. 9647, pp. 256–267). Cham: Springer Nature. https://doi.org/10.1007/978-3-319-32360-2_20' chicago: 'Biswas, Ranita, and Partha Bhowmick. “On Functionality of Quadraginta Octants of Naive Sphere with Application to Circle Drawing.” In Discrete Geometry for Computer Imagery, 9647:256–67. Cham: Springer Nature, 2016. https://doi.org/10.1007/978-3-319-32360-2_20.' ieee: R. Biswas and P. Bhowmick, “On functionality of quadraginta octants of naive sphere with application to circle drawing,” in Discrete Geometry for Computer Imagery, Nantes, France, 2016, vol. 9647, pp. 256–267. ista: 'Biswas R, Bhowmick P. 2016. On functionality of quadraginta octants of naive sphere with application to circle drawing. Discrete Geometry for Computer Imagery. DGCI: International Conference on Discrete Geometry for Computer Imagery, LNCS, vol. 9647, 256–267.' mla: Biswas, Ranita, and Partha Bhowmick. “On Functionality of Quadraginta Octants of Naive Sphere with Application to Circle Drawing.” Discrete Geometry for Computer Imagery, vol. 9647, Springer Nature, 2016, pp. 256–67, doi:10.1007/978-3-319-32360-2_20. short: R. Biswas, P. Bhowmick, in:, Discrete Geometry for Computer Imagery, Springer Nature, Cham, 2016, pp. 256–267. conference: end_date: 2016-04-20 location: Nantes, France name: 'DGCI: International Conference on Discrete Geometry for Computer Imagery' start_date: 2016-04-18 date_created: 2019-01-08T20:44:37Z date_published: 2016-04-09T00:00:00Z date_updated: 2022-01-28T08:10:11Z day: '09' department: - _id: HeEd doi: 10.1007/978-3-319-32360-2_20 extern: '1' intvolume: ' 9647' language: - iso: eng month: '04' oa_version: None page: 256-267 place: Cham publication: Discrete Geometry for Computer Imagery publication_identifier: eisbn: - 978-3-319-32360-2 isbn: - 978-3-319-32359-6 issn: - 0302-9743 - 1611-3349 publication_status: published publisher: Springer Nature quality_controlled: '1' status: public title: On functionality of quadraginta octants of naive sphere with application to circle drawing type: conference user_id: 8b945eb4-e2f2-11eb-945a-df72226e66a9 volume: 9647 year: '2016' ... --- _id: '5805' abstract: - lang: eng text: Discretization of sphere in the integer space follows a particular discretization scheme, which, in principle, conforms to some topological model. This eventually gives rise to interesting topological properties of a discrete spherical surface, which need to be investigated for its analytical characterization. This paper presents some novel results on the local topological properties of the naive model of discrete sphere. They follow from the bijection of each quadraginta octant of naive sphere with its projection map called f -map on the corresponding functional plane and from the characterization of certain jumps in the f-map. As an application, we have shown how these properties can be used in designing an efficient reconstruction algorithm for a naive spherical surface from an input voxel set when it is sparse or noisy. alternative_title: - LNCS article_processing_charge: No author: - first_name: Nabhasmita full_name: Sen, Nabhasmita last_name: Sen - first_name: Ranita full_name: Biswas, Ranita id: 3C2B033E-F248-11E8-B48F-1D18A9856A87 last_name: Biswas orcid: 0000-0002-5372-7890 - first_name: Partha full_name: Bhowmick, Partha last_name: Bhowmick citation: ama: 'Sen N, Biswas R, Bhowmick P. On some local topological properties of naive discrete sphere. In: Computational Topology in Image Context. Vol 9667. Cham: Springer Nature; 2016:253-264. doi:10.1007/978-3-319-39441-1_23' apa: 'Sen, N., Biswas, R., & Bhowmick, P. (2016). On some local topological properties of naive discrete sphere. In Computational Topology in Image Context (Vol. 9667, pp. 253–264). Cham: Springer Nature. https://doi.org/10.1007/978-3-319-39441-1_23' chicago: 'Sen, Nabhasmita, Ranita Biswas, and Partha Bhowmick. “On Some Local Topological Properties of Naive Discrete Sphere.” In Computational Topology in Image Context, 9667:253–64. Cham: Springer Nature, 2016. https://doi.org/10.1007/978-3-319-39441-1_23.' ieee: 'N. Sen, R. Biswas, and P. Bhowmick, “On some local topological properties of naive discrete sphere,” in Computational Topology in Image Context, vol. 9667, Cham: Springer Nature, 2016, pp. 253–264.' ista: 'Sen N, Biswas R, Bhowmick P. 2016.On some local topological properties of naive discrete sphere. In: Computational Topology in Image Context. LNCS, vol. 9667, 253–264.' mla: Sen, Nabhasmita, et al. “On Some Local Topological Properties of Naive Discrete Sphere.” Computational Topology in Image Context, vol. 9667, Springer Nature, 2016, pp. 253–64, doi:10.1007/978-3-319-39441-1_23. short: N. Sen, R. Biswas, P. Bhowmick, in:, Computational Topology in Image Context, Springer Nature, Cham, 2016, pp. 253–264. conference: end_date: 2016-06-17 location: Marseille, France name: 'CTIC: Computational Topology in Image Context' start_date: 2016-06-15 date_created: 2019-01-08T20:44:24Z date_published: 2016-06-02T00:00:00Z date_updated: 2022-01-28T08:01:22Z day: '02' department: - _id: HeEd doi: 10.1007/978-3-319-39441-1_23 extern: '1' intvolume: ' 9667' language: - iso: eng month: '06' oa_version: None page: 253-264 place: Cham publication: Computational Topology in Image Context publication_identifier: eisbn: - 978-3-319-39441-1 eissn: - 1611-3349 isbn: - 978-3-319-39440-4 issn: - 0302-9743 publication_status: published publisher: Springer Nature quality_controlled: '1' status: public title: On some local topological properties of naive discrete sphere type: book_chapter user_id: 8b945eb4-e2f2-11eb-945a-df72226e66a9 volume: 9667 year: '2016' ... --- _id: '5809' abstract: - lang: eng text: A discrete spherical circle is a topologically well-connected 3D circle in the integer space, which belongs to a discrete sphere as well as a discrete plane. It is one of the most important 3D geometric primitives, but has not possibly yet been studied up to its merit. This paper is a maiden exposition of some of its elementary properties, which indicates a sense of its profound theoretical prospects in the framework of digital geometry. We have shown how different types of discretization can lead to forbidden and admissible classes, when one attempts to define the discretization of a spherical circle in terms of intersection between a discrete sphere and a discrete plane. Several fundamental theoretical results have been presented, the algorithm for construction of discrete spherical circles has been discussed, and some test results have been furnished to demonstrate its practicality and usefulness. article_processing_charge: No author: - first_name: Ranita full_name: Biswas, Ranita id: 3C2B033E-F248-11E8-B48F-1D18A9856A87 last_name: Biswas orcid: 0000-0002-5372-7890 - first_name: Partha full_name: Bhowmick, Partha last_name: Bhowmick - first_name: Valentin E. full_name: Brimkov, Valentin E. last_name: Brimkov citation: ama: 'Biswas R, Bhowmick P, Brimkov VE. On the connectivity and smoothness of discrete spherical circles. In: Combinatorial Image Analysis. Vol 9448. Cham: Springer Nature; 2016:86-100. doi:10.1007/978-3-319-26145-4_7' apa: 'Biswas, R., Bhowmick, P., & Brimkov, V. E. (2016). On the connectivity and smoothness of discrete spherical circles. In Combinatorial image analysis (Vol. 9448, pp. 86–100). Cham: Springer Nature. https://doi.org/10.1007/978-3-319-26145-4_7' chicago: 'Biswas, Ranita, Partha Bhowmick, and Valentin E. Brimkov. “On the Connectivity and Smoothness of Discrete Spherical Circles.” In Combinatorial Image Analysis, 9448:86–100. Cham: Springer Nature, 2016. https://doi.org/10.1007/978-3-319-26145-4_7.' ieee: 'R. Biswas, P. Bhowmick, and V. E. Brimkov, “On the connectivity and smoothness of discrete spherical circles,” in Combinatorial image analysis, vol. 9448, Cham: Springer Nature, 2016, pp. 86–100.' ista: 'Biswas R, Bhowmick P, Brimkov VE. 2016.On the connectivity and smoothness of discrete spherical circles. In: Combinatorial image analysis. vol. 9448, 86–100.' mla: Biswas, Ranita, et al. “On the Connectivity and Smoothness of Discrete Spherical Circles.” Combinatorial Image Analysis, vol. 9448, Springer Nature, 2016, pp. 86–100, doi:10.1007/978-3-319-26145-4_7. short: R. Biswas, P. Bhowmick, V.E. Brimkov, in:, Combinatorial Image Analysis, Springer Nature, Cham, 2016, pp. 86–100. conference: end_date: 2015-11-27 location: Kolkata, India name: 'IWCIA: International Workshop on Combinatorial Image Analysis' start_date: 2015-11-24 date_created: 2019-01-08T20:45:19Z date_published: 2016-01-06T00:00:00Z date_updated: 2022-01-28T08:13:03Z day: '06' department: - _id: HeEd doi: 10.1007/978-3-319-26145-4_7 extern: '1' intvolume: ' 9448' language: - iso: eng month: '01' oa_version: None page: 86-100 place: Cham publication: Combinatorial image analysis publication_identifier: eisbn: - 978-3-319-26145-4 eissn: - 1611-3349 isbn: - 978-3-319-26144-7 issn: - 0302-9743 publication_status: published publisher: Springer Nature quality_controlled: '1' status: public title: On the connectivity and smoothness of discrete spherical circles type: book_chapter user_id: 8b945eb4-e2f2-11eb-945a-df72226e66a9 volume: 9448 year: '2016' ... --- _id: '5804' abstract: - lang: eng text: We present here the first integer-based algorithm for constructing a well-defined lattice sphere specified by integer radius and integer center. The algorithm evolves from a unique correspondence between the lattice points comprising the sphere and the distribution of sum of three square numbers in integer intervals. We characterize these intervals to derive a useful set of recurrences, which, in turn, aids in efficient computation. Each point of the lattice sphere is determined by resorting to only a few primitive operations in the integer domain. The symmetry of its quadraginta octants provides an added advantage by confining the computation to its prima quadraginta octant. Detailed theoretical analysis and experimental results have been furnished to demonstrate its simplicity and elegance. author: - first_name: Ranita full_name: Biswas, Ranita id: 3C2B033E-F248-11E8-B48F-1D18A9856A87 last_name: Biswas orcid: 0000-0002-5372-7890 - first_name: Partha full_name: Bhowmick, Partha last_name: Bhowmick citation: ama: Biswas R, Bhowmick P. From prima quadraginta octant to lattice sphere through primitive integer operations. Theoretical Computer Science. 2015;624(4):56-72. doi:10.1016/j.tcs.2015.11.018 apa: Biswas, R., & Bhowmick, P. (2015). From prima quadraginta octant to lattice sphere through primitive integer operations. Theoretical Computer Science. Elsevier. https://doi.org/10.1016/j.tcs.2015.11.018 chicago: Biswas, Ranita, and Partha Bhowmick. “From Prima Quadraginta Octant to Lattice Sphere through Primitive Integer Operations.” Theoretical Computer Science. Elsevier, 2015. https://doi.org/10.1016/j.tcs.2015.11.018. ieee: R. Biswas and P. Bhowmick, “From prima quadraginta octant to lattice sphere through primitive integer operations,” Theoretical Computer Science, vol. 624, no. 4. Elsevier, pp. 56–72, 2015. ista: Biswas R, Bhowmick P. 2015. From prima quadraginta octant to lattice sphere through primitive integer operations. Theoretical Computer Science. 624(4), 56–72. mla: Biswas, Ranita, and Partha Bhowmick. “From Prima Quadraginta Octant to Lattice Sphere through Primitive Integer Operations.” Theoretical Computer Science, vol. 624, no. 4, Elsevier, 2015, pp. 56–72, doi:10.1016/j.tcs.2015.11.018. short: R. Biswas, P. Bhowmick, Theoretical Computer Science 624 (2015) 56–72. date_created: 2019-01-08T20:44:06Z date_published: 2015-04-18T00:00:00Z date_updated: 2021-01-12T08:03:36Z day: '18' doi: 10.1016/j.tcs.2015.11.018 extern: '1' intvolume: ' 624' issue: '4' language: - iso: eng month: '04' oa_version: None page: 56-72 publication: Theoretical Computer Science publication_identifier: issn: - 0304-3975 publication_status: published publisher: Elsevier quality_controlled: '1' status: public title: From prima quadraginta octant to lattice sphere through primitive integer operations type: journal_article user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 volume: 624 year: '2015' ...