Fractional scale transform and fractional Mellin transform for scale, rotation, and translation invariant pattern recognition to discriminate between 30 phytoplankton species
DOI:
https://doi.org/10.31349/RevMexFis.72.051301Keywords:
Computer vision, fractional Mellin transform, fractional Scale transform, pattern recognition, phytoplanktonAbstract
The fractional scale transform is introduced and studied to develop a new pattern recognition system invariant to scale, translation, and rotation. This system was also implemented with the fractional Mellin transform, and the results were compared with those obtained with the fractional scale transform. The analysis was performed to classify 30 phytoplankton species. For both transformations, optimal orders were found for each species. This study implemented nonlinear correlation and adaptive nonlinear correlation for the classification stage. The system achieved a mean accuracy of 0.998 with nonlinear correlation and 0.999 with adaptive nonlinear correlation.
Downloads
References
H. M. Ozaktas, M. A. Kutay, and D. Mendlovic, Introduction to the fractional Fourier transform and its applications, In Advances in imaging and electron physics, 106 (1999) 239, https://doi.org/10.1016/S1076-5670(08) 70272-6
L. F. López-Ávila and J. Álvarez-Borrego, Fractional Fourier transform for digital image recognition, International Journal of Pattern Recognition and Artificial Intelligence 34 (2020), https://doi.org/10.1142/S0218001420540257
E. Garza-Flores and J. Álvarez-Borrego, Pattern recognition using binary masks based on the fractional Fourier transform, Journal of Modern Optics 65 (2018) 1634, https://doi.org/10.1080/09500340.2018.1457186
O. Akay and G. F. Boudreaux-Bartels, Fractional Mellin transform: an extension of fractional frequency concept for scale, In Proc. 8th IEEE Digital Signal Processing Workshop, pages CD-ROM. IEEE (1998)
P. L. Butzer, A. A. Kilbas, and J. J. Trujillo, Fractional calculus in the Mellin setting and Hadamard-type fractional integrals, Journal of Mathematical Analysis and Applications 269 (2002) 1, https://doi.org/10.1016/S0022-247X(02)00001-X
P. L. Butzer, A. A. Kilbas, and J. J. Trujillo, Compositions of Hadamard-type fractional integration operators and the semigroup property, Journal of Mathematical Analysis and Applications 269 (2002) 387, https://doi.org/10.1016/S0022-247X(02)00049-5
P. L. Butzer, A. A. Kilbas, and J. J. Trujillo, Mellin transform analysis and integration by parts for Hadamard-type fractional integrals, Journal of Mathematical Analysis and Applications 270 (2002) 1, https://doi.org/10.1016/S0022-247X(02)00066-5
D. Casasent and D. Psaltis, Scale invariant optical transform, Optical Engineering 15 (1976) 258, https://doi.org/10.1117/12.7971962
D. Casasent and D. Psaltis, Position, rotation, and scale invariant optical correlation, Applied Optics 15 (1976) 1795, https://doi.org/10.1364/AO.15.001795
D. Sazbon et al., Using Fourier/Mellin-based correlators and their fractional versions in navigational tasks, Pattern Recognition 35 (2002) 2993, https://doi.org/10.1016/S0031-3203(02)00018-3
V. D. Sharma, On the relationship between the fourier-mellin and the fractional fourier-mellin transform, Bulletin of Pure & Applied Sciences-Mathematics and Statistics 30 (2011) 251
X. Zuo and N. Zhou, Spectrum analysis on Mellin Transform and fractional Mellin transform, In 2011 4th International Congress on Image and Signal Processing, 4 (2011) 2218-2222, https://doi.org/10.1109/CISP.2011.6100534
N. Zhou et al., Novel color image encryption algorithm based on the reality preserving fractional Mellin transform, Optics & Laser Technology 44 (2012) 2270, https://doi.org/10.1016/j.optlastec.2012.02.027
S. Vashisth et al., Devil’s vortex phase structure as frequency plane mask for image encryption using the fractional Mellin transform, International Journal of Optics 2014 (2014) 728056, https://doi.org/10.1155/2014/728056
V. D. Sharma and P. B. Deshmukh, Introduction to two dimensional fractional fourier-mellin transform, International Journal of Research in Engineering and Applied Sciences 5 (2015) 137
M. Baloch et al., The Relationship of Fractional Laplace Transform with Fractional Fourier, Mellin and Sumudu Transforms, American Scientific Research Journal for Engineering, Technology, and Sciences 55 (2019) 11
M. Wang et al., Image encryption scheme based on a Gaussian apertured reality-preserving fractional Mellin transform, Optica Applicata 50 (2020) 477, https://doi.org/10.37190/oa20012
L. Cohen, The scale representation, IEEE Transactions on Signal Processing 41 (1993) 3275, https://doi.org/10.1109/78.258073
A. Coronel-Beltrán and J. Álvarez-Borrego, Comparative analysis between different font types and letter styles using a nonlinear invariant digital correlation, Journal of Modern Optics 57 (2010) 58, https://doi.org/10.1080/09500340903511695
A. De Sena and D. Rocchesso, A fast Mellin and scale transform, EURASIP Journal on Advances in Signal Processing 2007 (2007) 089170, https://doi.org/10.1155/2007/89170
D. Casasent and D. Psaltis, New optical transforms for pattern recognition, Proceedings of the IEEE 65 (1977) 77, https://doi.org/10.1109/PROC.1977.10432
G. Cristobal, J. Cuesta, and L. Cohen, Image filtering and denoising through the scale transform, In Proceedings of the IEEE-SP International Symposium on TimeFrequency and Time-Scale Analysis (Cat. No. 98TH8380) (IEEE, 1998) pp. 617-620, https://doi.org/10.1109/TFSA.1998.721500
J. Álvarez-Borrego et al., Invariant recognition of polychromatic images of Vibrio cholerae O1, Optical Engineering 41 (2002) 827, https://doi.org/10.1117/1.1456539
V. D. Sharma and P. B. Deshmukh, Applications of Two Dimensional Fractional Mellin Transform, International Journal of Scientific and Innovative Mathematical Research 2 (2014) 794
J. L. Pech-Pacheco et al., Automatic object identification irrespective of geomatric changes, Optical Engineering 42 (2003) 551, https://doi.org/10.1117/1.1531189
S. Solorza and J. Álvarez-Borrego, Position and rotationinvariant pattern recognition system by binary rings masks, Journal of Modern Optics 62 (2015) 851, https://doi.org/10.1080/09500340.2015.1013579
A. S. Ventura, J. Álvarez-Borrego, and S. Solorza, Adaptive nonlinear correlation with a binary mask invariant to rotation and scale, Optics Communications 339 (2015) 185, https://doi.org/10.1016/j.optcom.2014.11.051
Downloads
Published
How to Cite
Issue
Section
License
Copyright (c) 2026 E. Garza-Flores, J. Álvarez-Borrego
This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.
Authors retain copyright and grant the Revista Mexicana de Física right of first publication with the work simultaneously licensed under a CC BY-NC-ND 4.0 that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
