Tropical algebra with high-order matrix for multiple-noise removal

The technology for multiple-noise removal has triggered skyrocketing interest in both mathematics and engineering, and the tropical algebra has laid the foundation for an abundance of noise filters. However, the denoising of the filter based on the traditional algebra is inextricably complex, and its algorithm is extremely intricate and awfully inefficient, so it is necessary to estimate the statistical characteristics of noise in a novel way. Now the tropical algebra has opened the path for a new way to design optimally a denoising method, which has obvious advantages over traditional ones in denoising efficiency and simple filtering algorithm. In this paper, the idempotent of the multiplicative semigroup of [Formula: see text] tropical matrices is studied. First, the tropical algebra and the tropical matrix multiplicative semigroup are introduced. Second, the idempotent classification of the [Formula: see text] tropical matrix multiplicative semigroup is given. Finally, an example is given, in which the filter based on tropical algebra is used for denoising, and an optimization method with tropical polynomials as constraints is given.