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.     

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.     

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.     

Two-Dimensional Image Transmission Based on the Ultrafast Optical Data Format Conversion between a Temporal Signal and a Two-Dimensional Spatial Signal

We have experimentally demonstrated a two-dimensional (2-D) image transmission based on the ultrafast optical data format conversion between a temporal signal and a spatial signal with an ultrashort optical pulse. In the proposed system we adopt a spectral holography technique to transmit a one-dimensional (1-D) spatial signal and use a spatial-domain time-frequency transform to realize a transform between 1-D and 2-D spatial signals. By use of these techniques, a low-optical-loss transmission system can be constructed. To demonstrate a 2-D image transmission with this technique, we achieved experimentally transmission of the alphabet letter T as a 3 x 3 pixel 2-D spatial image.     

Efficient computation of optical disk readout by use of the chirp z transform

Computation of the readout signal of an optical disk involves Fourier transforms from the objective lens pupil to the disk and, after interaction with the disk, from the disk to the objective pupil. Traditionally, the complex two-dimensional Fourier transform is numerically evaluated as a two-dimensional fast Fourier transform. To obtain sufficient resolution in the involved planes, one must choose sampling grid sizes of typically 1,024 x 1,024 or higher, resulting in a substantial computation time if the calculation is to be repeated many times. Discussed is an alternative method for evaluating the Fourier transform, based on the chirp z transform, by which a considerable improvement in efficiency can be obtained without loss of accuracy.     

Resolving quadrature fringes in real time

In many interferometers, two fringe signals can be generated in quadrature. The relative phase of the two fringe signals depends on whether the optical path length is increasing or decreasing. A system is developed in which two quadrature fringe signals are digitized and analyzed in real time with a digital signal processor to yield a linear, high-resolution, wide-dynamic-range displacement transducer. The resolution in a simple Michelson interferometer with inexpensive components is 5 x 10(-13) m Hz(-1/2) at 2 Hz.     

Proposal for optical implementation of the wigner distribution function

The Wigner distribution function (WDF) offers comprehensive insight into a signal, for it employs both space (or time) and frequency simultaneously. Whenever optical signals are involved, the importance of the WDF is significantly higher because of the diffraction (or dispersion) behavior of optical signals. Novel optical implementations of the WDF and of the inverse Wigner transform are proposed. Both implementations are based on bulk optics elements incorporating joint transform correlator architecture. A similar implementation is derived for the ambiguity function, which is related to the WDF through Fourier transformation.     

Two-dimensional wavelet transform for reliability-guided phase unwrapping in optical fringe pattern analysis

This paper theoretically discusses modulus of two-dimensional (2D) wavelet transform (WT) coefficients, calculated by using two frequently used 2D daughter wavelet definitions, in an optical fringe pattern analysis. The discussion shows that neither is good enough to represent the reliability of the phase data. The differences between the two frequently used 2D daughter wavelet definitions in the performance of 2D WT also are discussed. We propose a new 2D daughter wavelet definition for reliability-guided phase unwrapping of optical fringe pattern. The modulus of the advanced 2D WT coefficients, obtained by using a daughter wavelet under this new daughter wavelet definition, includes not only modulation information but also local frequency information of the deformed fringe pattern. Therefore, it can be treated as a good parameter that represents the reliability of the retrieved phase data. Computer simulation and experimentation show the validity of the proposed method.     

Efficient computation of joint fractional Fourier domain signal representation

A joint fractional domain signal representation is proposed based on an intuitive understanding from a time-frequency distribution of signals that designates the joint time and frequency energy content. The joint fractional signal representation (JFSR) of a signal is so designed that its projections onto the defining joint fractional Fourier domains give the modulus square of the fractional Fourier transform of the signal at the corresponding orders. We derive properties of the JFSR, including its relations to quadratic time-frequency representations and fractional Fourier transformations, which include the oblique projections of the JFSR. We present a fast algorithm to compute radial slices of the JFSR and the results are shown for various signals at different fractionally ordered domains.     

压缩对称嵌套阵列和稀疏信号重构的近场目标方位估计

针对现有近场源估计算法中近场源数量受限于阵元数的问题,提出了一种基于稀疏对称嵌套阵列和稀疏信号重构的近场欠定波达方向估计方法。首先利用四阶累积量,将二维空间参数估计问题转化为一维参数估计问题,同时得到差分阵列;为了进一步提高估计分辨率与减少估计误差,对虚拟阵列的接收信号在空间域进行稀疏表示;最后通过L1范数最小二乘法得到目标源的波达方向。相较于现有算法,该方法可以估计更多的目标源,并且有更低的均方误差与更高的分辨率。实验仿真验证了算法的有效性与优越性。     

基于复包络提取的波束形成方法

结合带通信号的特性,提出了一种基于复包络提取的波束形成方法。首先,提取各阵元接收的信号的同相分量和正交分量;其次,分别对上述同相分量和正交分量进行相移;最后再对移相之后的同相分量和正交分量进行延时相加,从而形成入射方向的波束。该方法有效降低了波束形成器的输入和输出采样率以及数字处理的复杂度。     

Characterization of Structural Noise Patterns and Echo Separation in the Time-Frequency Plane for Austenitic Stainless Steels

The aim of this study is to characterize the structural noise for a better flaw detection in heterogeneous materials (steels, weld, composites...) using ultrasonic waves. For this purpose, the continuous wavelet transform is applied to ultrasonic A-scan signals acquired using an ultrasonic non destructive testing (NDT) device. The time-scale representation provided, which highlights the temporal evolution of the spectral content of the A-scan signals, is relevant but can lead to misinterpretation. The problem is to identify if each pattern from the wavelet representation is due to the structural noise or the flaw. To solve this problem, a detection technique based on statistical significance testing in the time-scale plane is used. Information about the structural noise signals is injected into the decision process using an autoregressive model, which seems relevant according to the spectral content of the signal. The approach is tested on experimental signals, obtained by ultrasonic NDT of metallic materials (austenitic stainless steel) then on a weld in this steel and indeed enables to distinguish the components of the signal as flaw echoes, which differ from the structural noise.     

Optical sectioning for optical scanning holography using phase-space filtering with Wigner distribution functions

We propose a novel optical sectioning method for optical scanning holography, which is performed in phase space by using Wigner distribution functions together with the fractional Fourier transform. The principle of phase-space optical sectioning for one-dimensional signals, such as slit objects, and two-dimensional signals, such as rectangular objects, is first discussed. Computer simulation results are then presented to substantiate the proposed idea.     

Optical Generation of Gabor's Expansion Coefficients for Rastered Signals

Gabor's expansion of a signal into a discrete set of properly shifted and modulated versions of an elementary signal is introduced; several examples of elementary signals are considered. An arrangement is described which is able to generate Gabor's expansion coefficients of a one-dimensional signal by optical means. An important feature of the optical arrangement is that it accepts the one-dimensional signal on a raster format; hence, the two-dimensional nature of the optical processing system is fully utilized.     

Extracting one-dimensional wavelet features with a diffractive optical inner-product transform

A diffractive optical element (DOE) that performs a wavelet transform on one-dimensional signals is presented. The DOE performs an inner product between the input image and a set of wavelet basis functions. The input image is displayed in a spatial light modulator. It is constructed from a one-dimensional signal that is spread over the second dimension. Simulated results obtained by the use of elementary one-dimensional signals are presented.     

Experimental realization of optical lumped nanocircuits at infrared wavelengths

The integration of radiofrequency electronic methodologies on micro- as well as nanoscale platforms is crucial for information processing and data-storage technologies. In electronics, radiofrequency signals are controlled and manipulated by 'lumped' circuit elements, such as resistors, inductors and capacitors. In earlier work, we theoretically proposed that optical nanostructures, when properly designed and judiciously arranged, could behave as nanoscale lumped circuit elements--but at optical frequencies. Here, for the first time we experimentally demonstrate a two-dimensional optical nanocircuit at mid-infrared wavelengths. With the guidance of circuit theory, we design and fabricate arrays of Si3N4 nanorods with specific deep subwavelength cross-sections, quantitatively evaluate their equivalent impedance as lumped circuit elements in the mid-infrared regime, and by Fourier transform infrared spectroscopy show that these nanostructures can indeed function as two-dimensional optical lumped circuit elements. We further show that the connections among nanocircuit elements, in particular whether they are in series or in parallel combination, can be controlled by the polarization of impinging optical signals, realizing the notion of 'stereo-circuitry' in metatronics-metamaterials-inspired optical circuitry.     

信号的小波尺度-频率表示及其在机械故障诊断中的应用

基于对信号的连续小波变换特点的分析，提出信号特征的小波尺度一频率表示方法。该方法对污染的脉冲信号和正弦信号的应用结果表明该方法能有效提取强噪声背景下的周期成分。以变速器齿轮的故障诊断为例，用该方法对变速箱齿轮振动信号的尺度-频率表示进行分析，结果表明该方法能非常有效地将不同磨损状况的齿轮振动信号的特征表现出来。     

Joint transform correlator for optical temporal signals

Correlation is considered to be a fundamental operation in the field of signal processing. The fact that this operation can be implemented optically in a relatively simple manner is an important advantage of utilizing optical systems for signal processing. The VanderLugt 4-f system and the joint transform correlator (JTC) are the two most popular configurations for performing a spatial correlation operation optically. So far the JTC architecture has been used for performing correlation between two images, which are illuminated by a quasi-monochromatic light source. We propose a generalization of the JTC that performs a correlation between two temporal optical signals.     

Ultrasonic data compression via parameter estimation

Ultrasonic imaging in medical and industrial applications often requires a large amount of data collection. Consequently, it is desirable to use data compression techniques to reduce data and to facilitate the analysis and remote access of ultrasonic information. The precise data representation is paramount to the accurate analysis of the shape, size, and orientation of ultrasonic reflectors, as well as to the determination of the properties of the propagation path. In this study, a successive parameter estimation algorithm based on a modified version of the continuous wavelet transform (CWT) to compress and denoise ultrasonic signals is presented. It has been shown analytically that the CWT (i.e., time x frequency representation) yields an exact solution for the time-of-arrival and a biased solution for the center frequency. Consequently, a modified CWT (MCWT) based on the Gabor-Helstrom transform is introduced as a means to exactly estimate both time-of-arrival and center frequency of ultrasonic echoes. Furthermore, the MCWT also has been used to generate a phase x bandwidth representation of the ultrasonic echo. This representation allows the exact estimation of the phase and the bandwidth. The performance of this algorithm for data compression and signal analysis is studied using simulated and experimental ultrasonic signals. The successive parameter estimation algorithm achieves a data compression ratio of (1-5N/J), where J is the number of samples and N is the number of echoes in the signal. For a signal with 10 echoes and 2048 samples, a compression ratio of 96% is achieved with a signal-to-noise ratio (SNR) improvement above 20 dB. Furthermore, this algorithm performs robustly, yields accurate echo estimation, and results in SNR enhancements ranging from 10 to 60 dB for composite signals having SNR as low as -10 dB.     

S变换时频谱SVD降噪的冲击特征提取方法

为了从滚动轴承故障振动信号中提取出冲击特征,以进行轴承故障诊断,提出基于S变换时频谱奇异值分解(SVD)的信号降噪方法。S变换是一种信号时频表示方法,适合于处理与分析非平稳的冲击特征信号。在SVD降噪过程中,数据矩阵由信号的S变换谱系数构成;奇异值序列的置零阈值位置坐标可由奇异值差分谱最前面部分峰值群的最后一个峰值点序号来确定。最后对降噪的数据矩阵进行S逆变换,获得信号的时域冲击特征。仿真研究表明,基于S变换时频谱的SVD降噪方法可以成功地从低信噪比信号中提取出周期性的冲击特征。将本方法用于处理与分析滚动轴承故障振动信号,根据所提取出的冲击特征出现频率,能够方便有效地实现轴承相关故障的诊断。