June  2020, 28(2): 1095-1106. doi: 10.3934/era.2020060

Rough semi-uniform spaces and its image proximities

Department of Mathematics, Motilal Nehru National Institute of Technology Allahabad, Prayagraj-211004, India

* Corresponding author: Surabhi Tiwari

Received  February 2020 Revised  May 2020 Published  June 2020

In this paper, we introduce the concept of rough semi-uniform spaces as a supercategory of rough pseudometric spaces and approximation spaces. A completion of approximation spaces has been constructed using rough semi-uniform spaces. Applications of rough semi-uniform spaces in the construction of proximities of digital images is also discussed.

Citation: Surabhi Tiwari, Pankaj Kumar Singh. Rough semi-uniform spaces and its image proximities. Electronic Research Archive, 2020, 28 (2) : 1095-1106. doi: 10.3934/era.2020060
References:
[1]

B. Batíková, Completion of semi-uniform spaces, Appl. Categ. Structures, 15 (2007), 483-491.  doi: 10.1007/s10485-007-9092-5.  Google Scholar

[2]

A. D. ConcilioC. GuadagniJ. F. Peters and S. Ramanna, Descriptive proximities. Properties and interplay between classical proximities and overlap, Math. Comput. Sci., 12 (2018), 91-106.  doi: 10.1007/s11786-017-0328-y.  Google Scholar

[3]

S. DevF. M. SavoyY. H. Lee and S. Winkler, Rough-set-based color channel selection, IEEE Geoscience and Remote Sensing Letters, 14 (2017), 52-56.  doi: 10.1109/LGRS.2016.2625303.  Google Scholar

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C. J. Henry and S. Ramanna, Signature-based perceptual nearness: Application of near sets to image retrieval, Math. Comput. Sci., 7 (2013), 71-85.  doi: 10.1007/s11786-013-0145-x.  Google Scholar

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S. A. Naimpally and B. Warrack, Proximity Spaces, Reprint of the 1970 original [MR0278261]. Cambridge Tracts in Mathematics, 59. Cambridge University Press, Cambridge, 2008. doi: 10.1017/CBO9780511569364.  Google Scholar

[6]

Z. Pawlak, Rough sets, Internat. J. Comput. Inform. Sci., 11 (1982), 341-356.  doi: 10.1007/BF01001956.  Google Scholar

[7]

A. S. Aguiar Pessoa, S. Stephany and L. M. Garcia Fonseca, Feature selection and image classification using rough sets theory, 2011 IEEE International Geoscience and Remote Sensing Symposium, (2011), 2904–2907. doi: 10.1109/IGARSS.2011.6049822.  Google Scholar

[8]

J. F. Peters, Computational Proximity. Excursions in the Topology of Digital Images, Intelligent Systems Reference Library, 102. Springer, [Cham], 2016. doi: 10.1007/978-3-319-30262-1.  Google Scholar

[9]

J. F. Peters, Topology of Digital Images: Visual Pattern Discovery in Proximity Spaces, Intell. Syst. Ref. Libr., Springer, 2014. doi: 10.1007/978-3-642-53845-2.  Google Scholar

[10]

J. F. PetersJ. Stepaniuk and A. Skowron, Nearness of visual objects. Application of rough sets in proximity spaces, Fund. Inform., 128 (2013), 159-176.  doi: 10.3233/FI-2013-939.  Google Scholar

[11]

J. F. Peters and P. Wasilewski, Foundation of near sets, Inform. Sci., 179 (2009), 3091-3109.  doi: 10.1016/j.ins.2009.04.018.  Google Scholar

[12]

P. K. Singh and S. Tiwari, A fixed point theorem in rough semi-linear uniform spaces, Submitted. Google Scholar

[13]

P. K. Singh and S. Tiwari, Topological structures in rough set theory: A survey, Hacet. J. Math. Stat., (2020), 1–25. doi: 10.15672/hujms.662711.  Google Scholar

[14]

S. Tiwari and P. K. Singh, Čech rough proximity spaces, Mat. Vesnik, 72 (2020), 6-16.   Google Scholar

[15]

S. Tiwari and P. K. Singh, An approach of proximity in rough set theory, Fund. Inform., 166 (2019), 251-271.  doi: 10.3233/FI-2019-1802.  Google Scholar

[16]

M. Vlach, Algebraic and topological aspects of rough set theory, Fourth International Workshop on Computational Intelligence and Application-IEEE, SMC, (2008), 23–30. Google Scholar

[17]

M. Wolski, Granular computing: Topological and categorical aspects of near and rough set approaches to granulation of knowledge, Lecture Notes in Comput. Sci., 7736 (2013), 34-52.  doi: 10.1007/978-3-642-36505-8_3.  Google Scholar

[18]

M. Wolski, Perception and classification. A note on near sets and rough sets, Fund. Inform., 101 (2010), 143-155.  doi: 10.3233/FI-2010-281.  Google Scholar

[19]

W. Z. Wu and J. S. Mi, Some mathematical structures of generalized rough sets in infinite universes of discourse, Lecture Notes in Comput. Sci., 6499 (2011), 175-206.   Google Scholar

[20]

Y. Y. Yao, Relational interpretations of neighborhood operators and rough set approximation operators, Inform. Sci., 111 (1998), 239-259.  doi: 10.1016/S0020-0255(98)10006-3.  Google Scholar

show all references

References:
[1]

B. Batíková, Completion of semi-uniform spaces, Appl. Categ. Structures, 15 (2007), 483-491.  doi: 10.1007/s10485-007-9092-5.  Google Scholar

[2]

A. D. ConcilioC. GuadagniJ. F. Peters and S. Ramanna, Descriptive proximities. Properties and interplay between classical proximities and overlap, Math. Comput. Sci., 12 (2018), 91-106.  doi: 10.1007/s11786-017-0328-y.  Google Scholar

[3]

S. DevF. M. SavoyY. H. Lee and S. Winkler, Rough-set-based color channel selection, IEEE Geoscience and Remote Sensing Letters, 14 (2017), 52-56.  doi: 10.1109/LGRS.2016.2625303.  Google Scholar

[4]

C. J. Henry and S. Ramanna, Signature-based perceptual nearness: Application of near sets to image retrieval, Math. Comput. Sci., 7 (2013), 71-85.  doi: 10.1007/s11786-013-0145-x.  Google Scholar

[5]

S. A. Naimpally and B. Warrack, Proximity Spaces, Reprint of the 1970 original [MR0278261]. Cambridge Tracts in Mathematics, 59. Cambridge University Press, Cambridge, 2008. doi: 10.1017/CBO9780511569364.  Google Scholar

[6]

Z. Pawlak, Rough sets, Internat. J. Comput. Inform. Sci., 11 (1982), 341-356.  doi: 10.1007/BF01001956.  Google Scholar

[7]

A. S. Aguiar Pessoa, S. Stephany and L. M. Garcia Fonseca, Feature selection and image classification using rough sets theory, 2011 IEEE International Geoscience and Remote Sensing Symposium, (2011), 2904–2907. doi: 10.1109/IGARSS.2011.6049822.  Google Scholar

[8]

J. F. Peters, Computational Proximity. Excursions in the Topology of Digital Images, Intelligent Systems Reference Library, 102. Springer, [Cham], 2016. doi: 10.1007/978-3-319-30262-1.  Google Scholar

[9]

J. F. Peters, Topology of Digital Images: Visual Pattern Discovery in Proximity Spaces, Intell. Syst. Ref. Libr., Springer, 2014. doi: 10.1007/978-3-642-53845-2.  Google Scholar

[10]

J. F. PetersJ. Stepaniuk and A. Skowron, Nearness of visual objects. Application of rough sets in proximity spaces, Fund. Inform., 128 (2013), 159-176.  doi: 10.3233/FI-2013-939.  Google Scholar

[11]

J. F. Peters and P. Wasilewski, Foundation of near sets, Inform. Sci., 179 (2009), 3091-3109.  doi: 10.1016/j.ins.2009.04.018.  Google Scholar

[12]

P. K. Singh and S. Tiwari, A fixed point theorem in rough semi-linear uniform spaces, Submitted. Google Scholar

[13]

P. K. Singh and S. Tiwari, Topological structures in rough set theory: A survey, Hacet. J. Math. Stat., (2020), 1–25. doi: 10.15672/hujms.662711.  Google Scholar

[14]

S. Tiwari and P. K. Singh, Čech rough proximity spaces, Mat. Vesnik, 72 (2020), 6-16.   Google Scholar

[15]

S. Tiwari and P. K. Singh, An approach of proximity in rough set theory, Fund. Inform., 166 (2019), 251-271.  doi: 10.3233/FI-2019-1802.  Google Scholar

[16]

M. Vlach, Algebraic and topological aspects of rough set theory, Fourth International Workshop on Computational Intelligence and Application-IEEE, SMC, (2008), 23–30. Google Scholar

[17]

M. Wolski, Granular computing: Topological and categorical aspects of near and rough set approaches to granulation of knowledge, Lecture Notes in Comput. Sci., 7736 (2013), 34-52.  doi: 10.1007/978-3-642-36505-8_3.  Google Scholar

[18]

M. Wolski, Perception and classification. A note on near sets and rough sets, Fund. Inform., 101 (2010), 143-155.  doi: 10.3233/FI-2010-281.  Google Scholar

[19]

W. Z. Wu and J. S. Mi, Some mathematical structures of generalized rough sets in infinite universes of discourse, Lecture Notes in Comput. Sci., 6499 (2011), 175-206.   Google Scholar

[20]

Y. Y. Yao, Relational interpretations of neighborhood operators and rough set approximation operators, Inform. Sci., 111 (1998), 239-259.  doi: 10.1016/S0020-0255(98)10006-3.  Google Scholar

Figure 1.  Digital Image of a Butterfly
Figure 2.  Velocity contours describes the velocity contours of fluid flow past a circular cylinder (Plotted in Ansys 15.0)
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