# Cross-Correlation and Fractal Analysis in the Images Diatoms Symmetry
## 1 Introduction  
Diatoms (Bacillariophyceae) are microscopic photosynthetic algae that grow in freshwater or saltwater environments but are rich in dissolved silica. Volcanic activity is a decisive factor for dissolved silica in a body of water. The frustule or skeleton of diatoms is composed of opaline or biogenic silica (comprised of two shells = \"valves,\" joined by parallel overlapping strips of silica = \"copulae\") and serves as a morphological synapomorphy to define the clade [1]. It is symmetric in shape, with a large proportion of voids, decorated with a unique pattern of nanometer-sized features (pores, channels, spines). The most common forms are cylindrical, spherical, or circular discs; with dimensions between 1 and 500 microns. Diatoms are part of the phylum Ochrophytas, from the Chromista kingdom [2,3]. It is one of the most diverse eukaryotic algae. There are more than 200 genera of living diatoms [4], with an overall number of around 200,000 species having complex variability in dimensions and shapes [5]. Most live pelagically in fresh and marine water, wastewater, rivers, and oceans, although some live at the water-sediment interface (benthic) as surface films or even under damp atmospheric conditions [4]. Taken together, the diatoms constitute the most abundant group of marine eukaryotic phytoplankton. They are essential for both nutrient cycling and for the leading primary producers of the tropic chains in aquatic ecosystems (accounts for >40% of global primary productivity, which is equivalent biomass to that of all tropical rain forests on earth) [4,6,7]. Moreover, diatoms play important roles in the marine ecosystem; they are fundamental in carbon fixation from the atmosphere and 20% oxygen production [5,8,9] and emit a wide range of chemically active trace gases, including dimethyl sulfide, organohalogens, and volatile isoprenoids [10]. As a primary producer in the marine ecosystem, as a significantcarbon carrier to the deep ocean is one of the major components of the so-called biological carbon pump, and additionally as a significant source of petroleum deposits [4]. Evolutionarily, diatoms were derived from a secondary endosymbiotic event, which occurred between an unknown eukaryotic cell and a red alga. In this process, the diatom acquired genes from two eukaryotic genomes. In addition, there is an increasing body of evidence that suggests the occurrence of a considerable level of horizontal gene transfer among marine organisms [11; 12]. Such an evolutionary background renders algae a \"melting pot\" of genes and provides them with a particular opportunity to evolve new gene and metabolic networks [13]. Despite this fact, a large number of diatom proteins are shared by many groups of photosynthetic eukaryotes, namely red alga, green plant, and cyanobacterium, and many of them are likely involved in chloroplast functions [4]. Currently, the interest in and biotechnological uses of microalgae in the agri-food and pharmaceutical fields are increasing [14; 15]. It is due to the importance of these species in the ecosystem's maintenance, as well as their photosynthetic capacity and to produce active metabolites. Furthermore, the diversity of diatoms represents a variety of environments because they are used to estimate the integrity of ecosystems and assess eutrophication and water quality [16; 17]. Likewise, diatoms are used in other applications such as forensic analysis for location estimates, exploration of oil reserves, silica biomineralization [17; 18], biofuel production, drug delivery systems, analytical tools, biosensors and nanomaterials and nanoparticles. However, diatoms also have other fascinating applications in environment monitoring, animal feed, and aquaculture, which indeed have a considerable potential considering climate change and the devastating impacts of globalization on ecology and environment [5]. These species are found grouped in colonies or unicellular form. Additionally, they can be reproduced in a sexual or asexual way.