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Modulation measurement profilometry applies coaxial and coimage plane optical systems to reconstruct the surface shape of the measured object, in which, the direction of the grating projection is the same as that of CCD to capture the images. This technology completes the reconstruction by the extraction of modulation values instead of phase information. It can effectively avoid the shadows, shutoff and calculation of phase unwrapping and accomplish the shape measurement for object with complex surface. However, in the practical application, the measurement precision is usually affected by the nonlinearity of CCD’s photoelectric response. In order to avoid the influence of high-order harmonics on fundamental spectrum, Wavelet Transform is introduced into fringe analysis and modulation extraction. With the merits of local analysis and multi-resolution, this method can obtain higher measurement precision by comparing with the Fourier transform method. Both computer simulation and practical experiment testified the validity of the proposed method. [ABSTRACT FROM AUTHOR]

This paper presents an information authentication scheme using sparse representation of double random phase encoding in fractional Fourier transform domain. Different from traditional cryptographic applications, only sparse version of the ciphertext is available at the receiver end. The decrypted image is unrecognizable and of no meaningful information of the plaintext, which can enhance the resistance against various attacks. The decrypted image can be authenticated using nonlinear optical correlation approach. Numerical simulations have been carried out, and the results prove the effectiveness and noise resistance of the proposed scheme. [ABSTRACT FROM AUTHOR]

A compact, static Fourier transform imaging spectropolarimeter based on quarter-wave plate array is conceptually described. It improves a Savart interferometer by replacing front polarizer with a Wollaston prism and an aperture divided by a quarter-wave plate array, and can simultaneously acquire six interferograms corresponding to six linearly polarized lights on a single CCD. The spectral dependence of all Stokes parameters can be recovered by Fourier transform. Since there is no rotating or moving part, the system is relatively robust. The interference model is proved. The performance of the system is demonstrated through a numerical simulation. The orientation of quarter-wave plate array is optimized. Alignment errors of the elements respect to the ideal position are discussed and a procedure for compensating the alignment errors is also developed. [ABSTRACT FROM AUTHOR]

Based on interferogram frequency domain self-filtering methods, a simple, robust and effective carrier-removal approach for carrier interferogram analysis is proposed. The spatial carrier interferogram is firstly extrapolated to increase the frequency resolution, and a synthetic referencing interferogram distilled from the extrapolation interferogram with frequency domain self-filtering method. The carrier phase component is removed by subtracting the carrier phase extracted from the synthetic referencing interferogram with fast Fourier transform (FFT) method. Compared with existing carrier removal methods, numerical simulations and experiments, the proposed method is effective and accurate for suppressing the carrier-removal error caused by the digitization of the interferogram in the Fourier transform method (FTM) for carrier interferogram analysis. [ABSTRACT FROM AUTHOR]

The convolution theorem for the linear canonical transform (LCT) is of importance in signal processing theory and application. Recently, some attempts at extending the convolution theorem in the Fourier transform (FT) domain to the LCT domain have derived many important results, which are very useful and effective in filter design and signal reconstruction, but none of them generalize very nicely and simply the classical result for the FT. In this paper, we formulate a new kind of convolution structure for the LCT, which has the elegance and simplicity in both time and LCT domains comparable to that of the FT and preserves the commutative and associative properties. Then with the new convolution theorem, it is easy to implement in the designing of multiplicative filters through both the new convolution in the time domain and the product in the LCT domain, and it is convenient to deduce the Shannon-type reconstruction formula for bandlimited signals in the LCT domain. Theoretical analyses and numerical simulations are also presented to show the correctness and effectiveness of the proposed techniques. [ABSTRACT FROM AUTHOR]

Fractional Fourier Transform (FRFT) is regarded as an effective method for detection of Linear Frequency Modulation (LFM) signals in recent years. However, the performance of FRFT detection will deteriorate sharply in the weak noise scene. An alternative solution is the use of bistable system, which can generate stochastic resonance (SR), and is relatively easy to achieve. This paper proposes a novel LFM weak signal detection algorithm based on bistable system and FRFT, named BSFRFT. We use the SR effect of bistable system to amplify LFM signal, and then apply FRFT to the previous output. We also present an evaluation criteria to measure the ability of detection methods and to compare performance of FRFT and BSFRFT. Numerical simulations show that the algorithm is effective for two cases – the LFM signal covered by white noise or colored noise (including pink noise). [ABSTRACT FROM AUTHOR]

This research aims to make a comparison between the analytical and numerical methods of simulation of the Gaussian beam propagation in free space. This comparison brings evaluation of the accuracy and applicability limitations of the angular spectrum diffraction algorithm. In this approach, planar harmonic waves are assumed as basic components of a propagating wave, so the algorithm combines the angular spectrum technique and the 2D fast Fourier transformation procedures. It means a quite different approach than the well-established Huygens–Sommerfeld one, based on spherical waves. The Gaussian beam is a rare case of the exact analytical 3D solution of a specific wave diffraction problem, in paraxial scalar approximation. Gaussian beam form has been chosen here as an evaluation reference. It allows to avoid comparisons between two numerical procedures, both of which may suffer from imperfections. In this comparison, the respective 2D cross-sections of calculated 3D complex valued wave fields distributions are presented. Also values of specific Gaussian beam parameters like full phase and Gouy phase shift and wave-front curvature radius are presented. The comparison between that form and the results of the angular spectrum algorithm brings information about the performance quality of that algorithm. [ABSTRACT FROM AUTHOR]

In this paper, a modified multiple-parameter fractional Fourier transform (MMPFRFT) is proposed which is another form of fractional Fourier transform in term of its degree of freedom. The defect has been found in encryption technique, illustrated in (R. Tao, J. Lang, Y. Wang, Opt. Lett. 33 (06) (2008) 581–583), dissolved and verification has been done on MATLAB ® . A novel technique for single channel, four-image encryption is also proposed based on MMPFRFT and pixel scrambling technique. The low frequency components which are extracted from four primitive images and placed in the center position, have utilized pixel scrambling technique and then encrypted based on MMPFRFT. Scrambling technology is utilized to enhance the security of the system. Numerical simulations are presented and it is claimed that this technique is superior to the existing techniques (Z. Liu et al., Optik 122 (11) (2011) 1010–1013, J. Wu, X. Luo, N. Zhou, Opt. Laser Technol. 45 (2013) 571–577). [ABSTRACT FROM AUTHOR]

Based on the theory of scalar diffraction, this paper makes research on the process of hologram reconstruction of true color calculation by lensless Fourier transform, and gives its algorithm on the basis of the angular spectrum theory as well as the instances of color image reconstruction under the red, green and blue colors wavelength. In order to eliminate the influence of color difference on the reconstructive image, this paper analyzes the causes of the color difference, and puts forward the method of interpolation reconstruction to the object information in the process of hologram reconstruction, so as to eliminate the color difference. With the advantages of computer generated holographic, the high-quality color hologram by lensless Fourier transform based on the angular spectrum theory has been realized, and the results of corresponding numerical simulation are given. The results show that the method in the paper can effectively eliminate the color difference caused by different wavelength, and the experimental results validate its feasibility. [ABSTRACT FROM AUTHOR]

Abstract: The fringe pattern phase analysis method is proposed for the leveling of mask and wafer in proximity lithography. The tilt between mask and wafer in the space is reflected in the tilted fringe pattern. The method combining the 2-D Fourier transform and 2-D Hanning window is proposed for processing the tilted fringe pattern. The offset and angle of tilt are extracted through phase analysis. Computer simulation and experiment are both performed to verify this method. The results indicate that the tilt of the mask and wafer in the space can be extracted with high accuracy through this method. [Copyright &y& Elsevier]

Abstract: In this paper, a new digital method has been developed for eliminating the zero-order and conjugate image in off-axis digital holography. The method consists of filtering the spatial frequencies associated with these undesired terms (the zero-order and conjugate images) in the computed Fourier transform plane of the hologram multiplied with computer simulated original reference wave (written as (RI h )). An important conclusion is that, without the need for additional shift-spectrum process, the spectrum of real image always appears in the center of the spectrum plane and never shifts. This brings us some convenient to design a filtering window. The theoretical analysis and the corresponding simulation and experiment results are given. As the zero-order and conjugate image are eliminated, both the contrast and brightness of the real image are enhanced and the image quality is improved. Meanwhile, the process of designing a filtering window is simple and convenient. [Copyright &y& Elsevier]

Abstract: A new method for double image encryption is proposed that is based on amplitude-phase hybrid encoding and iterative random phase encoding in fractional Fourier transform (FrFT) domains. In the iterative random phase encoding operation, a binary random matrix is defined to encode two original images to a single complex-valued image, which is then converted into a stationary white noise image by the iterative phase encoding with FrFTs. Compared with the previous schemes that uses fully phase encoding, the proposed method reduces the difference between two original images in key space and sensitivity to the FrFT orders. The primitive images can be retrieved exactly by applying correct keys with initial conditions of chaotic system, the pixel scrambling operation and the FrFT orders. Computer simulations demonstrate that the encryption method has impressively high security level and certain robustness against data loss and noise interference. [Copyright &y& Elsevier]

Abstract: Three frames of deformed patterns are needed at least in conventional Phase Measuring Profilometry (PMP). It is generally applied to the profile measurement of a static object. For on-line measurement, because of the moving object, it is difficult to capture three or more frames of deformed patterns that satisfy the requirement of PMP. A new method of on-line 3-D shape measurement through only one frame deformed pattern in PMP is presented. A static sinusoidal fringe is projected and any one frame of real-time deformed patterns is captured by CCD. N frames of sub-images are extracted from the captured deformed pattern. Improved Stoilov algorithm is proposed to get phase. Computer simulation and experimental results show its feasibility and affectivity to reconstruct a 3-D shape and the accuracy is higher than Fourier Transform Profilemetry (FTP) which is also a method that only needs one frame of deformed pattern. [Copyright &y& Elsevier]

Abstract: Based on fractional Fourier transform, an image encryption algorithm is proposed and researched. A local random phase encoding is introduced into this algorithm. The data at the local area of complex function is converted by fractional Fourier transform. The local random phase encoding is performed many times. Moreover only one set of random phase data is used in image encryption process. Compare to double random phase encoding, the parameter defining local area can be regarded as the additional key to increase the security of the encryption scheme. Some numerical simulations are achieved to demonstrate the performance of the image encryption scheme. [Copyright &y& Elsevier]

Abstract: In the present paper, we describe the encryption and decryption of two-dimensional images. The encryption is done by employing a sandwich phase diffuser made by using two elongated speckle patterns, and placed in the Fourier plane of a double random phase encoding system. After encryption, the two constituent phase diffusers of such a sandwich diffuser are separated. During decryption, if the conjugate of either of the two elongated phase speckle patterns is used, the image cannot be retrieved. Correct decryption is also not possible if a sandwich diffuser with any of the phase speckle patterns is shifted in position with respect to the other. For decryption, the encrypted image is Fourier transformed and multiplied by the conjugate of the sandwich diffuser, and then the product is further Fourier transformed. It is possible to generate the image only if both the elongated phase speckle patterns are matched point-to-point and then the proper conjugate is made. The use of elongated speckle patterns in constituting a sandwich phase diffuser makes the system less complicated as compared to the use of a sandwich diffuser made with normal speckle patterns, enabling an easy alignment of the random phase diffuser at the time of rejoining the constituent diffusers for making the right key. The ease of alignment is due to the reduction of the requirement of 360° scanning at the time of rejoining these diffusers with little reduction in the security of the system. Simulation results are presented in support of the proposed idea. For optical implementation, the decrypted image may be obtained by generating a phase conjugate wave by the phase conjugation technique, and passing through the same sandwich phase diffuser. To evaluate the reliability of the technique, mean square error (MSE) between the decrypted and original image has been calculated. [Copyright &y& Elsevier]

In this paper, we discuss the problem of phase retrieval in discrete windowed special affine Fourier transform domain, i.e., the problem of recovering a signal from the magnitudes of discrete windowed special affine Fourier transform. We first provide some appropriate conditions under which discrete windowed special affine Fourier transform allows phase retrieval. Then above conditions can be weaken for non-vanishing signals. Finally, the results of numerical simulations illustrate our theoretical analysis and provide some directions for future work. [ABSTRACT FROM AUTHOR]

Abstract: Using the accurate expression of the Fourier transform of the pure Voigt spectral line profile, we present the principle of the passive optical method to measure temperature, velocity, and pressure of atmospheric wind for the Voigt profile. Through discussion, from the Gaussian, Lorentzian and Voigt profiles, we find that velocity will not be affected, pressure can be derived for Lorentzian and Voigt profiles, and temperature will be affected greatly. By simulation and comparison, we find that temperature derived from the Voigt profile is greater than that from the Gaussian profile, and it is less than that from the Lorentzian profile. This discussion will be significant in accurate upper atmospheric wind measurement. [Copyright &y& Elsevier]