# Cross-Correlation and Fractal Analysis in the Images Diatoms Symmetry
## 1 Introduction
In addition, diatoms are characterized by having a silica capsule (frustule) that allows them to be preserved as microfossils [3; 19]. The frustule has rounded or elongated shapes and bilateral or radial symmetry, and they are formed by two valve faces or theca (epitheca and hypotheca) joined at their margins. They can be pinnate; some have a central longitudinal groove named \"the raphe,\" which forms stigmas [3]. The frustules have ornaments and mark often used as distinctive features for their classification [20; 21]. In the theca, they have regular or irregular strize (stigmas), and they can also present pores or chambers and thin transverse ribs [3]. Common methods for the physical classification of diatoms are based on optical microscopy, which uses quantitative and qualitative morphological traits for their identification [21]. Through optical microscopy, Campo determines the length and width of the frustules, the density of striae, symmetry and the presence or absence of raphe and ornaments [20; 22], followed by other characteristics, such as ultrastructure, sexual reproduction, and the development of auxospores [23]. Other methods, such as electron microscopy and DNA sequence analysis, allow better differentiation between species with similar morphology or homoplastic [22], the combination of morphological, molecular, and chemical data, referred to as the polyphasic or integrative approach [23]. Additionally, molecular techniques are used by the diatom's identification through the sequence amplification and analysis of the 18S ribosomal RNA gene, the large subunit of RuBisCO rbcl gene and psb C, being the metagenomic analysis, a fundamental approach to understanding better the diversity of these species [1; 24], another approach is the untargeted UPLC-HR-MS/MS metabolomic approach, used for detection of key biomarkers and particularly, metabolic fingerprinting has been shown to be a promising method in the classification and taxonomy of filamentous fungi, yeast and microalgae. Secondary metabolites profiling has a high differentiation at order, genus and species levels in seaweeds [25], microalgae [26] and diatoms [23]. However, despite efficient molecular and biochemical methods, morphological identification cannot be ruled out because it provides essential and more accessible information. Therefore, there is interest in creating AI algorithms or using machine learning techniques that could rapidly and efficiently determine internal and external specific traits of microalgae rapidly and efficiently [27]. Bioprospecting and multiple uses of diatoms generate the need to develop new rapid, sensitive, and inexpensive identification methods. Methodologies using neural networks have been published, in the case of [PERSON] et al., four algorithms with relatively balanced complexity and accuracy among deep learning image classification algorithms for automatic diatom recognition; the inception-dependent model was more accurate; this has stronger directivity and targeting in feature extraction of diatom and the best performance on diatom identification [28]. A fractal dimension is an index for characterizing fractal patterns or sets by quantifying their complexity as a ratio of the change in detail to the change in scale. Several types of fractal dimensions can be measured theoretically and empirically. Fractal dimensions are used to characterize a wide spectrum of objects ranging from abstract to practical phenomena, including turbulence river networks, urban growth, human physiology, medicine, and market trends [29; 30; 31]. Cross-correlation is a measure of the similarity of two series as a function of the displacement of one relative to the other and this is used in signal processing. This is also known as a sliding dot product or sliding inner product. In statistics, cross-correlation is sometimes used to refer to the covariance cov(X, Y) between two random vectors X and Y.