# Cross-Correlation and Fractal Analysis in the Images Diatoms Symmetry

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Diatom Images

#### 2.2. Fractal Dimension Analysis

^{®}[36], and it was applied to the different diatoms images used in this study.

#### 2.3. Power Spectra Analysis

^{®}, the FFT function computes the Fourier transform using a fast Fourier transform algorithm. Use FFT to compute the discrete Fourier transform of the signal. y = fft(x); plot the power spectrum as a function of frequency.

#### 2.4. Two-Dimensional Cross-Correlation

#### 2.5. Software

## 3. Results and Discussion

#### 3.1. Self-Similarity and Symmetry

#### 3.2. Power Spectrum in the Self-Similarity and Symmetry

#### 3.3. CC between Diatoms

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Ashworth, M.P.; Lobban, C.S.; Witkowski, A.; Theriot, E.C.; Sabir, M.J.; Baeshen, M.N.; Hajarah, N.H.; Baeshen, N.A.; Sabir, J.S.; Jansen, R.K. Molecular and Morphological Investigations of the Stauros-Bearing, Raphid Pennate Diatoms (Bacillariophyceae): Craspedostauros E.J. Cox, and Staurotropis T.B.B. Paddock, and Their Relationship to the Rest of the Mastogloiales. Protist
**2017**, 168, 48–70. [Google Scholar] [CrossRef] - Cavalier-Smith, T. Kingdom Chromista and Its Eight Phyla: A New Synthesis Emphasising Periplastid Protein Targeting, Cytoskeletal and Periplastid Evolution, and Ancient Divergences. Protoplasma
**2018**, 255, 297–357. [Google Scholar] [CrossRef] [PubMed][Green Version] - Sánchez, C.; Cristóbal, G.; Bueno, G. Diatom Identification Including Life Cycle Stages through Morphological and Texture Descriptors. PeerJ
**2019**, 7, e6770. [Google Scholar] [CrossRef] - Obata, T.; Fernie, A.R.; Nunes-Nesi, A. The Central Carbon and Energy Metabolism of Marine Diatoms. Metabolites
**2013**, 3, 325–346. [Google Scholar] [CrossRef] [PubMed] - Sharma, N.; Simon, D.P.; Diaz-Garza, A.M.; Fantino, E.; Messaabi, A.; Meddeb-Mouelhi, F.; Germain, H.; Desgagné-Penix, I. Diatoms Biotechnology: Various Industrial Applications for a Greener Tomorrow. Front. Mar. Sci.
**2021**, 8, 636613. [Google Scholar] [CrossRef] - Mann, D.G.; Droop, S.J.M. Biodiversity, Biogeography and Conservation of Diatoms. In Biogeography of Freshwater Algae; Springer: Dordrecht, The Netherlands, 1996; pp. 19–32. [Google Scholar]
- Guiry, M.D. How Many Species of Algae Are There? J. Phycol.
**2012**, 48, 1057–1063. [Google Scholar] [CrossRef] [PubMed] - Saxena, A.; Tiwari, A.; Kaushik, R.; Iqbal, H.M.N.; Parra-Saldívar, R. Diatoms Recovery from Wastewater: Overview from an Ecological and Economic Perspective. J. Water Process Eng.
**2021**, 39, 101705. [Google Scholar] [CrossRef] - Benoiston, A.-S.; Ibarbalz, F.M.; Bittner, L.; Guidi, L.; Jahn, O.; Dutkiewicz, S.; Bowler, C. The Evolution of Diatoms and Their Biogeochemical Functions. Philos. Trans. R. Soc. B Biol. Sci.
**2017**, 372, 20160397. [Google Scholar] [CrossRef][Green Version] - Srikanta Dani, K.G.; Silva Benavides, A.M.; Michelozzi, M.; Peluso, G.; Torzillo, G.; Loreto, F. Relationship between Isoprene Emission and Photosynthesis in Diatoms, and Its Implications for Global Marine Isoprene Estimates. Mar. Chem.
**2017**, 189, 17–24. [Google Scholar] [CrossRef] - Coleman, M.L.; Sullivan, M.B.; Martiny, A.C.; Steglich, C.; Barry, K.; DeLong, E.F.; Chisholm, S.W. Genomic Islands and the Ecology and Evolution of Prochlorococcus. Science
**2006**, 311, 1768–1770. [Google Scholar] [CrossRef][Green Version] - Keeling, P.J.; Palmer, J.D. Horizontal Gene Transfer in Eukaryotic Evolution. Nat. Rev. Genet.
**2008**, 9, 605–618. [Google Scholar] [CrossRef] [PubMed] - Curtis, B.A.; Tanifuji, G.; Burki, F.; Gruber, A.; Irimia, M.; Maruyama, S.; Arias, M.C.; Ball, S.G.; Gile, G.H.; Hirakawa, Y.; et al. Algal Genomes Reveal Evolutionary Mosaicism and the Fate of Nucleomorphs. Nature
**2012**, 492, 59–65. [Google Scholar] [CrossRef][Green Version] - Morais, W.G., Jr.; Gorgich, M.; Corrêa, P.S.; Martins, A.A.; Mata, T.M.; Caetano, N.S. Microalgae for Biotechnological Applications: Cultivation, Harvesting and Biomass Processing. Aquaculture
**2020**, 528, 735562. [Google Scholar] [CrossRef] - Machado Sierra, E.; Serrano, M.C.; Manares, A.; Guerra, A.; Aranguren Díaz, Y. Microalgae: Potential for Bioeconomy in Food Systems. Appl. Sci.
**2021**, 11, 11316. [Google Scholar] [CrossRef] - Blanco, S. Diatom Taxonomy and Identification Keys. In Modern Trends in Diatom Identification; Springer International Publishing: Cham, Switzerland, 2020; pp. 25–38. [Google Scholar]
- Jamali, A.A.; Akbari, F.; Ghorakhlu, M.M.; de la Guardia, M.; Khosroushahi, A.Y. Applications of Diatoms as Potential Microalgae in Nanobiotechnology. BioImpacts
**2012**, 2, 83–89. [Google Scholar] [CrossRef] - Sun, X.W.; Zhang, Y.X.; Losic, D. Diatom Silica, an Emerging Biomaterial for Energy Conversion and Storage. J. Mater. Chem. A
**2017**, 5, 8847–8859. [Google Scholar] [CrossRef] - Passy, S.I. Environmental Randomness Underlies Morphological Complexity of Colonial Diatoms. Funct. Ecol.
**2002**, 16, 690–695. [Google Scholar] [CrossRef] - Rajkumar, R.; Yaakob, Z. The Biology of Microalgae. In Biotechnological Applications of Microalgae; Bux, F., Ed.; CRC Press: Boca Raton, FL, USA, 2013; pp. 7–16. ISBN 978-0-42908-711-0. [Google Scholar]
- Cristóbal, G.; Blanco, S.; Bueno, G. Modern Trends in Diatom Identification: Fundamentals and Applications; Springer: Berlin/Heidelberg, Germany, 2020; ISBN 978-3-03039-211-6. [Google Scholar]
- Blanco, S.; Borrego-Ramos, M.; Olenici, A. Disentangling Diatom Species Complexes: Does Morphometry Suffice? PeerJ
**2017**, 5, e4159. [Google Scholar] [CrossRef] [PubMed][Green Version] - Trentin, R.; Moschin, E.; Duarte Lopes, A.; Schiaparelli, S.; Custódio, L.; Moro, I. Molecular, Morphological and Chemical Diversity of Two New Species of Antarctic Diatoms, Craspedostauros ineffabilis Sp. Nov. and Craspedostauros zucchellii Sp. Nov. J. Mar. Sci. Eng.
**2022**, 10, 1656. [Google Scholar] [CrossRef] - Liu, M.; Zhao, Y.; Sun, Y.; Li, Y.; Wu, P.; Zhou, S.; Ren, L. Comparative Study on Diatom Morphology and Molecular Identification in Drowning Cases. Forensic Sci. Int.
**2020**, 317, 110552. [Google Scholar] [CrossRef] - Hughes, A.H.; Magot, F.; Tawfike, A.F.; Rad-Menéndez, C.; Thomas, N.; Young, L.C.; Stucchi, L.; Carettoni, D.; Stanley, M.S.; Edrada-Ebel, R.; et al. Exploring the Chemical Space of Macro- and Micro-Algae Using Comparative Metabolomics. Microorganisms
**2021**, 9, 311. [Google Scholar] [CrossRef] - Silva, S.G.; Paula, P.; da Silva, J.P.; Mil-Homens, D.; Teixeira, M.C.; Fialho, A.M.; Costa, R.; Keller-Costa, T. Insights into the Antimicrobial Activities and Metabolomes of Aquimarina (Flavobacteriaceae, Bacteroidetes) Species from the Rare Marine Biosphere. Mar. Drugs
**2022**, 20, 423. [Google Scholar] [CrossRef] [PubMed] - Xu, Z.; Jiang, Y.; Ji, J.; Ji, J.; Forsberg, E.; Li, Y.; He, S.; He, S. Classification, Identification, and Growth Stage Estimation of Microalgae Based on Transmission Hyperspectral Microscopic Imaging and Machine Learning. Opt. Express OE
**2020**, 28, 30686–30700. [Google Scholar] [CrossRef] - Zhu, Y.-Z.; Zhang, J.; Cheng, Q.; Yu, H.-X.; Deng, K.-F.; Zhang, J.-H.; Qin, Z.-Q.; Zhao, J.; Sun, J.-H.; Huang, P. Comparison among Four Deep Learning Image Classification Algorithms in AI-Based Diatom Test. Fa Yi Xue Za Zhi
**2022**, 38, 31–39. [Google Scholar] [CrossRef] - Chen, Y. Modeling Fractal Structure of City-Size Distributions Using Correlation Functions. PLoS ONE
**2011**, 6, e24791. [Google Scholar] [CrossRef] [PubMed][Green Version] - Popescu, D.P.; Flueraru, C.; Mao, Y.; Chang, S.; Sowa, M.G. Signal Attenuation and Box-Counting Fractal Analysis of Optical Coherence Tomography Images of Arterial Tissue. Biomed. Opt. Express
**2010**, 1, 268–277. [Google Scholar] [CrossRef] [PubMed][Green Version] - King, R.D.; George, A.T.; Jeon, T.; Hynan, L.S.; Youn, T.S.; Kennedy, D.N.; Dickerson, B. Characterization of Atrophic Changes in the Cerebral Cortex Using Fractal Dimensional Analysis. Brain Imaging Behav.
**2009**, 3, 154–166. [Google Scholar] [CrossRef] [PubMed][Green Version] - Pedraza, A.; Bueno, G.; Deniz, O.; Cristóbal, G.; Blanco, S.; Borrego-Ramos, M. Automated Diatom Classification (Part B): A Deep Learning Approach. Appl. Sci.
**2017**, 7, 460. [Google Scholar] [CrossRef][Green Version] - Du Buf, H.; Bayer, M.; Droop, S.; Head, R.; Juggins, S.; Fischer, S.; Bunke, H.; Wilkinson, M.; Roerdink, J.; Pech-Pacheco, J.; et al. Diatom Identification: A Double Challenge Called ADIAC. In Proceedings of the 10th International Conference on Image Analysis and Processing, Venice, Italy, 27–29 September 1999; pp. 734–739. [Google Scholar]
- Wu, J.; Jin, X.; Mi, S.; Tang, J. An Effective Method to Compute the Box-Counting Dimension Based on the Mathematical Definition and Intervals. Results Eng.
**2020**, 6, 100106. [Google Scholar] [CrossRef] - Baranov, M.; Velichko, E.; Greshnevikov, K. Analysis of Fractal Structures in Dehydrated Films of Protein Solutions. Symmetry
**2021**, 13, 123. [Google Scholar] [CrossRef] - Alceu Costa Hausdorff (Box-Counting) Fractal Dimension. Available online: https://www.mathworks.com/matlabcentral/fileexchange/30329-hausdorff-box-counting-fractal-dimension (accessed on 27 December 2021).
- Wang, X.; Lu, Y.; Liu, J. A Fast Image Alignment Approach for 2D Classification of Cryo-EM Images Using Spectral Clustering. Curr. Issues Mol. Biol.
**2021**, 43, 1652–1668. [Google Scholar] [CrossRef] [PubMed] - Azar, R.Z.; Goksel, O.; Salcudean, S.E. Comparison between 2-D Cross Correlation with 2-D Sub-Sampling and 2-D Tracking Using Beam Steering. IEEE Trans. Ultrason. Ferroelectr. Freq. Control
**2011**, 58, 1534–1537. [Google Scholar] [CrossRef] [PubMed] - Woodard, K.; Neustupa, J. Geometric Morphometrics of Bilateral Asymmetry in Eunotia Bilunaris (Eunotiales, Bacillariophyceae) as a Tool for the Quantitative Assessment of Teratogenic Deviations in Frustule Shapes. Symmetry
**2022**, 14, 42. [Google Scholar] [CrossRef] - Pham, D.T.; Musielak, Z.E. Spectra of Reduced Fractals and Their Applications in Biology. arXiv
**2021**, arXiv:2111.01719. [Google Scholar] [CrossRef] - Mandelbrot, B.B.; Wheeler, J.A. The Fractal Geometry of Nature. Am. J. Phys.
**1983**, 51, 286–287. [Google Scholar] [CrossRef] - van der Schaaf, A.; van Hateren, J.H. Modelling the Power Spectra of Natural Images: Statistics and Information. Vis. Res.
**1996**, 36, 2759–2770. [Google Scholar] [CrossRef][Green Version] - Kloster, M.; Langenkämper, D.; Zurowietz, M.; Beszteri, B.; Nattkemper, T.W. Deep Learning-Based Diatom Taxonomy on Virtual Slides. Sci. Rep.
**2020**, 10, 14416. [Google Scholar] [CrossRef]

**Figure 1.**Diatom images for each species taken by optical microscopy from ADIAC diatoms image database.

**Figure 3.**Calculated 2D cross-correlation and preprocessing steps using symmetry operations from the (

**a**) imagen I3 and (

**b**) imagen I34. The red lines are the CCauto.

**Figure 4.**Polar plot for all the diatom images where the radius is Nf and the angle is the projection for each symmetry operation (

**a**) PSv, (

**b**) R180, (

**c**) PSo and (

**d**) score plot PC 1 vs. PC2 for all diatoms images using Nf, and projection parameters.

**Figure 6.**(

**a**) Projection for each symmetry operation (

**b**) score plot PC 1 vs. PC2 for all diatoms images using Nf, and projections parameters, inset: loading plot of PC1 vs. PC2.

Cod | Name | Cod | Name | Cod | Name | Cod | Name | Cod | Name |
---|---|---|---|---|---|---|---|---|---|

I1 | Actinoptychus senarius | I11 | Achnanthes | I21 | Martyana martyi | I31 | Brachysira brebissonii | I41 | Neidium affine |

I2 | Auliscus sculptus | I12 | Caloneis amphisbaena | I22 | Melosira nummuloides | I32 | Craticula ambigua | I42 | Nitzschia amphibia |

I3 | Cyclostephanos dubius | I13 | Denticula subtilis | I23 | Opephora marina | I33 | Ctenophora pulchella | I43 | Plagiotropis tayrecta |

I4 | Cyclotella radiosa | I14 | Didymosphenia geminata | I24 | Parlibellus berkeleyi | I34 | Cymbella cornuta | I44 | Staurophora amphioxys |

I5 | Hyalodiscus scoticus | I15 | Dimeregramma minor | I25 | Plagiogramma staurophorum | I35 | Encyonopsis microcephala | I45 | Stenopterobia densestriata |

I6 | Cavinula jaernefeltii | I16 | Tryblionella coarctata | I26 | Pseudostaurosira brevistriata | I36 | Eunotia bilunaris | I46 | Surirella angusta |

I7 | Delphineis surirella | I17 | Diploneis boldtiana | I27 | Stauroforma inermis | I37 | Eunotia incisa | I47 | Meridion circulare |

I8 | Luticola mutica | I18 | Encyonema elginense | I28 | Thalassionema nitzschioides | I38 | Gyrosigma acuminatum | I48 | Odontella aurita |

I9 | Psammodiscus nitidus | I19 | Epithemia sorex | I29 | Amphora acutiuscula | I39 | Hannaea arcus | I49 | Stauroneis legumen |

I10 | Rhaphoneis amphiceros | I20 | Lyrella abrupta | I30 | Berkeleya rutilans | I40 | Licmophora oedipus | I50 | Tabellaria flocculosa |

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**MDPI and ACS Style**

Pestana-Nobles, R.; Villarreal-González, R.; Galan-Freyle, N.J.; Aranguren-Díaz, Y.; Machado-Sierra, E.; Yime-Rodríguez, E.; Pacheco-Londoño, L.C. Cross-Correlation and Fractal Analysis in the Images Diatoms Symmetry. *Appl. Sci.* **2023**, *13*, 4909.
https://doi.org/10.3390/app13084909

**AMA Style**

Pestana-Nobles R, Villarreal-González R, Galan-Freyle NJ, Aranguren-Díaz Y, Machado-Sierra E, Yime-Rodríguez E, Pacheco-Londoño LC. Cross-Correlation and Fractal Analysis in the Images Diatoms Symmetry. *Applied Sciences*. 2023; 13(8):4909.
https://doi.org/10.3390/app13084909

**Chicago/Turabian Style**

Pestana-Nobles, Roberto, Reynaldo Villarreal-González, Nataly J. Galan-Freyle, Yani Aranguren-Díaz, Elwi Machado-Sierra, Eugenio Yime-Rodríguez, and Leonardo C. Pacheco-Londoño. 2023. "Cross-Correlation and Fractal Analysis in the Images Diatoms Symmetry" *Applied Sciences* 13, no. 8: 4909.
https://doi.org/10.3390/app13084909