Research On Fractional Calculus And Its Application In Digital Watermarking

Fractional calculus includes fractional derivatives and fractional integrals. It means to generalize the differentiation and integration into fractional order. It is a more than 300-years-old topic from the first raised, but because of the high computation complexity, it was restricted in the theoretical research field by mathematicians. With the high speed development of computer science and the increasing ability of calculation in recent years, the realization of fractional calculus becomes feasible and fractional calculus was noticed and researched by more and more engineers.The following is the main research contents:(1) Analysis about the basic theory of fractional calculus: introduced the different forms of fractional calculus definition, and the simulation analysis of popular fractional calculus methods. First, validated the different definition of fractional calculus, then simulated fractional calculus of the same signal under different definitions, to get further analysis of the difference between the definitions.(2) Fractional differential equation was researched: first reviewed the traditional differential equation solving steps and results, and then compares the two differential equations for the similarities and differences, in order to get a solution of the fractional differential equation; then, by using computer's numerical capabilities, combined with fractional calculus method, presented the numerical simulation solution of the fractional differential equations.(3) The application of fractional calculus in digital watermarking technology: the operational order is used as the secrete key for the watermarking information embedding and recovering. The watermarking system performance is analyzed, particularly discussed the sensitivity of the operational order in-depth, concluded that using the operational order as the secrete key has a high degree of confidentiality and good performance against attack.