Читать книгу Optical Cryptosystems - Naveen K. Nishchal - Страница 27

Mellin transform is closely related to the Laplace and Fourier transforms and exhibits a certain invariance to object magnification. Considering 1D form, the Mellin transform of a function f(x) is defined by [4]

M(s)=∫0∞f(x)xs−1dx(2.37)

where, in the most general case, s is a complex variable. Considering the complex variable s = i2πf_X and substituting x = e^−ξ, the Mellin transform is expressed as

M(i2πfX)=∫−∞∞f(e−ξ)e−i2πfXξdξ(2.38)

Equation (2.34) represents the Fourier transform of the function f(e^−ξ). The Mellin transform can be implemented with an optical Fourier transforming system. The most important property of this transform is that its magnitude is independent of scale-size changes in the input. This property has attracted the use of the Mellin transform to image processing applications including image encryption [36]. Similar to the concept of FRT, fractional Mellin transform has also been used in developing a nonlinear encryption scheme to survive the conventional attacks. Also, extending the properties of FRT, fractional Hartley transform has been proposed for image encryption [37]. In all the cases, the basic DRPE framework has been combined with other optically implementable transforms to enhance the level of security. The attributes of transforms provide attractive features in security applications.