Читать книгу Optical Cryptosystems - Naveen K. Nishchal - Страница 26

The discrete cosine transform (DCT) expresses a finite sequence of data points in terms of cosine functions at different frequencies. Since fewer cosine functions are required to approximate a typical signal, DCT finds application in data/image compression. This property attracted the use of DCT and discrete fractional cosine transform in image encryption applications. The 2D DCT of an image f(x,y) of size M × N is defined as [35]

B(ξ,η)=2LKγ(ξ)γ(η)∑u=0L−1∑v=0K−1A(u,v)cos(2u+1)ξπ2Lcos(2v+1)ηπ2K(2.34)

The corresponding inverse DCT is expressed as

A(u,v)=2LK∑ξ=0L−1∑η=0K−1γ(ξ)γ(η)B(ξ,η)cos(2u+1)ξπ2Lcos(2v+1)ηπ2K(2.35)

γ(ξ)=12forξ=01forξ=1,2,…,L−1,γ(η)=12forη=01forη=1,2,…,K−1(2.36)

The discrete fractional cosine transform has a similar relationship with the discrete FRT. In this case, fractional order is an added property to the encryption scheme.