Phase retrieval from fringe pattern with 1-D discrete wavelet transform

This paper proposes a novel method to obtain frequency modulation (FM) signals from a single fringe pattern for phase retrieval. First, a 1D discrete Meyer wavelet is employed to decompose the pattern image signal row by row and the soft-thresholding approach is applied to remove noise. The low frequency coefficients of the wavelet decomposition are then set to 0, and the signal is reconstructed. Moreover, the optimal wavelet decomposition level is adaptively determined using a cost function-based method. The reconstructed signal, which no longer contains a background component, is normalized using a nonlinear and piecewise normalization method. The proposed method is faster and more accurate than some other phase retrieval approaches, which is illustrated with two test cases.     

Phase retrieval of speckle fringe pattern with carriers using 2D wavelet transform

A novel two-dimensional (2D) Gabor continuous wavelet transform (CWT) for phase retrieval and fringe filtering of speckle fringe patterns with spatial carriers is proposed. Theoretical analysis of 2D Gabor CWT is presented and results are compared with advanced fan 2D CWT using both the computer simulated and experimental speckle fringe patterns. It is shown that noise reduction by 2D Gabor CWT demonstrates better results than that of the advanced fan 2D CWT. Two-dimensional Gabor CWT is also compared with 2D Fourier transform and results show that 2D Gabor CWT algorithm has better noise immunity.     

A new wavelet transform for reliability-guided phase unwrapping of optical fringe patterns

A new daughter wavelet definition for reliability-guided phase unwrapping of the optical fringe pattern is introduced. The new daughter wavelet definition is given by normalized mother wavelet with a new factor including a Gaussian function. The modulus of the wavelet transform coefficients, obtained by using 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 which represents the reliability of the reconstructed phase data. Mathematical demonstration of this parameter is given. Both the computer simulation and experiment reveal the validity of the proposed method.     

Analysis of phase distortion in phase-shifted fringe projection

This paper describes the analysis of phase distortion in phase-shifted fringe projection method. A phase distortion occurs when the phase shifting technique is applied to extract the phase values from projected fringe patterns in surface contouring. The phase distortion will induce measurement errors especially in the measurement of micro-components. The cause of such phase distortion is investigated and the influence of phase distortion on the measurement of micro-components is discussed. To eliminate the phase distortion, a continuous wavelet transform (CWT) is employed to extract phase values from object surface modulated fringe patterns. Principle of the proposed CWT phase extraction method is described and experiments are conducted to verify the proposed method. It is shown that by the use of CWT phase extraction method phase distortion induced in conventional phase-shifting technique can be completely eliminated.     

A numerical spatial carrier for single fringe pattern analysis algorithm

Hilbert transform and continuous wavelet transform are combined to form an algorithm for the analysis of a single fringe pattern with open or closed fringes. Only one recorded fringe pattern is needed to extract the phase information. A second π/2 phase shifted fringe pattern should first be generated from the other using Hilbert transform to superpose numerically the spatial carrier. Several phase extraction techniques are conceived to demodulate fringe patterns. In this paper, the wavelet technique is used to obtain the required phase distribution. The algorithm offers an advantage that it is appropriate when the spatial carrier is impossible to be added experimentally. Finally, a simulation was carried out to validate the algorithm, giving good results.     

Demodulation of a single interferogram based on continuous wavelet transform and phase derivative

A novel method based on continuous wavelet transform (CWT) and guidance of phase derivative is developed to measure the phase of a single fringe pattern which contains closed fringes. Wrapped phase values are retrieved by ridge extraction algorithms based on CWT which has the capability of better noise reduction and thus increases the resolution of measurement significantly. To further reduce the noise, the scales detected by maximum ridge algorithm are filtered iteratively before retrieval of wrapped phase. The proposed method also identifies any ambiguous point in a non-monotonous fringe pattern by directly tracking an inflexion point from an unwrapped phase map without the use of a carrier. The algorithm developed is validated by computer simulation and experimental results. Based on micro interferometry the experimental results for both static and dynamic deformations of a micro structure demonstrate that the proposed method is an effective tool for the analysis of closed fringe patterns and subsequent deformation measurement. However, the proposed technique is limited to measurement of surface which is relatively smooth compared to the mean wavelength of the light source. In addition, prior knowledge of the sign of surface slope is required in cases where a spatial carrier is not available or adaptable.     

Digital hologram encryption using discrete wavelet packet transform

In this paper, key components of a digital hologram were tracked using discrete wavelet packet transform to propose an encryption. Information needed to handle digital holograms were obtained by analyzing the characteristic of digital holograms in the Fresnel and frequency domains. By integrating the information that was experimentally obtained, an encryption method that uses the wavelet transform and subband packetization was proposed. Thus it is possible to change the encryption strength by changing the level of wavelet transform and the energy value of the coefficients to be encrypted. Optimal parameters can be found by analyzing the encryption effect numerically and visually. Therefore, it is possible to execute the encryption efficiently using the parameters presented in this paper without a separate analysis process. The experimental results showed that encrypting only 0.032% of the entire data was enough to hide the contents of the hologram. Here, we regarded the key to encrypt the subband packetization information as the overall encryption key.     

An improved windowed Fourier transform for fringe demodulation

A modified algorithm of windowed Fourier transform (WFT) for phase retrieval from electronic speckle-shearing fringe patterns with carriers is proposed. The algorithm is based on the introduction of a fast Fourier transform (FFT) in WFT to reduce computation time for fringe demodulation. Since boundary effects in FFT will influence the accuracy of phase retrieval, the Gerchberg method is employed to extrapolate the fringe pattern at the boundaries to reduce boundary effects. A theoretical analysis of the algorithm is presented. Both simulated and experimental results show that the proposed method has reduced the computation time significantly compared with the convolution method of WFT without sacrificing measurement accuracy.     

Absolute phase recovery of three fringe patterns with selected spatial frequencies

A new temporal approach is presented for the recovery of the absolute phase maps from their wrapped versions based on the use of fringe patterns of three different spatial frequencies. In contrast to the two-frequency method recently published, the method proposed is characterized by better anti-error capability as measured by phase error tolerance bound. A general rule for the selection of the three frequencies is presented, and its relationship to the phase error tolerance bound is derived. Theoretical analysis and experimental results are also presented to validate the effectiveness of the proposed three frequency technique.     

Phase retrieval of optical fringe patterns from the ridge of a wavelet transform

A new method for phase retrieval of optical fringe patterns is presented. This method is based on a wavelet transform and is capable of extracting the full 2D phase distribution from a single fringe pattern. An important conclusion that the phase of the optical fringe pattern is equal to the phase of its wavelet transform on the ridge of the wavelet transform is theoretically clarified. The method is compared with the Fourier transform and the integration methods. A numerical simulation and an experimental example of phase retrieval are shown.     

瞬态信号的小波变换波达方向估计

针对瞬态信号方位估计问题,提出了基于连续小波变换的多重信号分类测向算法(CWT_MUSIC)。首先由信号特征确定小波尺度参数,构造Morlet小波,对信号进行小波变换,利用获得的小波变换系数建立多分辨时频阵列信号模型,并据此模型设计基于子空间的MUSIC算法以实现瞬态信号的波达方向估计;然后对该算法的多分辨与误差性能进行分析,最后仿真实验和实际爆炸试验验证了所提出的CWT_MUSIC算法能有效地提高空间谱的分辨率和DOA估计性能。     

光学可分离小波变换的研究

为改进光学小波变换,提出了光学可分离小波变换的概念。通过对光学小波变换的必要条件进行分析,采用层叠算法和二维可分离小波变换构造方法计算出了尺度和小波函数的二维离散近似序列。将新的母小波引入到光学小波变换中,获得了光学可分离小波变换。通过与现有光学小波变换及二维可分离离散小波变换进行比较说明了其优点。     

Continuous wavelet transform for micro-component profile measurement using vertical scanning interferometry

White-light interferometric techniques have been widely used in three-dimensional (3D) profiling. This paper presents a new method based on vertical scanning interferometry (VSI) for the 3D profile measurement of a micro-component that contains sharp steps. The use of a white-light source in the system overcomes the phase ambiguity problem often encountered in monochromatic interferometry and also reduces speckle noises. A new algorithm based on the continuous wavelet transform (CWT) is used to retrieve the phase of an interferogram. The algorithm accurately determines local fringe peak and improves the vertical resolution of the measurement. The proposed method is highly resistant to noise and is able to achieve high accuracy. A micro-component (lamellar grating) fabricated by sacrificial etching technique is used as a test specimen to verify the proposed method. The measurement uncertainty of the experimental results is discussed.     

Analysis of Lau interferometric fringe for measurement of focal length by using wavelet filtering techniques

New filtering scheme is investigated and implemented on Lau interferometric fringes to enhance the signal-to-noise ratio (SNR) in the Lau interferograms. Towards this, the applicability of Haar, Daubechies and Symlet wavelet based filtering schemes have been tested. Of these schemes, the Symlet based filtering scheme gives the best results in terms of improved signal to noise ratio and speckle index. This scheme is then used to analyse Lau interferograms recorded in the experimental arrangement for focal length measurement. The processed interferograms yield improved accuracy and precision of measurement.     

Statistical searching of deformation phases on wavelet transform maps of fringe patterns

A wavelet transform (WT) analysis is presented to obtain the deformation phases from the fringes with non-uniform carrier frequency, which may appear in the pattern of varied-periodic fringes generated in displacement measurement. Based on the phase maps of the Morlet WT coefficients distributed in a space-scale spectrum, a statistical processing is carried out to search the compacted density of the phase intervals over the scale, and from that the phase modulations related to the object deformation can be determined. Numerical simulation demonstrates the validity of the pattern-processing technique, and the experimental results show the applications to the measurement of the in-plane displacement by the digital speckle pattern interferometry (DSPI) and the measurement of the out-of-plane deflection by the projecting Moiré fringes.     

小波相位解调轮廓术

针对傅里叶变换轮廓术因混频难以准确提取基频的问题,可将小波变换用于三维形貌相位直接解调。对其原理进行了研究,通过编程实现了小波相位解调。取Morlet复小波作为小波母函数,对调制栅线图逐行进行连续小波变换,从各位置的沿尺度方向的小波变换系数幅值的极值中可直接求取对应的相位数据。由于小波函数具有空域_频率两域的局部化特性,因此它对变形栅具有很强的自适应能力。为验证新方法,对其进行了仿真分析,同时还对石膏半球模型和化妆品瓶进行了实际测量,并分别用傅里叶变换和小波变换进行了处理。结果表明,新方法有效地克服了混频的问题,改善了相位解调效果,提高了测量精度,特别适合于复杂物体的形貌测量。     

A two-dimensional fast Fourier transform method for measuring the inclination angle of parallel fringe patterns

In this paper we describe a technique based on two-dimensional fast Fourier fringe pattern analysis for the automated evaluation of the inclination angle of parallel fringe patterns. Analysis of noise-free theoretical parallel fringe patterns is presented and numerical results of the angle evaluation error using this algorithm are discussed. We also demonstrate an application of the method to the measurement of the inclination angle of experimental parallel fringe patterns.     

A simple method for phase wraps elimination or reduction in spatial fringe patterns

In this paper, we propose a simple method for processing a 2D wrapped phase map that contains a spatial carrier signal in order to completely eliminate, or greatly reduce, the number of phase wraps in the image. The 2D Fourier transform of the wrapped phase map is calculated. Then the spectrum is shifted to the origin in frequency space. After that, the inverse 2D Fourier transform is computed. Finally, a four-quadrant arctangent function is used to calculate the angle of the complex array that was produced by the inverse 2D Fourier transform. This produces a phase map with a smaller number of 2π phase jumps than the original phase map. In some cases, all of the phase wraps are eliminated and there is therefore no need to unwrap the resultant phase map. The reduction of the number of 2π phase jumps can reduce the execution time and improve the noise performance of some phase unwrapping algorithms such as the Flynn method. The validation of the proposed algorithm is demonstrated experimentally and also via computer-simulation.     

Background removal from polarized Raman spectra of tooth enamel using the wavelet transform

A wavelet transformation method is introduced to remove the large fluorescence background from polarized Raman spectra of stained tooth enamel. This method exploits the wavelet multiresolution decomposition where the experimental Raman spectrum is decomposed into signals with different frequency components, and where the lowest frequency background and highest frequency noise are removed. This method is optimized using a simulated collection of parallel‐polarized and cross‐polarized Raman spectra of the enamel and then applied to a set of experimental data. The results show that the wavelet transform technique can extract the pure spectra from background and noise, with the depolarization ratio used to discriminate between early dental caries and sound enamel preserved. Copyright © 2010 Crown in the right of Canada. Published by John Wiley & Sons, Ltd.