In this work, we developed a method to decipher the shape message imprinted in single sediment particles, a method that would reflect the detail information of surface morphology. It uses 2D mathematical descriptions to restore and characterize 3D shape and surface morphology of sediment particles in terms of the "mathematical sediment", so as to better understand 3D geometric characteristics reflected by single sediment particles. To prevent the simplification of morphology description and overcome the deficiencies of lower-dimensional characteristics, we proposed a concept of "mathematical sediment" in this paper. The mathematical sediment uses image analysis of scanning electron microscope photographs and complex Fourier shape analysis to describe the planar projection shape of sediment particles and further restore the 3D morphology of sediment particles through a certain combination ways. The shapes of sediment particles are controlled using Fourier coefficients to generate a variety of mathematical sediments with various shapes, which allow the realization of the description and analysis of the 3D morphology of sediment particles. The fractal theory is further used to verify the rationality of mathematical sediment. Compared with the traditional method, the mathematical sediment overcomes the lack of particle system and smooth sphere systems and can reproduce 3D irregular surfaces for the morphology analysis. The rationality verification showed that the complex surface morphology of mathematical sediment is basically similar to the surface morphology of natural sediment. The mathematical sediment can reflect the true surface characteristics of sediment particles. The characteristics of shape properties and surface morphology it conveys are consistent with the natural sediment and can be used as a research basis for the further study both in fresh water and marine.
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