Model-Driven Applications of Fractional Derivatives and Integrals
This work offers a logical extension of current techniques for signal processing, but it is incremental as it builds on existing fractional calculus concepts.
The paper tackles the problem of extending integral and differential operators through fractional order derivatives and integrals, using a frequency-domain perspective to provide computational tools and sample applications for one- and two-dimensional signals.
Fractional order derivatives and integrals (differintegrals) are viewed from a frequency-domain perspective using the formalism of Riesz, providing a computational tool as well as a way to interpret the operations in the frequency domain. Differintegrals provide a logical extension of current techniques, generalizing the notion of integral and differential operators and acting as kind of frequency-domain filtering that has many of the advantages of a nonlocal linear operator. Several important properties of differintegrals are presented, and sample applications are given to one- and two-dimensional signals. Computer code to carry out the computations is made available on the author's website.