Finding the largest empty cuboid inside a 3D digital object

The largest empty cuboid (LEC) which is axis-parallel, placed inside a 3D digital object against the background grid, captures the characteristics of the object and depicts the centrality of its shape. An efficient algorithm to determine the LEC inside a 3D digital object has been proposed here. The...

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Vydané v:	Multimedia tools and applications Ročník 80; číslo 28-29; s. 35521 - 35541
Hlavní autori:	Mondal, Sharmistha, Biswas, Arindam, Sarkar, Apurba
Médium:	Journal Article
Jazyk:	English
Vydavateľské údaje:	New York Springer US 01.11.2021 Springer Nature B.V
Predmet:	1166: Advances of machine learning in data analytics and visual information processing Algorithms Computer Communication Networks Computer Science Data Structures and Information Theory Intersections Modular construction Multimedia Information Systems Polygons Special Purpose and Application-Based Systems Three dimensional printing Slicing Largest rectangle Largest empty cuboid Isothetic inner cover Intersection polygon
ISSN:	1380-7501, 1573-7721
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Abstract	The largest empty cuboid (LEC) which is axis-parallel, placed inside a 3D digital object against the background grid, captures the characteristics of the object and depicts the centrality of its shape. An efficient algorithm to determine the LEC inside a 3D digital object has been proposed here. The LEC, analogous to the largest rectangle (LR) in a 2D polygon, reveals the structure of an object in 3D. Primarily, we find the intersection polygon of the slice polygons, which are obtained from the isothetic inner cover of the digital object, at two or more consecutive levels. Starting with the bottom-most slice polygon, first we find its intersection polygon with the slice polygon at the next higher level, then the intersection of the resulting intersection polygon with the next higher level is obtained, and so on, until we reach the topmost slice. Each time, the largest rectangle inscribed inside the intersection polygon is determined and the volume of the corresponding candidate LEC is obtained by multiplying the area with the corresponding height. The above procedure is repeated starting with the slice polygon at each level. The maximum of these candidate LECs is reported. The best case computational complexity of the proposed algorithm is O ( n 3/2 ), where n is the number of voxels on the surface of the object. The experimental results on a variety of objects demonstrate that the central portion of the shape is captured by the LEC. Apart from the digital object, the algorithm can be used to find the LEC where we deal with isothetic polytope like 3D printing and modular construction.
AbstractList	The largest empty cuboid (LEC) which is axis-parallel, placed inside a 3D digital object against the background grid, captures the characteristics of the object and depicts the centrality of its shape. An efficient algorithm to determine the LEC inside a 3D digital object has been proposed here. The LEC, analogous to the largest rectangle (LR) in a 2D polygon, reveals the structure of an object in 3D. Primarily, we find the intersection polygon of the slice polygons, which are obtained from the isothetic inner cover of the digital object, at two or more consecutive levels. Starting with the bottom-most slice polygon, first we find its intersection polygon with the slice polygon at the next higher level, then the intersection of the resulting intersection polygon with the next higher level is obtained, and so on, until we reach the topmost slice. Each time, the largest rectangle inscribed inside the intersection polygon is determined and the volume of the corresponding candidate LEC is obtained by multiplying the area with the corresponding height. The above procedure is repeated starting with the slice polygon at each level. The maximum of these candidate LECs is reported. The best case computational complexity of the proposed algorithm is O ( n 3/2 ), where n is the number of voxels on the surface of the object. The experimental results on a variety of objects demonstrate that the central portion of the shape is captured by the LEC. Apart from the digital object, the algorithm can be used to find the LEC where we deal with isothetic polytope like 3D printing and modular construction. The largest empty cuboid (LEC) which is axis-parallel, placed inside a 3D digital object against the background grid, captures the characteristics of the object and depicts the centrality of its shape. An efficient algorithm to determine the LEC inside a 3D digital object has been proposed here. The LEC, analogous to the largest rectangle (LR) in a 2D polygon, reveals the structure of an object in 3D. Primarily, we find the intersection polygon of the slice polygons, which are obtained from the isothetic inner cover of the digital object, at two or more consecutive levels. Starting with the bottom-most slice polygon, first we find its intersection polygon with the slice polygon at the next higher level, then the intersection of the resulting intersection polygon with the next higher level is obtained, and so on, until we reach the topmost slice. Each time, the largest rectangle inscribed inside the intersection polygon is determined and the volume of the corresponding candidate LEC is obtained by multiplying the area with the corresponding height. The above procedure is repeated starting with the slice polygon at each level. The maximum of these candidate LECs is reported. The best case computational complexity of the proposed algorithm is O(n3/2), where n is the number of voxels on the surface of the object. The experimental results on a variety of objects demonstrate that the central portion of the shape is captured by the LEC. Apart from the digital object, the algorithm can be used to find the LEC where we deal with isothetic polytope like 3D printing and modular construction.
Author	Biswas, Arindam Mondal, Sharmistha Sarkar, Apurba
Author_xml	– sequence: 1 givenname: Sharmistha orcidid: 0000-0003-1811-7154 surname: Mondal fullname: Mondal, Sharmistha email: sharmistha28101990@gmail.com organization: Department of Information Technology, Indian Institute of Engineering Science and Technology – sequence: 2 givenname: Arindam surname: Biswas fullname: Biswas, Arindam organization: Department of Information Technology, Indian Institute of Engineering Science and Technology – sequence: 3 givenname: Apurba surname: Sarkar fullname: Sarkar, Apurba organization: Department of Computer Science and Technology, Indian Institute of Engineering Science and Technology
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Cites_doi	10.1016/j.jmva.2009.02.010 10.1007/978-3-030-01267-0_12 10.1109/CVPR42600.2020.00312 10.1109/ICPR.2000.905491 10.1109/ICCV.2007.4409028 10.1016/j.jcss.2017.05.006 10.1016/B978-155860861-0/50005-5 10.1007/3-540-47967-8_50 10.1201/b14786 10.1007/978-3-642-34732-0_2 10.1016/B978-0-12-811889-4.00007-5 10.1145/1026711.1026743 10.1080/00207160.2012.734813 10.1016/j.neucom.2019.04.062
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References	Vajramushti N, Kakadiaris IA, Theoharis T, Papaioannou G (2004) Efficient 3d object retrieval using depth images. In: Proceedings of the 6th ACM SIGMM international workshop on Multimedia information retrieval, MIR ’04. ACM, New York, pp 189–196 Klette R., Rosenfeld A (2004) Digital geometry: Geometric Methods for Digital Picture Analysis. Morgan Kaufmann San Francisco Gu X, Wang S, Kim J, Zeng Y, Wang Y, Qin H, Samaras D (2007) Ricci flow for 3d shape analysis. In: 2007 IEEE 11Th international conference on computer vision, IEEE, pp 1–8 PatrangenaruVLiuXSugathadasaSA nonparametric approach to 3d shape analysis from digital camera images—iJ Multivar Anal201010111131255761510.1016/j.jmva.2009.02.010 Laga H (2018) A survey on nonrigid 3d shape analysis. In: Academic press library in signal processing, vol 6, Elsevier, pp 261–304 FuKZhaoQGuIY-HYangJDeepside: a general deep framework for salient object detectionNeurocomputing2019356698210.1016/j.neucom.2019.04.062 žunić J, Rosin PL (2002) A rectilinearity measurement for polygons. In: European conference on computer vision, Springer, pp 746–758 Fan D-P, Cheng M-M, Liu J-J, Gao S-H, Hou Q, Borji A (2018) Salient objects in clutter: Bringing salient object detection to the foreground. In: Proceedings of the European conference on computer vision (ECCV), pp 186–202 De M, Nandy SC (2011) Space-efficient algorithms for empty space recognition among a point set in 2d and 3d. In: Proceedings of the 23rd annual canadian conference on computational geometry, Toronto, Ontario, Canada, August 10-12, 2011 Papaioannou G, Karabassi E-A, Theoharis T (2000) Segmentation and surface characterization of arbitrary 3d meshes for object reconstruction and recognition. In: Proceedings 15th international conference on pattern recognition. ICPR-2000, vol 1, IEEE, pp 734–737 Fu K, Fan D-P, Ji G-P, Zhao Q (2020) Jl-dcf: Joint learning and densely-cooperative fusion framework for rgb-d salient object detection. In: Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, pp 3052–3062 Mukhopadhyay J, Das PP, Chattopadhyay S, Bhowmick P, Chatterji BN (2016) Digital Geometry in Image Processing. IIT Kharagpur Research Monograph Series. CRC Press SarkarABiswasADuttMBhattacharyaAFinding a largest rectangle inside a digital object and rectangularizationJ Comput Syst Sci201895204217378758910.1016/j.jcss.2017.05.006 Karmakar N, Biswas A, Bhowmick P (2012) Fast slicing of orthogonal covers using dcel. In: International workshop on combinatorial image analysis, Springer, pp 16–30 KarmakarNBiswasABhowmickPBhattacharyaBBA combinatorial algorithm to construct 3d isothetic coversInt J Comput Math20139081571160610.1080/00207160.2012.734813 10490_CR15 10490_CR14 10490_CR11 10490_CR1 10490_CR10 N Karmakar (10490_CR7) 2013; 90 K Fu (10490_CR3) 2019; 356 A Sarkar (10490_CR13) 2018; 95 10490_CR4 10490_CR5 10490_CR2 10490_CR8 V Patrangenaru (10490_CR12) 2010; 101 10490_CR9 10490_CR6
References_xml	– reference: Gu X, Wang S, Kim J, Zeng Y, Wang Y, Qin H, Samaras D (2007) Ricci flow for 3d shape analysis. In: 2007 IEEE 11Th international conference on computer vision, IEEE, pp 1–8 – reference: Papaioannou G, Karabassi E-A, Theoharis T (2000) Segmentation and surface characterization of arbitrary 3d meshes for object reconstruction and recognition. In: Proceedings 15th international conference on pattern recognition. ICPR-2000, vol 1, IEEE, pp 734–737 – reference: SarkarABiswasADuttMBhattacharyaAFinding a largest rectangle inside a digital object and rectangularizationJ Comput Syst Sci201895204217378758910.1016/j.jcss.2017.05.006 – reference: Fan D-P, Cheng M-M, Liu J-J, Gao S-H, Hou Q, Borji A (2018) Salient objects in clutter: Bringing salient object detection to the foreground. In: Proceedings of the European conference on computer vision (ECCV), pp 186–202 – reference: Laga H (2018) A survey on nonrigid 3d shape analysis. In: Academic press library in signal processing, vol 6, Elsevier, pp 261–304 – reference: Fu K, Fan D-P, Ji G-P, Zhao Q (2020) Jl-dcf: Joint learning and densely-cooperative fusion framework for rgb-d salient object detection. In: Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, pp 3052–3062 – reference: Mukhopadhyay J, Das PP, Chattopadhyay S, Bhowmick P, Chatterji BN (2016) Digital Geometry in Image Processing. IIT Kharagpur Research Monograph Series. CRC Press – reference: PatrangenaruVLiuXSugathadasaSA nonparametric approach to 3d shape analysis from digital camera images—iJ Multivar Anal201010111131255761510.1016/j.jmva.2009.02.010 – reference: FuKZhaoQGuIY-HYangJDeepside: a general deep framework for salient object detectionNeurocomputing2019356698210.1016/j.neucom.2019.04.062 – reference: Karmakar N, Biswas A, Bhowmick P (2012) Fast slicing of orthogonal covers using dcel. In: International workshop on combinatorial image analysis, Springer, pp 16–30 – reference: KarmakarNBiswasABhowmickPBhattacharyaBBA combinatorial algorithm to construct 3d isothetic coversInt J Comput Math20139081571160610.1080/00207160.2012.734813 – reference: Klette R., Rosenfeld A (2004) Digital geometry: Geometric Methods for Digital Picture Analysis. Morgan Kaufmann San Francisco – reference: Vajramushti N, Kakadiaris IA, Theoharis T, Papaioannou G (2004) Efficient 3d object retrieval using depth images. In: Proceedings of the 6th ACM SIGMM international workshop on Multimedia information retrieval, MIR ’04. ACM, New York, pp 189–196 – reference: De M, Nandy SC (2011) Space-efficient algorithms for empty space recognition among a point set in 2d and 3d. In: Proceedings of the 23rd annual canadian conference on computational geometry, Toronto, Ontario, Canada, August 10-12, 2011 – reference: žunić J, Rosin PL (2002) A rectilinearity measurement for polygons. In: European conference on computer vision, Springer, pp 746–758 – ident: 10490_CR1 – volume: 101 start-page: 11 issue: 1 year: 2010 ident: 10490_CR12 publication-title: J Multivar Anal doi: 10.1016/j.jmva.2009.02.010 – ident: 10490_CR2 doi: 10.1007/978-3-030-01267-0_12 – ident: 10490_CR4 doi: 10.1109/CVPR42600.2020.00312 – ident: 10490_CR11 doi: 10.1109/ICPR.2000.905491 – ident: 10490_CR5 doi: 10.1109/ICCV.2007.4409028 – volume: 95 start-page: 204 year: 2018 ident: 10490_CR13 publication-title: J Comput Syst Sci doi: 10.1016/j.jcss.2017.05.006 – ident: 10490_CR8 doi: 10.1016/B978-155860861-0/50005-5 – ident: 10490_CR15 doi: 10.1007/3-540-47967-8_50 – ident: 10490_CR10 doi: 10.1201/b14786 – ident: 10490_CR6 doi: 10.1007/978-3-642-34732-0_2 – ident: 10490_CR9 doi: 10.1016/B978-0-12-811889-4.00007-5 – ident: 10490_CR14 doi: 10.1145/1026711.1026743 – volume: 90 start-page: 1571 issue: 8 year: 2013 ident: 10490_CR7 publication-title: Int J Comput Math doi: 10.1080/00207160.2012.734813 – volume: 356 start-page: 69 year: 2019 ident: 10490_CR3 publication-title: Neurocomputing doi: 10.1016/j.neucom.2019.04.062
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