Fast numerical algorithm for the linear canonical transform

The linear canonical transform (LCT) describes the effect of any quadratic phase system (QPS) on an input optical wave field. Special cases of the LCT include the fractional Fourier transform (FRT), the Fourier transform (FT), and the Fresnel transform (FST) describing free-space propagation. Currently there are numerous efficient algorithms used (for purposes of numerical simulation in the area of optical signal processing) to calculate the discrete FT, FRT, and FST. All of these algorithms are based on the use of the fast Fourier transform (FFT). In this paper we develop theory for the discrete linear canonical transform (DLCT), which is to the LCT what the discrete Fourier transform (DFT) is to the FT. We then derive the fast linear canonical transform (FLCT), an N log N algorithm for its numerical implementation by an approach similar to that used in deriving the FFT from the DFT. Our algorithm is significantly different from the FFT, is based purely on the properties of the LCT, and can be used for FFT, FRT, and FST calculations and, in the most general case, for the rapid calculation of the effect of any QPS.     

Fractional derivatives-analysis and experimental implementation

The fractional derivative spatial-filtering operator is useful for image-processing applications, particularly for examination of phase objects. Experimental implementation is difficult because the mask function combines both amplitude and phase. We present a simple one-dimensional analysis of the fractional derivative operation and note similarities with the fractional Hilbert transform. We demonstrate how to encode these amplitude and phase masks using a phase-only liquid-crystal spatial light modulator and present experimental results. Finally, we introduce a radially symmetric extension of this operation that is more useful for objects having an arbitrary shape.     

Geometry and dynamics of squeezing in finite systems

Squeezing and its inverse magnification form a one-parameter group of linear canonical transformations of continuous signals in paraxial optics. We search for corresponding unitary matrices to apply on signal vectors in N-point finite Hamiltonian systems. The analysis is extended to the phase space representation by means of Wigner quasi-probability distribution functions on the discrete torus and on the sphere. Together with two previous studies of the fractional Fourier and Fresnel transforms, we complete the finite counterparts of the group of linear canonical transformations.     

Adaptive phase-shifting algorithm for temporal phase evaluation

Most standard temporal-phase-shifting (TPS) algorithms evaluate the phase by computing a windowed Fourier transform (WFT) of the intensity signal at the carrier frequency of the system. However, displacement of the specimen during image acquisition may cause the peak of the transform to shift away from the carrier frequency, leading to phase errors and even unwrapping failure. We present a novel TPS method that searches for the peak of the WFT and evaluates the phase at that frequency instead of at the carrier frequency. The performance of this method is compared with that of standard algorithms by using numerical simulations. Experimental results from high-speed speckle interferometry studies of carbon fiber panels are also presented.     

Noniterative boundary-artifact-free wavefront reconstruction from its derivatives

Wavefront sensors are usually based on measuring the wavefront derivatives. The most commonly used approach to quantitatively reconstruct the wavefront uses discrete Fourier transform, which leads to artifacts when phase objects are located at the image borders. We propose here a simple approach to avoid these artifacts based on the duplication and antisymmetrization of the derivatives data, in the derivative direction, before integration. This approach completely erases the border effects by creating continuity and differentiability at the edge of the image. We finally compare this corrected approach to the literature on model images and quantitative phase images of biological microscopic samples, and discuss the effects of the artifacts on the particular application of dry mass measurements.     

Eigenfunctions of the offset Fourier,fractional Fourier,and linear canonical transforms

The offset Fourier transform (offset FT), offset fractional Fourier transform (offset FRFT), and offset linear canonical transform (offset LCT) are the space-shifted and frequency-modulated versions of the original transforms. They are more general and flexible than the original ones. We derive the eigenfunctions and the eigenvalues of the offset FT, FRFT, and LCT. We can use their eigenfunctions to analyze the self-imaging phenomena of the optical system with free spaces and the media with the transfer function exp[j(h2x2 + h1x + h0)] (such as lenses and shifted lenses). Their eigenfunctions are also useful for resonance phenomena analysis, fractal theory development, and phase retrieval.     

Accurate estimation of the boundaries of a structured light pattern

Depth recovery based on structured light using stripe patterns, especially for a region-based codec, demands accurate estimation of the true boundary of a light pattern captured on a camera image. This is because the accuracy of the estimated boundary has a direct impact on the accuracy of the depth recovery. However, recovering the true boundary of a light pattern is considered difficult due to the deformation incurred primarily by the texture-induced variation of the light reflectance at surface locales. Especially for heavily textured surfaces, the deformation of pattern boundaries becomes rather severe. We present here a novel (to the best of our knowledge) method to estimate the true boundaries of a light pattern that are severely deformed due to the heavy textures involved. First, a general formula that models the deformation of the projected light pattern at the imaging end is presented, taking into account not only the light reflectance variation but also the blurring along the optical passages. The local reflectance indices are then estimated by applying the model to two specially chosen reference projections, all-bright and all-dark. The estimated reflectance indices are to transform the edge-deformed, captured pattern signal into the edge-corrected, canonical pattern signal. A canonical pattern implies the virtual pattern that would have resulted if there were neither the reflectance variation nor the blurring in imaging optics. Finally, we estimate the boundaries of a light pattern by intersecting the canonical form of a light pattern with that of its inverse pattern. The experimental results show that the proposed method results in significant improvements in the accuracy of the estimated boundaries under various adverse conditions.     

Generalized prolate spheroidal wave functions for optical finite fractional Fourier and linear canonical transforms

Prolate spheroidal wave functions (PSWFs) are known to be useful for analyzing the properties of the finite-extension Fourier transform (fi-FT). We extend the theory of PSWFs for the finite-extension fractional Fourier transform, the finite-extension linear canonical transform, and the finite-extension offset linear canonical transform. These finite transforms are more flexible than the fi-FT and can model much more generalized optical systems. We also illustrate how to use the generalized prolate spheroidal functions we derive to analyze the energy-preservation ratio, the self-imaging phenomenon, and the resonance phenomenon of the finite-sized one-stage or multiple-stage optical systems.     

Natural demodulation of two-dimensional fringe patterns. I. General background of the spiral phase quadrature transform

It is widely believed, in the areas of optics, image analysis, and visual perception, that the Hilbert transform does not extend naturally and isotropically beyond one dimension. In some areas of image analysis, this belief has restricted the application of the analytic signal concept to multiple dimensions. We show that, contrary to this view, there is a natural, isotropic, and elegant extension. We develop a novel two-dimensional transform in terms of two multiplicative operators: a spiral phase spectral (Fourier) operator and an orientational phase spatial operator. Combining the two operators results in a meaningful two-dimensional quadrature (or Hilbert) transform. The new transform is applied to the problem of closed fringe pattern demodulation in two dimensions, resulting in a direct solution. The new transform has connections with the Riesz transform of classical harmonic analysis. We consider these connections, as well as others such as the propagation of optical phase singularities and the reconstruction of geomagnetic fields.     

一种改进的DDTFA方法及其在齿轮箱故障诊断中的应用

针对数据驱动时频分析(DDTFA)方法的初始相位函数选取问题,提出一种可准确、快速且自适应优选初始相位函数的改进DDTFA方法。引入数学中函数求极值的思想,将信号的初始相位函数选取问题转换为初始解集的连续寻优问题,通过对DDTFA中的高斯牛顿迭代算法进行精简,以初始解集中的初始相位函数迭代一次斜率的变化量为导数获得初始解集的连续导数集,进而求得局部极大值,并以局部极大值对应信号分量的能量最强为准则优选信号的初始相位函数,进而完成信号分解。仿真分析与齿轮箱故障诊断实例表明,该方法可准确、快速且自适应地优选初始相位函数,并有效提取故障特征,且具有一定抗噪性。     

Digital transform processing of carrier fringe patterns from speckle-shearing interferometry

Abstract

The Fourier transform (FT) and the wavelet transform (WT) methods are used to process the fringe carrier pattern resulting from speckle-shearing inteferometry, in which the carrier frequency is modulated by deformation of a bending plate. Both the amount and the sign of the first derivative of the out-of-plane displacement can be obtained by these two transform techniques in the whole field. Phase distributions of the deflection slope are compared, which shows the wavelet analysis gives a better solution with noise reduction and without deficiency of filter window choice as for that in the Fourier transform. Meanwhile, the phase values in the path along the maximum WT amplitudes give a direct map of the second derivative patterns of the deflection, which presents the same image as that given by the shearing subtraction of the phase patterns from the inverse Fourier transformation but avoids the processing of unwrapping for the phase reconstruction.     

General n-dimensional quadrature transform and its application to interferogram demodulation

Quadrature operators are useful for obtaining the modulating phase phi in interferometry and temporal signals in electrical communications. In carrier-frequency interferometry and electrical communications, one uses the Hilbert transform to obtain the quadrature of the signal. In these cases the Hilbert transform gives the desired quadrature because the modulating phase is monotonically increasing. We propose an n-dimensional quadrature operator that transforms cos(phi) into -sin(phi) regardless of the frequency spectrum of the signal. With the quadrature of the phase-modulated signal, one can easily calculate the value of phi over all the domain of interest. Our quadrature operator is composed of two n-dimensional vector fields: One is related to the gradient of the image normalized with respect to local frequency magnitude, and the other is related to the sign of the local frequency of the signal. The inner product of these two vector fields gives us the desired quadrature signal. This quadrature operator is derived in the image space by use of differential vector calculus and in the frequency domain by use of a n-dimensional generalization of the Hilbert transform. A robust numerical algorithm is given to find the modulating phase of two-dimensional single-image closed-fringe interferograms by use of the ideas put forward.     

Complex signal representation, Mandel's theorem, and spiral phase quadrature transform

Complex (or analytic) signal representation as introduced by Gabor plays an important role in optical signal processing and in coherence theory of optical fields. Several definitions for extending the notion of complex signal representation to two dimensions have appeared in the literature. These definitions differ in their choice of the quadrature transform for a two-dimensional signal. We study the problem of determining the complex representation for two-dimensional real signals (or images) using a least-square minimization framework first used by Mandel [J. Opt. Soc. Am.57, 613 (1967)JOSAAH0030-3941]. In particular, we seek a suitable quadrature transform such that the resultant complex image has the least fluctuating envelope in an ensemble-averaged sense. It is observed that the spiral phase quadrature transform for two-dimensional signals is a solution of this analysis.     

Fresnel transform phase retrieval from magnitude

This report presents a generalized projection method for recovering the phase of a finite support, two-dimensional signal from knowledge of its magnitude in the spatial position and Fresnel transform domains. We establish the uniqueness of sampled monochromatic scalar field phase given Fresnel transform magnitude and finite region of support constraints for complex signals. We derive an optimally relaxed version of the algorithm resulting in a significant reduction in the number of iterations needed to obtain useful results. An advantage of using the Fresnel transform (as opposed to Fourier) for measurement is that the shift-invariance of the transform operator implies retention of object location information in the transformed image magnitude. As a practical application in the context of ultrasound beam measurement we discuss the determination of small optical phase shifts from near field optical intensity distributions. Experimental data are used to reconstruct the phase shape of an optical field immediately after propagating through a wide bandwidth ultrasonic pulse. The phase of each point on the optical wavefront is proportional to the ray sum of pressure through the ultrasound pulse (assuming low ultrasonic intensity). An entire pressure field was reconstructed in three dimensions and compared with a calibrated hydrophone measurement. The comparison is excellent, demonstrating that the phase retrieval is quantitative.     

光子多普勒测速系统信号解调方法比较

光子多普勒测速系统具有抗干扰能力强、测速范围大等优点,适用于信噪比低、信号质量差的测量场合。介绍光子多普勒测速系统的工作原理,详细阐述常用的四种信号解调方法——条纹法、相位解调法、短时傅里叶变换法、小波变换法的原理、特点和近几年的研究现状。利用上述四种信号解调方法对简谐振动调制的多普勒信号进行解调仿真,直观地展示不同信号解调方法的优缺点和适用性。实验结果表明,相位解调法最适合用于光子多普勒测速系统振动信号解调。最后讨论了采用递归希尔伯特变换的方法减小相位解调法的正交性误差的可行性,通过仿真实验验证了此方法的有效性。     

On the phase interpolation problem — A brief review and some new results

Reconstruction of a signal from the phase of its Fourier transform is important and useful in a variety of practical applications. In this paper, we give a brief review of the previous results and develop some new ones pertaining to the phase-only reconstruction problem. The review includes conditions under which the signal is uniquely specified by its Fourier transform phase and some algorithms for performing the reconstruction. New results consist of an extension of a previous technique and an explicit formula for reconstruction. A number of potential applications of the reconstruction techniques are briefly mentioned in the concluding section. Throughout this paper, reference to the phase or magnitude of the signal should be interpreted, respectively, as the phase or magnitude of theft of the signal.     

超声波测振信号的能量算子解调方法

连续超声波束遇到振动物体表面会产生多普勒效应,反射超声波信号是受振动信号调制的非线性调相信号。对反射波信号求导获得调幅调频信号,再采用能量算子对称差分法,求取该调幅调频信号的瞬时幅值及瞬时频率。鉴于超声波反射回波信号存在幅值衰减现象,而超声波频率不易受外界干扰,故通过调幅调频信号的瞬时频率提取被测物体的振动速度,并由振动速度求导得到振动加速度。同时,从幅值及频率两个方面探讨振动测量范围。仿真及实验结果表明:基于能量算子的超声波测振信号解调方法能有效地提取振动信号,与传统的相位解调方法相比,具有更大的测量范围。     

Complex-Valued Wavelet Lifting and Applications

Signals with irregular sampling structures arise naturally in many fields. In applications such as spectral decomposition and nonparametric regression, classical methods often assume a regular sampling pattern, thus cannot be applied without prior data processing. This work proposes new complex-valued analysis techniques based on the wavelet lifting scheme that removes “one coefficient at a time.” Our proposed lifting transform can be applied directly to irregularly sampled data and is able to adapt to the signal(s)’ characteristics. As our new lifting scheme produces complex-valued wavelet coefficients, it provides an alternative to the Fourier transform for irregular designs, allowing phase or directional information to be represented. We discuss applications in bivariate time series analysis, where the complex-valued lifting construction allows for coherence and phase quantification. We also demonstrate the potential of this flexible methodology over real-valued analysis in the nonparametric regression context. Supplementary materials for this article are available online.     

基于二进小波的雷达干涉图条纹探测相位解绕

以二进小波和相位解绕理论为基础，利用二进小波变换的快速算法，对雷达干涉条纹图进行多尺度相位突变边缘探测，进而根据信号和噪声的奇异性在二进小波变换下的局部模极大值随尺度变化的不同演化规律，提出了对雷达干涉条纹图的噪声有一定压制的基于二进小波的干涉条纹检测的相位解绕算法，结果表明，在噪声没有严重破坏相位突变边缘的情况下，相位解绕结果是令人满意的。     

Spectral ballistic imaging: a novel technique for viewing through turbid or obstructing media

We propose a new method for viewing through turbid or obstructing media. The medium is illuminated with a modulated cw laser and the amplitude and phase of the transmitted (or reflected) signal is measured. This process takes place for a set of wavelengths in a certain wide band. In this way we acquire the Fourier transform of the temporal output. With this information we can reconstruct the temporal shape of the transmitted signal by computing the inverse transform. The proposed method benefits from the advantages of the first-light technique: high resolution, simple algorithms, insensitivity to boundary condition, etc., without suffering from its main deficiencies: complex and expensive equipment.