Asymmetric multiple-image encryption based on the cascaded fractional Fourier transform

A multiple-image cryptosystem is proposed based on the cascaded fractional Fourier transform. During an encryption procedure, each of the original images is directly separated into two phase masks. A portion of the masks is subsequently modulated into an interim mask, which is encrypted into the ciphertext image; the others are used as the encryption keys. Using phase truncation in the fractional Fourier domain, one can use an asymmetric cryptosystem to produce a real-valued noise-like ciphertext, while a legal user can reconstruct all of the original images using a different group of phase masks. The encryption key is an indivisible part of the corresponding original image and is still useful during decryption. The proposed system has high resistance to various potential attacks, including the chosen-plaintext attack. Numerical simulations also demonstrate the security and feasibility of the proposed scheme.     

Double-image self-encoding and hiding based on phase-truncated Fourier transforms and phase retrieval

We propose a method to encrypt two covert images into an overt image based on phase-truncated Fourier transforms and phase retrieval. In this method, the two original images are self-encoded in the manner that one of the two images is directly separated into two phase masks (PMs) and used as keys for encryption, and then multiplied by a PM which is generated by using phase retrieval algorithm. At last, the whole encryption process is completed by a Fourier transform operation. In the decryption process, the image without a separation and the two PMs used as keys for encryption are all treated as encoded data. The cryptosystem is asymmetric which means the keys for encryption are different from those for decryption. Numerical simulations are presented to show the viability and good performance of the proposed method.     

Multiple-image encryption based on nonlinear amplitude-truncation and phase-truncation in Fourier domain

The classical double random phase encoding technique (DRPE) is vulnerable to chosen ciphertext attacks, known-plaintext attacks and chosen-plaintext attacks for its linearity. In order to avoid the disadvantages originated from the linearity and symmetric, an improved method for multiple-image encryption based on nonlinear operations in Fourier domain is proposed. The random phase masks (RPMs) for encryption and additive keys which are determined by the original images and generated by the nonlinear operations in encryption process, are necessary for image decoding. As a result of the nonlinear operations, the increase in the number of keys, removal of linearity and high robustness could be achieved in this cryptosystem. Computer simulations are presented to demonstrate its good performance, and the security is analyzed as well.     

Image recovery from double amplitudes in fractional Fourier domain

The classical Gerchberg--Saxton algorithm is introduced into the image recovery in fractional Fourier domain after adaptation. When this algorithm is applied directly, its performance is good for smoothed image, but bad for unsmoothed image. Based on the diversity of fractional Fourier transform on its orders, this paper suggests a novel iterative algorithm, which extracts the information of the original image from amplitudes of its fractional Fourier transform at two orders. This new algorithm consists of two independent Gerchberg--Saxton procedures and an averaging operation in each circle. Numerical simulations are carried out to show its validity for both smoothed and unsmoothed images with most pairs of orders in the interval [0, 1].     

分数傅里叶域中二维复图像相位恢复的混合输入输出算法

通过推广经典傅里叶域内的混合输入输出算法,研究分数傅里叶域内二维复图像的相位恢复问题。给出算法的具体实施过程,并进行了数值模拟。结果表明：对于级次在0与2之间的分数傅里叶变换,当级次在1/2与3/2之间时,该算法较好;对于其余的级次,该算法的效果不能令人满意。     

Image encryption based on extended fractional Fourier transform and digital holography technique

We present a new optical image encryption algorithm that is based on extended fractional Fourier transform (FRT) and digital holography technique. We can perform the encryption and decryption with more parameters compared with earlier similar methods in FRT domain. In the extended FRT encryption system, the input data to be encrypted is extended fractional Fourier transformed two times and random phase mask is placed at the output plane of the first extended FRT. By use of an interference with a wave from another random phase mask, the encrypted data is stored as a digital hologram. The data retrieval is operated by all-digital means. Computer simulations are presented to verify its validity and efficiency.     

Image encryption using fractional Mellin transform,structured phase filters,and phase retrieval

An image encryption scheme has been presented by using two structured phase masks in the fractional Mellin transform (FrMT) plane of a system, employing a phase retrieval technique. Since FrMT is a non-linear integral transform, its use enhances the system security. We also add further security features by carrying out spatial filtering in the frequency domain by using a combination of two phase masks: a toroidal zone plate (TZP) and a radial Hilbert mask (RHM). These masks together increase the key space making the system more secure. The phase key used in decryption has been obtained by applying an iterative phase retrieval algorithm based on the fractional Fourier transform. The algorithm uses amplitude constraints of secret target image and the ciphertext (encrypted image) obtained from multiplication of fractional Mellin transformed arbitrary input image and the two phase masks (TZP and RHM). The proposed encryption scheme has been validated for a few grayscale images, by numerical simulations. The efficacy of the scheme has been evaluated by computing mean-squared-error (MSE) between the secret target image and the decrypted image. The sensitivity analysis of the decryption process to variations in various encryption parameters has also been carried out.     

Displacement measurement based on joint fractional Fourier transform correlator

A displacement measurement technology based on joint fractional Fourier transform is firstly proposed. Contrast to conventional displacement measurement based on joint Fourier transform correlator, the position of cross correlation peak in the proposed technology could be fixed arbitrarily according to the order of fractional Fourier transform. The optical setup in the proposed technology is more flexible and easier to implement. Simulation and experiment results are given out to verify the analysis.     

Image watermarking by using phase retrieval algorithm in gyrator transform domain

We propose an image watermarking scheme based on the phase retrieval algorithm in gyrator domain. The watermark is converted into a noise-like image by Arnold transform. The scrambled image is regarded as the amplitude of gyrator spectrum. The Gerchberg-Saxton algorithm is employed to obtain the unknown phase function in gyrator pair, in which the host image is the amplitude of input function. The phase information and the parameters of the two transforms serve as the key of watermarking algorithm. The numerical simulation has demonstrated the performance of the proposed algorithm.     

Single phase encoding method based on the fractional Fourier transform

We propose a single phase encoding scheme for encrypting image by using fractional Fourier transform. Single phase mask is designed in order to be symmetrical about certain direction, which can be used in the process of both encryption and decryption. A conjugate mask is not required in the image decryption process, which is very convenient for the practical application in optics. Moreover, the optical implementation of the image encryption and decryption is given. The implementing structure is composed of lens and spherical mirror. Numerical simulations have demonstrated the validity and security of the encryption algorithm.     

Optical image encryption based on multichannel fractional Fourier transform and double random phase encoding technique

The optical image encryption based on multichannel fractional Fourier transform (FRT) and double random phase encoding technique is proposed. Optical principles of encoding and decoding are analyzed in detail. With this method, one can encrypt different parts of input image, respectively. The system security can be improved to some extent, not only because fractional orders and random phase masks in every channel can be set with freedom, but also because the system parameters among all channels are independent. Numerical simulation results of optical image encryption based on four channel FRT and double random phase encoding are given to verify the feasibility of the method.     

An optical waveform pre-distortion method based on time domain fractional Fourier transformation

An optical waveform pre-distortion method is proposed in reducing the temporal broadening of pulse based on the optical implementation of time domain fractional Fourier transform (FRFT). Moreover, a new analysis of the dispersion and self-phase modulation effects is investigated. The simulation results show that prechirp coefficient and proper fractional orders of FRFT can be beneficial in mitigating pulse broadening.     

Novel optical image encryption scheme based on fractional Mellin transform

A novel nonlinear image encryption scheme is proposed by introducing the fractional Mellin transform (FrMT) into the field of image security. As a nonlinear transform, FrMT is employed to get rid of the potential insecurity of the optical image encryption system caused by the intrinsic object-image relationship between the plaintext and the ciphertext. Different annular domains of the original image are transformed by FrMTs of different orders, and then the outputs are further encrypted by comprehensively using fractional Fourier transform (FrFT), amplitude encoding and phase encoding. The keys of the encryption algorithm include the orders of the FrMTs, the radii of the FrMT domains, the order of the FrFT and the phases generated in the further encryption process, thus the key space is extremely large. An optoelectronic hybrid structure for the proposed scheme is also introduced. Numerical simulations demonstrate that the proposed algorithm is robust with noise immunity, sensitive to the keys, and outperforms the conventional linear encryption methods to counteract some attacks.     

Parameter estimation of optical fringes with quadratic phase using the fractional Fourier transform

Optical fringes with a quadratic phase are often encountered in optical metrology. Parameter estimation of such fringes plays an important role in interferometric measurements. A novel method is proposed for accurate and direct parameter estimation using the fractional Fourier transform (FRFT), even in the presence of noise and obstacles. We take Newton׳s rings fringe patterns and electronic speckle pattern interferometry (ESPI) interferograms as classic examples of optical fringes that have a quadratic phase and present simulation and experimental results demonstrating the performance of the proposed method.     

Spectrum measurement in the fractional Fourier domain

Fractional Fourier transform (FRFT) plays an important role in many fields of optics and signal processing. This paper considers the problem of real-time measurement of the spectrum of a signal in the FRFT domain. In this paper, we propose two approaches for real-time measurement of the FRFT of a signal based on modulation and bandpass filtering systems. The relation is established between the linear frequency modulation (LFM or chirp) spectrum and the FRFT of its envelope. In addition, two applications for spectrum measurement are presented in the FRFT domain. The LFM signal can be bandlimited in the Fourier transform (FT) domain through spectrum measurement associated with bandpass filtering method. The results can also be useful in the problems related to swept-frequency filter for measurement in the FRFT domain.     

Fully phase-encrypted memory using cascaded extended fractional Fourier transform

In this paper, we implement a fully phase-encrypted memory system using cascaded extended fractional Fourier transform (FRT). We encrypt and decrypt a two-dimensional image obtained from an amplitude image. The full phase image to be encrypted is fractional Fourier transformed three times and random phase masks are placed in the two intermediate planes. Performing the FRT three times increases the key size, at an added complexity of one more lens. The encrypted image is holographically recorded in a photorefractive crystal and is then decrypted by generating through phase conjugation, the conjugate of the encrypted image. A lithium niobate crystal has been used as a phase contrast filter to reconstruct the decrypted phase image, alleviating the need of alignment in the Fourier plane making the system rugged.     

Secret shared multiple-image encryption based on row scanning compressive ghost imaging and phase retrieval in the Fresnel domain

A multiple-image encryption method is proposed that is based on row scanning compressive ghost imaging, (t, n) threshold secret sharing, and phase retrieval in the Fresnel domain. In the encryption process, after wavelet transform and Arnold transform of the target image, the ciphertext matrix can be first detected using a bucket detector. Based on a (t, n) threshold secret sharing algorithm, the measurement key used in the row scanning compressive ghost imaging can be decomposed and shared into two pairs of sub-keys, which are then reconstructed using two phase-only mask (POM) keys with fixed pixel values, placed in the input plane and transform plane 2 of the phase retrieval scheme, respectively; and the other POM key in the transform plane 1 can be generated and updated by the iterative encoding of each plaintext image. In each iteration, the target image acts as the input amplitude constraint in the input plane. During decryption, each plaintext image possessing all the correct keys can be successfully decrypted by measurement key regeneration, compression algorithm reconstruction, inverse wavelet transformation, and Fresnel transformation. Theoretical analysis and numerical simulations both verify the feasibility of the proposed method.     

Improved acoustic signals discrimination using fractional Fourier transform based phase-space representations

In this communication we propose performing two-dimensional correlation operation between phase-space representations based on the fractional Fourier transform, instead of correlating the signals themselves. A numerical examples clearly indicates superior discrimination performance.     

Position multiplexing multiple-image encryption using cascaded phase-only masks in Fresnel transform domain

A position multiplexing method based on the modified Gerchberg-Saxton algorithm (MGSA) and a cascaded phase modulation scheme in the Fresnel transform domain is proposed in the multiple-image-encryption framework. First of all, each plain image is encoded into a complex function using the MGSA. The phase components of the created complex functions are then multiplexed with different position parameters, and summed. The phase part of the summation result is recorded in the first phase-only mask (POM). The MGSA is applied on the amplitude part of the summation result to determine another phase only function which is then recorded in the second POM. The simulation results show that the crosstalk between multiplexed images is significantly reduced compared with an existing similar method [20]. Therefore, the multiplexing capacity in encrypting multiple grayscale images can be increased accordingly.     

Asymmetric multiple-image encryption based on coupled logistic maps in fractional Fourier transform domain

A multiple-image encryption scheme is proposed based on the asymmetric technique, in which the encryption keys are not identical to the decryption ones. First, each plain image is scrambled based on a sequence of chaotic pairs generated with a system of two symmetrically coupled identical logistic maps. Then, the phase-only function of each scrambled image is retrieved with an iterative phase retrieval process in the fractional Fourier transform domain. Second, all phase-only functions are modulated into an interim, which is encrypted into the ciphertext with stationary white noise distribution by using the fractional Fourier transform and chaotic diffusion. In the encryption process, three random phase functions are used as encryption keys to retrieve the phase-only functions of plain images. Simultaneously, three decryption keys are generated in the encryption process, which make the proposed encryption scheme has high security against various attacks, such as chosen plaintext attack. The peak signal-to-noise is used to evaluate the quality of the decrypted image, which shows that the encryption capacity of the proposed scheme is enhanced considerably. Numerical simulations demonstrate the validity and efficiency of the proposed method.