Phase Derivative Estimation Research Articles

Simultaneous estimation of multiple order phase derivatives using deep learning method in digital holographic interferometry

For non-contact deformation testing, digital holographic interferometry is a prominent optical technique where the first and second order interference phase derivatives directly embed information about the strain and curvature distributions of a deformed object. Hence, reliable extraction of multiple order phase derivatives is of great practical significance; however, this problem is marred by several challenges such as the need of multiple differentiation operations, complex shearing operations and performance degradation due to noise. In this paper, we introduce a deep learning approach for the direct and simultaneous estimation of first and second order phase derivatives in digital holographic interferometry. Our method's performance is demonstrated via rigorous numerical simulations exhibiting wide range of additive white Gaussian noise and speckle noise. Moreover, we substantiate the practical efficacy of our proposed method for processing deformation fringes acquired via digital holographic interferometry.

Phase derivative estimation in digital holographic interferometry using a deep learning approach.

In digital holographic interferometry, reliable estimation of phase derivatives from the complex interference field signal is an important challenge since these are directly related to the displacement derivatives of a deformed object. In this paper, we propose an approach based on deep learning for direct estimation of phase derivatives in digital holographic interferometry. Using a Y-Net model, our proposed approach allows for simultaneous estimation of phase derivatives along the vertical and horizontal dimensions. The robustness of the proposed approach for phase derivative extraction under both additive white Gaussian noise and speckle noise is shown via numerical simulations. Subsequently, we demonstrate the practical utility of the method for deformation metrology using experimental data obtained from digital holographic interferometry.

Non-invasive deformation metrology using subspace analysis in digital holographic interferometry

For non-invasive precision measurement of deformation using digital holographic interferometry (DHI), reliable estimation of interference phase from complex fringe signal is required. This article introduces a method for estimation of interference phase in DHI. The proposed method is based on computing roots of a frequency estimation function obtained from noise subspace of the fringe signal. The main advantage of the proposed method is robustness against severe noise. In addition, the method also offers the capability of direct phase derivative estimation. The performance of the proposed method is validated using numerical simulations. Our simulation results show that even for signal to noise ratio as low as −5 dB, the proposed method offers root mean square errors of less than 1 radian for phase retrieval. The practical utility of the method for non-invasive deformation analysis is demonstrated using experimental results in DHI.

Wiener Teager–Kaiser energy method for phase derivative estimation: application to speckle interferometry

We present a Wiener Teager–Kaiser approach for phase derivative estimation from a single speckle correlation fringe. In principle, the Teager–Kaiser operator estimates the energy of the fringe pattern and extracts its phase derivatives using an energy separation algorithm. However, in the estimation of the energy, this operator presents a computation error mainly due to a high frequency component. In this work, we addressed this error in mean square error sense by applying the Wiener filter on the operator prior to phase derivative computation. The performance of our proposed method on simulated and real fringe improves significantly the accuracy of the Teager–Kaiser operator.

Estimation of Phase and Its Higher Order Derivatives from a Single Complex Interferogram

This paper reports a method for the simultaneous estimation of unwrapped phase and higher order phase derivatives from a single phase fringe pattern recorded in an optical interferometric setup, thereby overcoming substantial barriers to achieving such measurements. The proposed method considers the interference phase as a weighted linear combination of Gaussian radial basis functions defined along a given row or column at a time. The Gaussian radial basis functions are defined with a constant standard deviation and equally spaced centers. Unscented Kalman filter is employed for the accurate estimation of the weights of the basis functions using the state space representation of the spatial evolution of the interferogram. The estimated weights along with the numerically computed gradients of the basis functions also provide the estimations of phase derivatives of arbitrary order. The proposed representation of interference phase along with the unscented Kalman filter provides high robustness against the speckle noise. Simulation study is preformed to evaluate the dependence of the phase and phase derivative estimation accuracy on the selection of basis dimension and the noise level. Experimental results demonstrate the practical applicability of the proposed method.

Determination of phase derivatives from a single fringe pattern using Teager Hilbert Huang transform

In this paper, a novel sequential algorithm for the estimation of phase derivatives from a single fringe pattern using electronic speckle pattern interferometry (ESPI) is proposed. The algorithm is based on empirical mode decomposition (EMD), vortex operator (VO) and Teager–Kaiser energy operator (TKEO). The empirical mode decomposition normalizes the fringe pattern; while vortex operator provides a 2D complex image and the phase derivatives are obtained using a novel image demodulation method called discrete higher order image demodulation algorithm (DHODA). Unlike phase shifting and Fourier transform methods, the proposed method does not require complex experimental setup or more than one fringe pattern for each deformation state. The proposed method is also able to provide phase derivatives in both thex andydirections from a single fringe pattern, which is difficult to achieve using shearography. Since the algorithm provides unwrapped phase derivatives directly, it does not require separate phase unwrapping process. Hence it is suitable for dynamic strain and curvature measurement. The proposed algorithm is validated by both simulation and experiment. The results are found to be accurate and the method requires less computation time than existing phase demodulation techniques.

Acquisition of multiple phase derivatives from a single frame in digital holographic interferometry

This paper proposes a technique to extract two-dimensional multiple phase derivatives (strains, slopes, and shearing strains) from a single frame of the interference fringe pattern recorded in an optical setup with multiple illumination beams. In the proposed method, the interference field is modeled as a multicomponent complex sinusoid in an analysis window around each pixel. The frequencies of these sinusoids provide the estimates of the spatially varying phase derivatives. Matrix enhancement and matrix pencil-based technique is utilized to accurately estimate the multiple frequencies. A numerical example is provided to test the performance of the proposed method in the presence of noise. Experimental results are given to substantiate its applicability in the simultaneous estimation of multiple phase derivatives in a multiwave digital holographic interferometry setup.

Phase derivative estimation from a single interferogram using a Kalman smoothing algorithm.

We report a technique for direct phase derivative estimation from a single recording of a complex interferogram. In this technique, the interference field is represented as an autoregressive model with spatially varying coefficients. Estimates of these coefficients are obtained using the Kalman filter implementation. The Rauch-Tung-Striebel smoothing algorithm further improves the accuracy of the coefficient estimation. These estimated coefficients are utilized to compute the spatially varying phase derivative. Stochastic evolution of the coefficients is considered, which allows estimating the phase derivative with any type of spatial variation. The simulation and experimental results are provided to substantiate the noise robustness and applicability of the proposed method in phase derivative estimation.

Matrix pencil based phase derivative estimation in digital holographic interferometry

A novel technique is proposed for the direct two-dimensional phase derivatives estimation based on the matrix pencil approach in digital holographic interferometry. The interference field is represented as a two-dimensional complex sinusoidal signal in an analysis window around each pixel. The method provides high resolution frequency estimates and does not require the two-dimensional search in the spectral domain. A simple denoising procedure is proposed based on singular value decomposition of the interference field. The uncertainty in the frequency estimate is utilized as a parameter to identify the regions containing high fringe concentration on the object surface. Simulation and experimental results are provided to show the noise robustness and the practical applicability of the proposed method.

Multiple phase derivative estimation using autoregressive modeling in holographic interferometry

A novel technique is proposed for the direct and simultaneous estimation of multiple phase derivatives from a deformation modulated carrier fringe pattern in a multi-wave holographic interferometry set-up. The fringe intensity is represented as a spatially-varying autoregressive (SVAR) model. The spatially-varying coefficients of the SVAR model are derived using a forward–backward approach of linear estimation of the fringe intensity. Using these coefficients, the poles of the SVAR model transfer function are computed and the angles of these poles provide the estimation of phase derivatives. The estimation of carrier frequency is performed by the proposed method using a reference interferogram. Simulation results are provided in the presence of noise and fringe amplitude modulation.

Direct phase derivative estimation using difference equation modeling in holographic interferometry

A new method is proposed for the direct phase derivative estimation from a single spatial frequency modulated carrier fringe pattern in holographic interferometry. The fringe intensity in a given row/column is modeled as a difference equation of intensity with spatially varying coefficients. These coefficients carry the information on the phase derivative. Consequently, the accurate estimation of the coefficients is obtained by approximating the coefficients as a linear combination of the predefined linearly independent basis functions. Unlike Fourier transform based fringe analysis, the method does not call for performing the filtering of the Fourier spectrum of fringe intensity. Moreover, the estimation of the carrier frequency is performed by applying the proposed method to a reference interferogram. The performance of the proposed method is insensitive to the fringe amplitude modulation and is validated with the simulation results.

Digital holographic moiré for the direct and simultaneous estimation of strain and slope fields.

A novel method is proposed for the direct and simultaneous estimation of multiple phase derivatives corresponding to strain and slope fields from a single moiré fringe pattern in digital holographic moiré. The interference field in a given row/column is a multicomponent complex exponential signal and is represented as a spatially-varying autoregressive (SVAR) process. The spatially-varying coefficients of the SVAR model are computed by approximating them as the linear combination of linearly independent basis functions. Further, the spatially varying poles of the transfer function corresponding to the SVAR model are computed which provide the accurate estimation of the multiple phase derivatives. The simulation and experimental results are provided to substantiate the effectiveness of the proposed method.

Estimation of phase derivatives using discrete energy separation algorithm in digital holographic interferometry.

This Letter proposes a new method for the estimation of the first- and second-order phase derivatives corresponding to strain and curvature from a single fringe pattern in digital holographic interferometry. The method is based on a discrete energy separation algorithm, which provides a biased phase derivative estimate in a noisy environment. Subsequently, the least-squares spline approximation with optimal number of knots selection technique is used to obtain the accurate estimation of phase derivatives. The accuracy and computational efficiency of the proposed method is validated with simulation and experimental results.

Application of complex-lag distributions for estimation of arbitrary order phase derivatives in digital holographic interferometry

This Letter proposes a method to estimate phase derivatives of arbitrary order in digital holographic interferometry. Based on the desired order, the generalized complex-lag distribution is computed from the reconstructed interference field. Subsequently, the phase derivative is estimated by tracing the peak of the distribution. Simulation and experimental results are presented to validate the method's potential.

Estimation of dynamically varying displacement derivatives using fringe projection technique

This paper presents a pseudo Wigner-Ville-distribution-based method in fringe projection for analyzing temporal behavior of the displacement derivative for a continuously deformed object. In the proposed method, a computer generated fringe pattern is projected on an object undergoing dynamic deformation, and the reflected intensity is recorded in the form of video, i.e., a stack of images are captured sequentially by a CCD camera. Each image represents a recorded fringe pattern at a particular time instant whose phase contains information about the instantaneous out-of-plane displacement or deformation with respect to the undeformed object, and the corresponding spatial phase derivative relates to the displacement derivative. Subsequently, pseudo Wigner-Ville distribution is used for instantaneous phase derivative estimation from the stack of images. Simulation and experimental results are presented to demonstrate the method's potential.

Estimation of displacement derivatives in digital holographic interferometry using a two-dimensional space-frequency distribution

The paper introduces a two-dimensional space-frequency distribution based method to directly obtain the unwrapped estimate of the phase derivative which corresponds to strain in digital holographic interferometry. In the proposed method, a two-dimensional pseudo Wigner-Ville distribution of the reconstructed interference field is evaluated and the peak of the distribution provides information about the phase derivative. The presence of a two-dimensional window provides high robustness against noise and enables simultaneous measurement of phase derivatives along both spatial directions. Simulation and experimental results are presented to demonstrate the method's applicability for phase derivative estimation.

Estimation of the phase derivative using an adaptive window spectrogram

The paper introduces an adaptive-window-spectrogram-based method to directly estimate the phase derivative from a single fringe pattern. The proposed method relies on estimating the phase derivative using spectrogram peak detection for a set of different window lengths. Then the optimal window length is selected from the set by resolving the estimator's bias variance trade-off using the intersection of confidence intervals rule. Finally, the phase derivative estimate corresponding to the optimum window is selected. The method's applicability to phase derivative estimation is demonstrated using simulation and experimental results.

Adaptive window Wigner-Ville-distribution-based method to estimate phase derivative from optical fringes

We introduce an adaptive window Wigner-Ville-distribution-based method to directly estimate the phase derivative from a single fringe pattern. In the proposed method, the phase derivative is estimated by using the peak detection of the pseudo-Wigner-Ville distribution for a set of different window lengths. Then the optimal window length is selected from the set by resolving the estimator's bias variance trade-off, using the intersection of confidence intervals rule. Finally, the phase derivative estimate corresponding to the optimum window is selected. Simulation and experimental results are presented to demonstrate the method's applicability for the phase derivative estimation.

Polynomial Wigner–Ville distribution-based method for direct phase derivative estimation from optical fringes

This paper proposes a polynomial Wigner–Ville distribution-based method to directly estimate phase derivative from a single fringe pattern. In the proposed method, we evaluate the polynomial Wigner–Ville distribution along each row/column of the given fringe pattern. The peak of the polynomial Wigner–Ville distribution is used as the phase derivative estimator. To improve the robustness of the distribution against the artifacts or interference terms, a windowed form of the polynomial Wigner–Ville distribution is used. Simulation and experimental results are presented which validate the method's applicability for direct phase derivative estimation.

Estimation of phase derivatives using discrete chirp-Fourier-transform-based method

Estimation of phase derivatives is an important task in many interferometric measurements in optical metrology. This Letter introduces a method based on discrete chirp-Fourier transform for accurate and direct estimation of phase derivatives, even in the presence of noise. The method is introduced in the context of the analysis of reconstructed interference fields in digital holographic interferometry. We present simulation and experimental results demonstrating the utility of the proposed method.