A comparison among a fuzzy algorithm for image rescaling with other methods of digital image processing

Abstract

The aim of this paper is to compare the fuzzy-type algorithm for image rescaling introduced by Jurio et al., 2011, quoted in the list of references, with some other existing algorithms such as the classical bicubic algorithm and the sampling Kantorovich (SK) one. Note that the SK algorithm is a recent tool for image rescaling and enhancement that has been revealed to be useful in several applications to real world problems, while the bicubic algorithm is widely known in the literature. A comparison among the abovementioned algorithms (all implemented in the MatLab programming language) was performed in terms of suitable similarity indices such as the Peak-Signal-to-Noise-Ratio (PSNR) and the likelihood index $S$.

Keywords

• Fuzzy-type algorithm
• SK algorithm
• bicubic algorithm
• PSNR
• S index
• image magnification

References

1. T. Acar, B. R. Draganov: A characterization of the rate of the simultaneous approximation by generalized sampling operators and their Kantorovich modification, J. Math. Anal. Appl., 530 (2) (2024), 127740.
2. T. Acar, M. Turgay: Approximation by Modified Generalized Sampling Series, Mediterr. J. Math., 21 (3) (2024), 107.
3. L. Angeloni, N. Nursel Cetin, D. Costarelli, A. R. Sambucini and G. Vinti: Multivariate sampling Kantorovich operators: quantitative estimates in Orlicz spaces, Constr. Math. Anal., 4 (2) (2021), 229–241.
4. L. Angeloni, D. Costarelli, M. Seracini, G. Vinti and L. Zampogni: Variation diminishing-type properties for multivariate sampling Kantorovich operators, Boll. dell’Unione Matem. Ital., 13 (4) (2020), 595–605.
5. L. Angeloni, G. Vinti: Multidimensional sampling-Kantorovich operators in BV-spaces, Open Math., 21 (1) (2023), 20220573,
6. V. Apollonio, R. D’Autilia, B. Scoppola, E. Scoppola and A. Troiani: Criticality of Measures on 2-d Ising Configurations: From Square to Hexagonal Graphs, J. Statistical Phys., 177 (5) (2019), 1009–1021.
7. K. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems, 20 (1986), 87–96.
8. C. Bardaro, P.L. Butzer, R.L. Stens and G. Vinti: Kantorovich-type generalized sampling series in the setting of Orlicz Spaces, Sampl. Theory Signal Process. Data Anal., 6 (1) (2007), 29–52

9. C. Bardaro, I. Mantellini: Asymptotic formulae for multivariate Kantorovich type generalized sampling series, Acta Mathematica Sinica (E.S.), 27 (7) (2011), 1247–1258.
10. C. Bardaro, I. Mantellini: On convergence properties for a class of Kantorovich discrete operators, Numer. Funct. Anal. Opt., 33 (2012), 374–396.
11. G. Biau, L. Devroye: Lectures on the nearest neighbor method, Springer International Publishing, Cham, Switzerland (2015).
12. A Boccuto, A. R. Sambucini: Some applications of modular convergence in vector lattice setting, Sampl. Theory Signal Process. Data Anal., 20 (2022), 12.
13. A Boccuto, A. R. Sambucini: Abstract integration with respect to measures and applications to modular convergence in vector lattice setting, Results Math., 78 (2023), 4.
14. P. Burillo, H. Bustince: Entropy on intuistionistic fuzzy sets and on interval-valued fuzzy sets, Fuzzy Sets and Systems, 78 (1996), 305–316.
15. P. Burillo, H. Bustince: Construction theorems for intuistionistic fuzzy sets, Fuzzy Sets and Systems, 84 (1996), 271–281.
16. H. Bustince, E. Barrenechea, M. Pagola and J. Fernandez: Interval-valued fuzzy sets constructed from matrices: Application to edge detection, Fuzzy Sets and Systems, 160 (13) (2009), 1819–1840.
17. H. Bustince, M. Pagola and E. Barrenechea: Construction of fuzzy indices from fuzzy DI-subsethood measures: Application to the global comparison of images, Inf. Sci., 177 (2007), 906–929.
18. P. L. Butzer, M. Schmidt and E.L. Stark: Observations on the history of central B-splines, Arch. Hist. Exact Sci., 39.2 (1988), 137–156.
19. D. Candeloro, A. R. Sambucini: Filter Convergence and Decompositions for Vector Lattice-Valued Measures, Mediterr. J. Math., 12 (2015), 621–637.
20. M. Cantarini, D. Costarelli and G. Vinti: Approximation of differentiable and not differentiable signals by the first derivative of sampling Kantorovich operators, J. Math. Anal. Appl., 509 (1) (2022), 125913.
21. M. Cantarini, D. Costarelli and G. Vinti: Approximation results in Sobolev and fractional Sobolov spaces by sampling Kantorovich operators, Fract. Calc. Appl. Anal., 26 (2023), 2493-2521.
22. M. Castro, D.M. Ballesteros and D. Renza: A dataset of 1050-tampered color and grayscale images (CG-1050), Data in brief, (2019).
23. N. Çetin, D. Costarelli, M. Natale and G. Vinti, Nonlinear multivariate sampling Kantorovich operators: quantitative estimates in functional spaces, Dolomites Res. Notes Approx., 15 (2022), 12–25.
24. F. Cluni, D. Costarelli, V. Gusella and G. Vinti, Reliability increase of masonry characteristics estimation by sampling algorithm applied to thermographic digital images, Probabilistic Eng. Mech., 60 (2020), 103022.
25. D. Costarelli, A. Croitoru, A. Gavrilu¸t, A. Iosif and A. R. Sambucini: The Riemann-Lebesgue integral of intervalvalued multifunctions, Mathematics, 8 (12) (2020), 1–17, 2250.
26. D. Costarelli, M. Natale and G. Vinti: Convergence results for nonlinear sampling Kantorovich operators in modular spaces, Numer. Funct. Anal. Optim., 44 (12) (2023), 1276–1299.
27. D. Costarelli, M. Piconi and G. Vinti: The multivariate Durrmeyer-sampling type operators: approximation in Orlicz spaces, Dolomites Res. Notes Approx., 15 (2022), 128–144.
28. D. Costarelli, M. Piconi and G. Vinti: Quantitative estimates for Durrmeyer-sampling series in Orlicz spaces, Sampl. Theory Signal Process. Data Anal., 21 (2023), 3.
29. D. Costarelli, A. R. Sambucini: Approximation results in Orlicz spaces for sequences of Kantorovich max-product neural network operators, Results Math., 73 (2018), 15.
30. D. Costarelli, A. R. Sambucini and G. Vinti: Convergence in Orlicz spaces by means of the multivariate max-product neural network operators of the Kantorovich type, Neural Comput. Appl., 31 (9) (2019), 5069–5078.
31. D. Costarelli, M. Seracini and G. Vinti: A comparison between the sampling Kantorovich algorithm for digital image processing with some interpolation and quasi-interpolation methods, Appl. Math. Comput., 374 (2020), 125046.
32. D. Costarelli, M. Seracini and G. Vinti: A segmentation procedure of the pervious area of the aorta artery from CT images without contrast medium, Math. Methods Appl. Sci., 43 (2020) 114-133.
33. D. Costarelli, G. Vinti: Approximation properties of the sampling Kantorovich operators: regularization, saturation, inverse results and Favard classes in Lp-spaces, J. Fourier Anal. Appl., 28 (2022), 49.
34. I. Couso, H. Bustince: From Fuzzy Sets to Interval-Valued and Atanassov Intuitionistic Fuzzy Sets: A Unified View of Different Axiomatic Measures, in IEEE Trans. Fuzzy Syst., 27 (2) (2019), 362–371.
35. A. Croitoru, A. Gavrilu¸t, A. Iosif and A. R. Sambucini: A note on convergence results for varying interval valued multisubmeasures, Math. Found. Comput., 4 (4) (2021), 299–310.
36. R. D’Autilia, L.N. Andrianaivo and A. Troiani: Parallel simulation of two-dimensional Ising models using probabilistic cellular automata, J. Stat. Phys., 184 (2021), 1–22.
37. L. Di Piazza, V. Marraffa, K. Musiał and A. R. Sambucini: Convergence for varying measures, J. Math. Anal. Appl., 518 (2023), 126782.
38. L. Di Piazza, V. Marraffa, K. Musiał and A. R. Sambucini: Convergence for varying measures in the topological case, Ann. Mat. Pura Appl., (4), 203 (2024), 71–86.
39. L. Di Piazza, V. Marraffa and B. Satco: Measure differential inclusions: existence results and minimum problems, Set-Valued Var. Anal., 29 (2) (2021), 361–382.
40. B. R. Draganov: A fast converging sampling operator, Constr. Math. Anal., 5 (4) (2022), 190-201.
41. R.C. Gonzalez, R.E. Woods: Digital Image Processing, Pearson Prenctice Hall (2007).
42. A. Jurio, D. Paternain, C. Lopez-Molina, H. Bustince, R. Mesiar and G. Beliakov: A construction method of interval-valued Fuzzy Sets for image processing, 2011 IEEE Symposium on Advances in Type-2 Fuzzy Logic Systems (T2FUZZ), (2011), 16–22.
43. Y. Kolomoitsev, M. Skopina: Approximation by sampling-type operators in Lp-spaces, Math. Methods Appl. Sci., 43 (16) (2020), 9358–9374.
44. Y. Kolomoitsev, M. Skopina: Quasi-projection operators in weighted Lp spaces, Appl. Comput. Harmon. Anal., 52 (2021), 165–197.
45. D. La Torre, F. Mendevil: The Monge-Kantorovich metric on multimeasures and self-similar multimeasures, Set-Valued Var. Anal., 23 (2015), 319–331.
46. J. Liu, Z. Gan and X. Zhu: Directional Bicubic Interpolation - A New Method of Image Super-Resolution, Proc. 3rd International Conf. Multimedia Technology (ICMT-13), In: Advances in Intelligent Systems Research, (2013).
47. A. Osowska-Kurczab, T. Les, T. Markiewicz, M. Dziekiewicz, M. Lorent, S. Cierniak, D. Costarelli, M. Seracini and G. Vinti: Improvement of renal image recognition through resolution enhancement, Expert Syst. Appl., 213 (A) (2023), 118836.
48. E. Pap, A. Iosif and A. Gavrilu¸t: Integrability of an Interval-valued Multifunction with respect to an Interval-valued Set Multifunction, Iran. J. Fuzzy Syst., 15 (3) (2018), 47–63.
49. B. Scoppola, A. Troiani and M. Veglianti: Shaken dynamics on the 3d cubic lattice, Electron. J. Probab., 27 (2022), 1–26.
50. A. Tanchenko: Visual-PSNR measure of image quality, J. Vis. Commun. Image Represent., 25 (5) (2014), 874–878.
51. The Vision and Image Processing Lab at University ofWaterloo, Greyscale Set 2, https://links.uwaterloo. ca/Repository.html
52. G. Vinti, L. Zampogni: Approximation results for a general class of Kantorovich type operators, Adv. Nonlinear Stud., 14 (4) (2014), 991–1011.