However, since R2(υ,μ) is a white noise uniformly distributed on [0,1],
<exp[i2π{R2(υ′,μ′)−R2(υ,μ)}]>R2=δυ−υ′δμ−μ′(2.11)
Substituting equation (2.11) in (2.10),
〈h*(x−ξ,y−η)h(p−ξ,q−η)〉R2=1N4∑υ=0N−1∑μ=0N−1exp[i2π{υ(p−x)+μ(q−y)}](2.12)
Applying the definition of discrete delta function,
∑υ=0N−1∑μ=0N−1exp[i2π{υ(p−x)+μ(q−y)}]=∑υ=0N−1exp[i2πυ(p−x)]∑μ=0N−1exp[i2πμ(q−y)]=N2δx−pδy−q(2.13)
Thus, the property stated in equation (2.8) is obtained. This property proves that the impulse response of the function exp[i2πR2(υ,μ)] is a stationary white noise.
Property 2:
The encrypted function ψ(x,y) is a stationary white noise with an autocorrelation function given as
<ψ*(x,y)ψ(ξ,η)>=1N2∑u=0N−1∑v=0N−1∣f(u,v)∣2δx−ξδy−η(2.14)
Proof:
ψ(x,y)=∑u=0N−1∑v=0N−1f(u,v)exp[i2πR1(u,v)h(x−u,y−v)](2.15)
Then,
<ψ*(x,y)ψ(ξ,η)>=∑u=0N−1∑v=0N−1∑p=0N−1∑q=0N−1f*(p,q)f(u,v)×<exp[i2π{R1(p,q)−R1(u,v)}]h*(x−p,y−q)h(ξ−u,η−v)>(2.16)
However,
<exp[i2π{R1(p,q)−R1(u,v)}]h*(x−p,y−q)h(ξ−u,η−v)>=<exp[i2π{R1(p,q)−R1(u,v)}]>R1×<h*(x−p,y−q)h(ξ−u,η−v)>R2(2.17)
R1(x,y) is a white noise uniformly distributed in [0,1], thus it can be written as
<exp[i2π{R1(p,q)−R1(u,v)}]>R1=δu−pδv−q(2.18)
Using Property 1 and equation (2.18),
<ψ*(x,y)ψ(ξ,η)>=∑u=0N−1∑v=0N−1∑p=0N−1∑q=0N−1f*(p,q)f(u,v)×δu−pδv−q×1N2δx−ξδy−η=∑u=0N−1∑v=0N−1f*(p,q)f(u,v)1N2δx−ξδy−η=1N2∑u=0N−1∑v=0N−1∣f(u,v)∣2δx−ξδy−η(2.19)
Property 2 helps establish the fact that although input plane RPM is not required for the retrieval of original information, the use of both the RPMs is important in converting the image to be encrypted into a white stationary noise. The use of input plane RPM makes the input image white but nonstationary and not encoded. Fourier plane RPM maintains whiteness and stationarizes and encodes the input image [1–3].
The publication of the pioneering ‘double random phase encoding’ attracted the attention of the research community and since then a large number of research papers have appeared in literature. This scheme has been implemented in different transform domains including fractional Fourier transform (FRT), Fresnel transform (FrT), gyrator transform (GT), wavelet transform (WT), and fractional Mellin transform. Also, other techniques such as asymmetric cryptosystems, phase-only encryption, multiple image encryption and color image encryption, have been reported. The brief introduction of some of the optical image encryption techniques in various transforms has been discussed in the following subsections.
2.2.2 Encryption using fractional Fourier transform
The fractional Fourier transform (FRT) is a generalization of the ordinary Fourier transform with an order parameter α. A Fourier transform is a first order FRT with α = 1. The FRT operator follows the mathematical properties such as linearity, continuity, self-imaging, partial convolution/correlation. The properties and applications of the ordinary Fourier transform are special cases of those of the FRT [12–16]. In every area where Fourier transform and frequency domain concepts are used, the potential exists for generalizations and improvement by using the fractional transform. Efficient algorithms exist in literature to compute FRTs in about the same time as discrete Fourier transform [13–15]. Optical implementations based on bulk systems can be performed using either a single lens system or with a two lens system [12]. Therefore, the generalization of Fourier transform to the FRT comes at no additional cost whether computed digitally or implemented optically. If FRT of order α is denoted by ℑα, then its inverse is denoted as ℑ−α. It has found many applications in optical and digital signal and image processing where the ordinary Fourier transform has traditionally played an important role. The FRT of a function f(x1,y1) is defined as [16]
g(x2,y2)=∬f(x1,y1)Bp(x2,y2;x1,y1)dx1dy1(2.20)
where Bp(·) is the kernel of the 2D FRT given by:
Bp(x2,y2;x1,y1)=Kexpiπx22+x12+y22+y12tanα−2iπx1x2y1y2sinα(2.21)
Here, α=pπ/2 where p denotes order of the FRT and
K=exp−j14πsgn(sinα)−α2∣sinα∣1/2(2.22)
where K represents a complex constant and sgn refers to the signum function. For optical implementation of FRT, two geometries have been proposed; through a two lens system and a single lens system [12]. Figure 2.3 shows the schematic for obtaining FRT through a single lens system. In this case, the input function could be placed at any location within the focal length of the lens (d < f). The distance to input and output planes from the lens are the same but less than the focal length. The symbol ‘f’ denotes the focal length of the lens. The optical implementation of FRT with a single lens corresponds to free-space propagation (shearing of Wigner distribution function in the x-direction), passage through a lens (shearing in the y-direction), and again free-space propagation (shearing in the x-direction). The distance parameter, d = f1 × tan(α/2), f1 = f × tan(α/2), where f is the focal length of the lens.
Figure 2.3. Schematic diagram for optical implementation of FRT with a single lens system.
The ciphertext generated by using DRPE in the FRT domain is written as [17–20]
E(x,y)=ℑβℑαf(x,y)×exp(i2πR1(x,y))×exp(i2πR2(u,v))(2.23)
Here, values of α and β are important for the retrieval of original information in addition to the use of respective RPMs. For decryption, the usual reverse process of encryption, as explained in section 2.2.1, is to be followed. Because the order of the FRT can take