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Recognition of spherical segments using number theoretic properties of isothetic covers

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Abstract

With the proliferation of cheap sensors and handheld devices, the amount of 3d data has grown exponentially and finds uses in the automated diagnosis of medical images, computer vision, and a host of other applications. Description and identification of geometrical primitives play an important role in computer vision and image processing. In this article, a definition of discrete spheres is given based on the dilation of euclidean spheres with a unit tetrahedron. It is shown in the article that the isothetic covers of spheres are equivalent to our definition of discrete spheres. Analysis of isothetic covers of spheres are presented, particularly its number-theoretic properties, and show that the bounding radius of isothetic cover faces is closely related to the distribution of the square number in integer intervals. Spherical segment recognition algorithms based on the number-theoretic properties of isothetic covers are proposed. Information content of the isothetic covers and computational load of the algorithm can be adjusted as per the requirements of the applications by changing the grid size. The computational complexities of the methods are determined and shows they are competitive to other related methods in the literature. The proposed methods are experimented with a large number of synthetic data to study its behavior and some of the results are presented in the article.

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All data generated or analysed during the current study are available from the corresponding author on reasonable request.

References

  1. Andres E (1994) Discrete circles, rings and spheres. Comput Graph 18:695–706

    Article  Google Scholar 

  2. Andres E, Roussillon T (2011) Analytical description of digital circles. In: Debled-Rennesson I, Domenjoud E, Kerautret B, Even P (eds) Discrete geometry for computer imagery, vol 6607 LNCS. Springer, Berlin Heidelberg, pp 235–246

  3. Badler NI (1977) Disk generators for a raster display device. Comput Graph Image Process 6:589–593

    Article  Google Scholar 

  4. Bera S, Bhowmick P, Bhattacharya B (2014) On covering a solid sphere with concentric spheres in {F}3. arXiv:1411.1

  5. Bera S, Bhowmick P, Bhattacharya B (2014) A digital-geometric algorithm for generating a complete spherical surface in \(\mathbb {Z}^{3}\). In: Applied algorithms. Springer International Publishing, Cham, pp 49–61

  6. Bhowmick P, Bhattacharya B (2008) Number-theoretic interpretation and construction of a digital circle. Discret Appl Math 156:2381–2399

    Article  MathSciNet  MATH  Google Scholar 

  7. Bhowmick P, Pal S (2014) Fast circular arc segmentation based on approximate circularity and cuboid graph. J Math Imaging Vision 49:98–122

    Article  MATH  Google Scholar 

  8. Birdal T, Busam B, Navab N, Ilic S, Sturm P (2019) Generic primitive detection in point clouds using novel minimal quadric fits. arXiv:1901.0

  9. Biswas SN, Chaudhuri BB (1985) On the generation of discrete circular objects and their properties. Comp Vision Graph Image Process 32:158–170

    Article  Google Scholar 

  10. Bresenham J (1977) A linear algorithm for incremental digital display of circular arcs. Commun ACM 20:100–106

    Article  MATH  Google Scholar 

  11. Brimkov VE, Barneva RP (2008) On the polyhedral complexity of the integer points in a hyperball. Theor Comput Sci 406:24–30

    Article  MathSciNet  MATH  Google Scholar 

  12. Camurri M, Vezzani R, Cucchiara R (2014) 3d hough transform for sphere recognition on point clouds. Mach Vis Appl 25:1877–1891

    Article  Google Scholar 

  13. Cao MY, Ye CH, Doessel O, Liu C (2006) Spherical parameter detection based on hierarchical hough transform. Pattern Recogn Lett 27:980–986

    Article  Google Scholar 

  14. Chamizo F (1998) Lattice points in bodies of revolution. Acta Arith 85:265–277

    Article  MathSciNet  MATH  Google Scholar 

  15. Chung WL (1977) On circle generation algorithms. Comput Graph Image Process 6:196–198

    Article  Google Scholar 

  16. Décoret X, Durand F, Sillion FX, Dorsey J (2003) Billboard clouds for extreme model simplification. ACM SIGGRAPH 2003 Papers, SIGGRAPH ’03 22:689–696

    Article  Google Scholar 

  17. Drost B, Ilic S (2015) Local hough transform for 3d primitive detection. In: Proceedings - 2015 international conference on 3d vision, 3DV 2015, pp 398–406

  18. Duan J, Lachhani K, Baghsiahi H, Willman E, Selviah DR (2014) Indoor rigid sphere recognition based on 3d point cloud data. In: Coleman S, Gardiner B, Kerr D (eds) IMVIP 2014: 2014 Irish machine vision and image processing. Irish Pattern Recognition & Classification Society, Derry-Londonderry, pp 28–33

  19. Fiorio C, Toutant JL (2006) Arithmetic discrete hyperspheres and separatingness. In: Kuba A, Nyúl LG, Palágyi K (eds) Discrete geometry for computer imagery, vol 4245 LNCS, pp 425–436

  20. Fiorio C, Jamet D, Toutant J-L (2006) Discrete circles: an arithmetical approach with non-constant thickness. In: Vision geometry XIV, vol 6066, p 60660C

  21. Fischler MA, Bolles RC (1981) Random sample consensus: a paradigm for model fitting with applications to image analysis and automated cartography. Commun ACM 24:381–395

    Article  MathSciNet  Google Scholar 

  22. Furukawa Y, Curless B, Seitz SM, Szeliski R (2010) Reconstructing building interiors from images. In: 2009 IEEE 12th international conference on computer vision, pp 80–87

  23. Georgiev K, Al-Hami M, Lakaemper R (2016) Real-time 3d scene description using spheres, cones and cylinders. arXiv:1603.0

  24. Hildenbrand D (2012) Foundations of geometric algebra computing. AIP Conf Proc 1479:27–30

    Article  Google Scholar 

  25. Horn BKP (1976) Circle generators for display devices. Comput Graph Image Process 5:280–288

    Article  Google Scholar 

  26. Hsu SY, Chow LR, Liu HC (1993) A new approach for the generation of circles. Comput Graph Forum 12:105–109

    Article  Google Scholar 

  27. Kharbat M, Aouf N, Tsourdos A, White B (2007) Sphere detection and tracking for a space capturing operation. In: 2007 IEEE conference on advanced video and signal based surveillance, AVSS 2007 proceedings, pp 182–187

  28. Li W, Cheng H, Zhang X (2021) Efficient 3d object recognition from cluttered point cloud. Sensors 21:5850

    Article  Google Scholar 

  29. Magyar A (2007) On the distribution of lattice points on spheres and level surfaces of polynomials. J Number Theory 122:69–83

    Article  MathSciNet  MATH  Google Scholar 

  30. Memiş A, Albayrak S, Bilgili F (2018) 3d detection of spheric and aspheric femoral heads in coronal mr images of patients with legg-calve-perthes disease using the spherical hough transform. In: ACM international conference proceeding series. ACM, pp 46–52

  31. Montani C, Scopigno R (1990) Spheres-to-voxels conversion, pp 327–334. Academic Press Professional, Inc., Cambridge

    Google Scholar 

  32. Pal S, Bhowmick P (2012) Determining digital circularity using integer intervals. J Math Imaging Vision 42:1–24

    Article  MathSciNet  MATH  Google Scholar 

  33. Pham S (1992) Digital circles with non-lattice point centers. Vis Comput 9:1–24

    Article  Google Scholar 

  34. Ren Z, Wang R, Snyder J, Zhou K, Liu X, Sun B, Sloan PP, Bao H, Peng Q, Guo B (2006) Real-time soft shadows in dynamic scenes using spherical harmonic exponentiation. ACM SIGGRAPH 2006 Papers, SIGGRAPH ’06 25:977–986

    Article  Google Scholar 

  35. Sandoval J, Tanaka K (2020) Paper robust sphere detection in unorganized 3d point clouds using an efficient hough voting scheme based on sliding voxels. IIEEJ Trans Image Electron Vis Comput 8

  36. Sommer C, Yumin S, Bylow E, Cremers D (2020) Primitect: fast continuous hough voting for primitive detection. Proc - IEEE Int Conf Robot Autom:8404–8410

  37. Song W, Zhang L, Tian Y, Fong S, Liu J, Gozho A (2020) Cnn-based 3d object classification using hough space of lidar point clouds. Hum-centric Comput Inf Sci 10:19

    Article  Google Scholar 

  38. Stelldinger P (2007) Image digitization and its influence on shape properties in finite dimenstions. IOS Press, Amsterdam

    Google Scholar 

  39. Surajkanta Y, Pal S (2020) Recognition of isothetic arc using number theoretic properties. Int J Image Graph 20:1–27

    Article  Google Scholar 

  40. Sveier A, Kleppe AL, Tingelstad L, Egeland O (2017) Object detection in point clouds using conformal geometric algebra. AACA 27:1961–1976

    Article  MathSciNet  MATH  Google Scholar 

  41. Thiery JM, Guy É, Boubekeur T (2013) Sphere-meshes: shape approximation using spherical quadric error metrics. ACM Trans Graph 32. Art. No. 178

  42. Thiery JM, Guy É, Boubekeur T, Eisemann E (2016) Animated mesh approximation with sphere-meshes. ACM Trans Graph 35:30:1–30:13

    Article  Google Scholar 

  43. Toutant JL, Andres E, Roussillon T (2013) Digital circles, spheres and hyperspheres: from morphological models to analytical characterizations and topological properties. Discret Appl Math 161:2662–2677

    Article  MathSciNet  MATH  Google Scholar 

  44. Toutant JL, Vacavant A, Kerautret B (2013) Arc recognition on irregular isothetic grids and its application to reconstruction of noisy digital contours, vol 7749 LNCS, pp 265–276. Springer, Berlin Heidelberg

    MATH  Google Scholar 

  45. Tsang KM (2000) Counting lattice points in the sphere. Bull Lond Math Soc 32:679–688

    Article  MathSciNet  MATH  Google Scholar 

  46. Wang L, Shen C, Duan F, Lu K (2016) Energy-based automatic recognition of multiple spheres in three-dimensional point cloud. Pattern Recognit Lett 83:287–293. Efficient Shape Representation, Matching, Ranking, and its Applications

    Article  Google Scholar 

  47. Wang L, Yan B, Duan F, Lu K (2020) Energy minimisation-based multi-class multi-instance geometric primitives extraction from 3d point clouds. IET Image Process 14:2660–2667

    Article  Google Scholar 

  48. Wang L, Li J, Fan D (2021) A graphical convolutional network-based method for 3d point cloud classification. In: 2021 33rd Chinese control and decision conference (CCDC). Institute of Electrical and Electronics Engineers Inc., pp 1686–1691

  49. Wright WE (1990) Parallelization of bresenham’s line and circle algorithms. IEEE Comput Graph Appl 10:60–67

    Article  Google Scholar 

  50. Wu X, Rokne JG (1987) Double-step incremental generation of lines and circles. Comput Vis Graph Image Process 37:331–344

    Article  Google Scholar 

  51. Yao C, Rokne JG (1995) Hybrid scan-conversion of circles. IEEE Trans Vis Comput Graph 1:311–318

    Article  Google Scholar 

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  1. National Institute of Technology, Silchar, India

    Yumnam Surajkanta & Shyamosree Pal

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  1. Yumnam Surajkanta
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Correspondence to Yumnam Surajkanta.

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Surajkanta, Y., Pal, S. Recognition of spherical segments using number theoretic properties of isothetic covers. Multimed Tools Appl 82, 19393–19416 (2023). https://doi.org/10.1007/s11042-022-14182-3

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  • DOI: https://doi.org/10.1007/s11042-022-14182-3

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