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

[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

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)
[1]

Byung-Soo Lee. Existence and convergence results for best proximity points in cone metric spaces. Numerical Algebra, Control & Optimization, 2014, 4 (2) : 133-140. doi: 10.3934/naco.2014.4.133

[2]

Joachim Escher, Piotr B. Mucha. The surface diffusion flow on rough phase spaces. Discrete & Continuous Dynamical Systems - A, 2010, 26 (2) : 431-453. doi: 10.3934/dcds.2010.26.431

[3]

Mateusz Krukowski. Arzelà-Ascoli's theorem in uniform spaces. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 283-294. doi: 10.3934/dcdsb.2018020

[4]

Martin Michálek, Dalibor Pražák, Jakub Slavík. Semilinear damped wave equation in locally uniform spaces. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1673-1695. doi: 10.3934/cpaa.2017080

[5]

Thuy N. T. Nguyen. Uniform controllability of semidiscrete approximations for parabolic systems in Banach spaces. Discrete & Continuous Dynamical Systems - B, 2015, 20 (2) : 613-640. doi: 10.3934/dcdsb.2015.20.613

[6]

Gaocheng Yue, Chengkui Zhong. Global attractors for the Gray-Scott equations in locally uniform spaces. Discrete & Continuous Dynamical Systems - B, 2016, 21 (1) : 337-356. doi: 10.3934/dcdsb.2016.21.337

[7]

Alexander A. Davydov, Massimo Giulietti, Stefano Marcugini, Fernanda Pambianco. Linear nonbinary covering codes and saturating sets in projective spaces. Advances in Mathematics of Communications, 2011, 5 (1) : 119-147. doi: 10.3934/amc.2011.5.119

[8]

Steve Hofmann, Dorina Mitrea, Marius Mitrea, Andrew J. Morris. Square function estimates in spaces of homogeneous type and on uniformly rectifiable Euclidean sets. Electronic Research Announcements, 2014, 21: 8-18. doi: 10.3934/era.2014.21.8

[9]

Luigi Ambrosio, Michele Miranda jr., Diego Pallara. Sets with finite perimeter in Wiener spaces, perimeter measure and boundary rectifiability. Discrete & Continuous Dynamical Systems - A, 2010, 28 (2) : 591-606. doi: 10.3934/dcds.2010.28.591

[10]

Ademir Fernando Pazoto, Lionel Rosier. Uniform stabilization in weighted Sobolev spaces for the KdV equation posed on the half-line. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1511-1535. doi: 10.3934/dcdsb.2010.14.1511

[11]

Gaocheng Yue. Attractors for non-autonomous reaction-diffusion equations with fractional diffusion in locally uniform spaces. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1645-1671. doi: 10.3934/dcdsb.2017079

[12]

Klaus Metsch. A note on Erdős-Ko-Rado sets of generators in Hermitian polar spaces. Advances in Mathematics of Communications, 2016, 10 (3) : 541-545. doi: 10.3934/amc.2016024

[13]

Liangwei Wang, Jingxue Yin, Chunhua Jin. $\omega$-limit sets for porous medium equation with initial data in some weighted spaces. Discrete & Continuous Dynamical Systems - B, 2013, 18 (1) : 223-236. doi: 10.3934/dcdsb.2013.18.223

[14]

Irene Benedetti, Luisa Malaguti, Valentina Taddei. Nonlocal problems in Hilbert spaces. Conference Publications, 2015, 2015 (special) : 103-111. doi: 10.3934/proc.2015.0103

[15]

Haim Brezis, Petru Mironescu. Composition in fractional Sobolev spaces. Discrete & Continuous Dynamical Systems - A, 2001, 7 (2) : 241-246. doi: 10.3934/dcds.2001.7.241

[16]

Bruce Hughes. Geometric topology of stratified spaces. Electronic Research Announcements, 1996, 2: 73-81.

[17]

Fritz Gesztesy, Rudi Weikard, Maxim Zinchenko. On a class of model Hilbert spaces. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 5067-5088. doi: 10.3934/dcds.2013.33.5067

[18]

Vincenzo Recupero. Hysteresis operators in metric spaces. Discrete & Continuous Dynamical Systems - S, 2015, 8 (4) : 773-792. doi: 10.3934/dcdss.2015.8.773

[19]

Alejandro Adem and Jeff H. Smith. On spaces with periodic cohomology. Electronic Research Announcements, 2000, 6: 1-6.

[20]

Sonja Cox, Arnulf Jentzen, Ryan Kurniawan, Primož Pušnik. On the mild Itô formula in Banach spaces. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2217-2243. doi: 10.3934/dcdsb.2018232

2018 Impact Factor: 0.263

Article outline

Figures and Tables

[Back to Top]