A discrete fractional angular transform

A new discrete fractional transform defined by two parameters (angle and fractional order) is presented. All eigenvectors of the transform are obtained by an angle using recursion method. This transform is named as discrete fractional angular transform (DFAT). The computational load of kernel matrix of the DFAT is minimum than all other transforms with fractional order. This characteristics has very important practical applications in signal and image processing. Numerical results and the mathematical properties of this transform are also given. As fractional Fourier transform, this transform can be applied in one and two dimensional signal processing.     

Eliminating the zero spectrum in Fourier transform profilometry using a two-dimensional continuous wavelet transform

In this paper, a filtering technique based upon two-dimensional continuous wavelet transform (2D-CWT) is used to eliminate the low frequency components of fringe patterns. The filtered fringe patterns are subsequently demodulated using a standard Fourier transform profilometry (FTP) algorithm. This image pre-filtering stage improves the noise performance of the FTP algorithm and enables the FTP method to demodulate fringe patterns with larger bandwidths. Also, the 2D-CWT technique reduces speckle noise significantly. Moreover, only a single fringe pattern is required in this technique. The 2D-CWT algorithm is capable of separating low frequency terms from the high frequency terms that contain phase-modulated fringe information, even when both interfere, greatly, in the frequency domain. The proposed algorithm is tested, both via computer simulation and using real fringe patterns. This revealed the robustness of this algorithm and also demonstrably enables the demodulation of a wider range of fringe patterns using the FTP technique.     

Improving the performance of DCT-based fragile watermarking using intelligent optimization algorithms

The performance of a fragile watermarking method based on discrete cosine transform (DCT) has been improved in this paper by using intelligent optimization algorithms (IOA), namely genetic algorithm, differential evolution algorithm, clonal selection algorithm and particle swarm optimization algorithm. In DCT based fragile watermarking techniques, watermark embedding can usually be achieved by modifying the least significant bits of the transformation coefficients. After the embedding process is completed, transforming the modified coefficients from the frequency domain to the spatial domain produces some rounding errors due to the conversion of real numbers to integers. The rounding errors caused by this transformation process were corrected by the use of intelligent optimization algorithms mentioned above. This paper gives experimental results which show the feasibility of using these optimization algorithms for the fragile watermarking and demonstrate the accuracy of these methods. The performance comparison of the algorithms was also realized.     

A new kind of double image encryption by using a cutting spectrum in the 1-D fractional Fourier transform domains

Based on 1-D fractional Fourier transform, we proposed an image encryption algorithm in order to hide two images simultaneously. When the fractional order is closed to 1, most energy in frequency domain is centralized in the center part of spectrum. The image can be recovered acceptable by using a half of spectrum, which locates in the middle part at x-direction or y-direction. Cutting operation is employed in order to combine two spectra. Double random phase encoding is employed for image encryption. The corresponding numerical simulations are performed to demonstrate the validity and efficiency of the algorithm.     

Double image encryption based on iterative fractional Fourier transform

We present an image encryption algorithm to simultaneously encrypt two images into a single one as the amplitudes of fractional Fourier transform with different orders. From the encrypted image we can get two original images independently by fractional Fourier transforms with two different fractional orders. This algorithm can be independent of additional random phases as the encryption/decryption keys. Numerical results are given to analyze the capability of this proposed method. A possible extension to multi-image encryption with a fractional order multiplexing scheme has also been given.     

Instantaneous velocity measurement of dynamic deformation by digital holographic interferometry

A novel method for the instantaneous velocity measurement of dynamic deformation by digital holographic interferometry is proposed. During dynamic deformation, a series of digital holograms is recorded by a high-speed camera. At each pixel of the phase difference maps, phase and amplitude information are combined as complex phasor (CP). Each pixel can be then considered as an independent sensor and a sequence of complex phasors of such a sensor is analyzed by short time Fourier transform (STFT) along the time axis. A fast iterative algorithm is developed for the computation of instantaneous velocity. The displacement of each pixel can also be obtained by integration of the instantaneous velocity over time and phase unwrapping process is thus avoided. The performance of the proposed CP method is compared experimentally with the commonly used digital phase subtraction method.     

Determination of displacement derivative in digital holographic interferometry

Digital holography (DH) and digital shearography (DS) both play an important role in non-destructive evaluation. In this paper, a novel method based on digital holographic interferometry (DHI) and complex phasor (CP) is proposed to determine displacement derivative. An algorithm is employed to filter the imaginary and real parts of complex values without the need of direct phase manipulation. Two-dimensional short time Fourier transform (STFT) is employed subsequently to process wrapped phase maps. An experiment is conducted to demonstrate the validity of the proposed method.     

Methods of Fourier optics in digital holographic microscopy

In this paper, processing methods of Fourier optics implemented in a digital holographic microscopy system are presented. The proposed methodology is based on the possibility of the digital holography in carrying out the whole reconstruction of the recorded wave front and consequently, the determination of the phase and intensity distribution in any arbitrary plane located between the object and the recording plane. In this way, in digital holographic microscopy the field produced by the objective lens can be reconstructed along its propagation, allowing the reconstruction of the back focal plane of the lens, so that the complex amplitudes of the Fraunhofer diffraction, or equivalently the Fourier transform, of the light distribution across the object can be known. The manipulation of Fourier transform plane makes possible the design of digital methods of optical processing and image analysis. The proposed method has a great practical utility and represents a powerful tool in image analysis and data processing. The theoretical aspects of the method are presented, and its validity has been demonstrated using computer generated holograms and images simulations of microscopic objects.     

Necessity analyses of phase masks in joint fractional Fourier transform correlator

In this paper, the mathematical expressions and the particularities of the joint fractional Fourier transform correlator (JFRTC) are detailedly analyzed. In a JFRTC, the distance between the two fractional correlation peaks can be adjusted flexibly by shifting the fractional orders. However, the intensity of the fractional correlation peaks decreases rapidly as the first fractional order p₁ drops. The former property is very useful in a practical recognition system, but the latter one seriously reduces the recognition sensitivity of a JFRTC. For this reason, phase masks should be added at the input plane of the JFRTC to enhance the intensities of the fractional correlation peaks and effectively overcome the default. Compared with the existing JFRTC with single phase modulation, a JFRTC with double phase modulation is suggested to more flexibly improve the performance of a JFRTC.     

Wavelet and model-based spectral analysis of color doppler optical coherence tomography

Color doppler optical coherence tomography (CD-OCT) uses time-frequency analysis (TFA) to extract motion-induced Doppler shifted in the interferometric OCT signal. In this paper, the performance of three TFAs are compared in a scattering flow phantom and in in vivo human retina: the short-time Fourier transform, the Morlet-wavelet transform, and the short-time MUSIC transform (STMT). The STMT is a new TFA that incorporates the MUSIC eigenfrequency estimator in a generalized short-time framework. The Morlet transform excels at identifying blood vessels, while the STMT is the most accurate predictor of Doppler shift frequency.     

Optical color image encryption with redefined fractional Hartley transform

We propose a new method for color image encryption by wavelength multiplexing on the basis of two-dimensional (2-D) generalization of 1-D fractional Hartley transform that has been redefined recently in search of its inverse transform. A color image can be considered as three monochromatic images and then divided into three components and each component is encrypted independently with different wavelength corresponding to red, green or blue light. The system parameters of fractional Hartley transform and random phase masks are keys in the color image encryption and decryption. Only when all of these keys are correct, can the image be well decrypted. The optical realization is then proposed and computer simulations are also performed to confirm the possibility of the proposed method.     

Double-image encryption based on iterative gyrator transform

A novel double-image encryption algorithm is proposed, which can simultaneously encrypt two images into a single one as the amplitude of gyrator transform with two different groups of angles. The two original images can be retrieved independently by gyrator transforms with two different groups of angles, one common phase mask, and two different private phase masks. The proposed approach can enlarge the key space, achieve faster convergence in iterative process, and avoid cross-talk between two images in reconstruction. Numerical simulations are presented to verify its validity and efficiency.     

Fractional Fourier transforms of elliptical Hermite-cosh-Gaussian beams

A new kind of laser beams named the elliptical Hermite-sinusoidal-Gaussian beams (EHSGBs) is introduced and defined by use of tensor method, and the elliptical Hermite-cosh-Gaussian beam (EHChGB) can be regarded as special case of EHSGBs. The fractional Fourier transform (FRFT) is applied to treat the propagation of EHChGBs. An analytical expression for an EHChGB passing through an FRFT system is derived by using vector integration. Some numerical simulations are illustrated for the propagation properties of EHChGBs through FRFT systems.     

Application of Cosine Zone Plates to Image Encryption

We analyse the diffraction result of optical field after Cosine zone plate, and theoretically deduce its transform matrix. Under some conditions, its diffraction distribution is a mixture of fractional Fourier spectra. Then we use Cosine zone plate and its diffraction result to image encryption. Possible optical image encryption and decryption implementations are proposed, and some numerical simulation results are also provided.     

Phase-Space Analysis of Wavefront Coding Imaging Systems

We explore the use of the Radon-Wigner transform, which is associated with the fractional Fourier transform of the pupil function, for determining the point spread function （PSF） of an incoherent defocused optical system. Then we introduce these phase-space tools to analyse the wavefront coding imaging system. It is shown that the shape of the PSF for such a system is highly invariant to the defocus-related aberrations except for a lateral shift. The optical transfer function of this system is also investigated briefly from a new understanding of ambiguity function.     

Fractional Gabor transform

A fractional Gabor transform (FRGT) is proposed. This new transform is a generalization of the conventional Gabor transform (GT) based on the Fourier transform to the windowed fractional Fourier transform (FRFT). The FRGT provides analyses of signals in both the real space and the FRFT frequency domain simultaneously. The space-FRFT frequency pattern can be rotated as the fractional order changes. The FRGT has an additional freedom, compared with the conventional GT, i.e., the transform order. The FRGT may offer a useful tool for guiding optimal filter design in the FRFT domain in signal processing.     

Triple image encryption scheme in fractional Fourier transform domains

We proposed a triple image encryption scheme by use of fractional Fourier transform. In this algorithm, an original image is encoded in amplitude part and other two images are encoded into phase information. The key of encryption algorithm is obtained from the difference between the third image and the output phase of transform. In general case, random phase encoding technology is not required in the proposed algorithm. Moreover, all information of images is preserved in theory when image are decrypted with correct key. The optical implementation of the algorithm is presented with an electro-optical hybrid structure. Numerical simulations have demonstrated the efficiency and the security of this algorithm. Based on this scheme a multiple image algorithm is expanded and designed.     

Height distribution on a rough plane and specularly diffracted light amplitude are Fourier transform pair

In this report we show that the amplitude of specularly diffracted light from a plane rough surface as a function of incident angle cosine is Fourier transform of the height distribution on the surface. Therefore, an even height distribution function, which is the case for many rough surfaces, can be obtained by measuring the specularly diffracted light intensities. Also, it is observed that for polychromatic illumination the spectrum of the specularly diffracted light is modified and the modification depends on roughness, incident angle, and wavelength. It is also shown that, for a fixed incident angle, the height distribution on the rough plane is Fourier transform of the spectral modifying function. Experimental studies on some surfaces of different roughnesses, prepared by grinding sheet-glasses by powders of different grain sizes, show that the corresponding height distributions are Gaussian and the rms heights obtained by the two approaches are quite consistent.     

Color image encoding in dual fractional Fourier-wavelet domain with random phases

A new cryptology in dual fractional Fourier-wavelet domain is proposed in this paper, which is calculated by discrete fractional Fourier transform and wavelet decomposition. Different random phases are used in different wavelet subbands in encryption. A new color image encoding method is also presented with basic color decomposition and encryption respectively. All the keys, including random phases and fractional orders in R, G and B three channels, should be correctly used in decryption, otherwise people cannot obtain the totally correct information. Some numerical simulations are presented to demonstrate the possibility of the method. It would have widely potential applications in digital color image processing and protection.     

WAVEWAT-improved solvent suppression in NMR spectra employing wavelet transforms

WAVEWAT is a new processing algorithm to suppress the on-resonance water signal in NMR spectra. It is based on a multiresolution analysis (MRA) of the free induction decay (FID) using a dyadic discrete wavelet transform (DWT). The width of the suppressed signal can be adjusted so that signals close to water are recovered without distortion of the signal shape and intensity. Computational efficiency is comparable to that of convolution filters employing a Fourier transform.