Ghost imaging is closely related to image encryption, since the random speckle patterns are often exploited. In the two-dimensional case, computational ghost imaging can be realized through LXR ^T= Y , where X is a two-dimensional object, Y is the bucket detection signals reshaped into two-dimensional form, and L and R are two random matrices. In this paper, a scenario of flexible image encryption in ghost imaging in two-dimensional case is proposed. The image is encrypted into the bucket detection signals, by arbitrarily arranging the two random matrices ( L and R ) and other two permutation matrices ( P ₁ and P ₂). The permutation matrices are used to disrupt the distribution of the bucket signals. Considering the specific size of the image, that the image may not be square but rectangular, eight ways of image encryption are investigated. Four of them use only one permutation matrix ( P ₁ or P ₂), and they are P ₁ LXR ^T= Y ₁, LP ₁ XR ^T= Y ₂, LXP ₂ R ^T= Y ₃, LXR ^T P ₂= Y 4 . The other four use two permutation matrices ( P ₁ and P ₂), and they are P ₁ LXP ₂ R ^T= Y ₅, P ₁ LXR ^T P ₂= Y ₆, LP ₁ XP ₂ R T= Y ₇, LP ₁ XR ^T P ₂= Y ₈.
In experiment, specifically, the measurement matrix is generated by the Kronecker product of the random matrices and permutation matrices. According to the 8 ways of image encryption, the 8 measurement matrices are A ₁=( P ₁ L )⊗R, A ₂=( LP ₁)⊗R, A ₃= L ⊗( RP ₂^T), A ₄= L ⊗( P ₂^T R ), A ₅=( P ₁ L )⊗( RP ₂^T), A ₆=( P ₁ L )⊗( P ₂^T R ), A ₇=( LP ₁)⊗( RP ₂^T), and A ₈=( LP ₁)⊗( P ₂^T R ), respectively. These measurement matrices are used to form the random speckle patterns which are then projected onto the object. A spatial light modulator (SLM) is employed to play the object and random speckle patterns. A charge coupled device (CCD) is used to obtain the bucket detection signals.
As truncated singular value decomposition (TSVD) is an effective way to denoise, it is performed to obtain the pseudoinverse matrices of the random matrices which are used in the decryption process. Only when the pseudoinverse matrices of the random matrices, as well as the correct sequences of the random and permutation matrices, are known in each way, can the image be successfully decrypted. Otherwise, image decryption will not be successful. The structural similarity (SSIM), peak signal-to-noise ratio (PSNR), and correlation coefficient (CC) are used to evaluate the quality of the decrypted images. The SSIMs of object and the two-dimensional bucket detection signals are very low, indicating the successful encryption. The PSNRs and CCs of the successful decrypted images are better than the unsuccessful ones. The successfully decrypted images clearly reconstruct the image of the object, while the unsuccessful images are in a mess.
Our method provides a new idea of image encryption in ghost imaging, and image encryption is therefore enhanced and made flexible. Moreover, the present protocol can be combined with other image encryption techniques, to form more flexible protocols of image encryption, and may also have some application prospects in other image processing such as watermarking, image hiding, and so on.