Читать книгу Optical Cryptosystems - Naveen K. Nishchal - Страница 24

The gyrator transform (GT) is a linear canonical integral transform, which produces the rotation in twisted position-spatial frequency planes of phase space [25]. Similar to FRT, gyrator transform is also a generalization of the ordinary Fourier transform with a parameter α. For α = 0, it corresponds to identity transform and for α = π/2, it corresponds to Fourier transform. The gyrator transform is periodic and additive with respect to parameter α.

Similar to FRT, gyrator transform has been used in image encryption applications [26, 27]. This is because of parameter α, which connects with the angle of gyrator transform and provides additional security to the encryption scheme. This is also optically implemented employing cylindrical lenses. Mathematically, GT of any function f(x,y) is defined as [25],

g(x2,y2)=1∣sinα∣∬f(x1,y1)expi2π(x2y2+x1y1)cosα−(x1y2+x2y1)sinαdx1dy1(2.30)

Here, (x1,y1) are the co-ordinates of the input function and (x2,y2) are the co-ordinates in the gyrator domain and α is the angle of the GT. The ciphertext generated by using DRPE in the gyrator domain is then written as:

E(x,y)=GTβGTαf(x,y)*exp(i2πR1(x,y))*exp(i2πR2(u,v))(2.31)

Here, GT^α{.} and GT^β{.} represent the gyrator transform operations applied for angles α and β, respectively. The functions R1(x,y) and R2(u,v) are two random phase value distribution, lying in the interval [0,1].