Читать книгу Optical Cryptosystems - Naveen K. Nishchal - Страница 21

It is important to note that the modulus of {f(x,y) × exp[i2πR₁(x,y)]} is same as the modulus of f(x,y). Therefore, the image is not encrypted in this case, although the RPM bonded input function {f(x,y) × exp[i2πR₁(x,y)]} is a white noise [1]. This is demonstrated by evaluating the ensemble average of this input function on the random function R₁(x,y):

<f(x,y)exp[i2πR1(x,y)]f(u,v)exp[−i2πR1(u,v)]>=f(x,y)f(u,v)δx−uδy−v(2.7)

since <exp[i2π[R1(x,y)−R1(u,v)]]>R1=δx−uδy−v where δx−u is the Kronecker symbol. The symbol ‘<>’ denotes the ensemble average. This white noise is nonstationary. If f(x,y) is filtered with a phase-only filter of transfer function exp[i2πR₂(u,v)] and impulse response h(x,y), then the obtained encrypted image is easy to decode.

In order to study the statistical properties of the encryption procedure, it is important to analyze the statistical property of the impulse response of a phase-only transfer function with a white noise. The following two properties are discussed.

Property 1:

If h(x,y) is the impulse response of a phase-only transfer function defined by H(υ,μ)=exp[i2πR2(υ,μ)] where R2(υ,μ) is a white noise uniformly distributed in [0,1], then, for all x,y,u,v,ξ,η:

〈h*(x−ξ,y−η)h(p−ξ,q−η)〉b=1N2δx−pδy−q(2.8)

where * denotes the complex conjugate and

δx−p1ifx−p=00otherwiseδy−q1ify−q=00otherwise

Proof:

For the proof of the property, the definition of Fourier transform of any function h(x, y) can be used [3] and hence it is written as

h(x,y)=1N2∑υ=0N−1∑μ=0N−1exp[i2πR2(υ,μ)]exp[i2π(υx+μy)](2.9)

Thus the correlation of h(x,y) is

〈h*(x−ξ,y−η)h(p−ξ,q−η)〉b=1N4∑υ=0N−1∑υ′=0N−1∑μ=0N−1∑μ′=0N−1×<exp[i2π{R2(υ′,μ′)−R2(υ,μ)}]>R2×exp[i2π{υ′(p−ξ)+μ′(q−η)}−{υ(x−ξ)+μ(y−η)}](2.10)

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.