Rajesh Langoju

Phase-shifting interferometry in the presence of nonlinear phase steps, harmonics, and noise

A phase-shifting piezo device commonly employed in phase-shifting interferometry exhibits a nonlinear response to applied voltage. Hence, a method for estimation of phase distribution in the presence of nonlinear phase steps is presented. The proposed method compensates for the harmonics present in the intensity fringe, allows the use of arbitrary phase-step values between 0 and tau rad, and does not impose constraints on the selection of particular phase-step values for minimizing nonlinearity and compensating for the harmonics. The comparison of the proposed method with other well-known benchmarking algorithms shows that our method is highly efficient and also works well in the presence of noise.

Statistical study of generalized nonlinear phase step estimation methods in phase-shifting interferometry

Signal processing methods based on maximum-likelihood theory, discrete chirp Fourier transform, and spectral estimation methods have enabled accurate measurement of phase in phase-shifting interferometry in the presence of nonlinear response of the piezoelectric transducer to the applied voltage. We present the statistical study of these generalized nonlinear phase step estimation methods to identify the best method by deriving the Cramér-Rao bound. We also address important aspects of these methods for implementation in practical applications and compare the performance of the best-identified method with other bench marking algorithms in the presence of harmonics and noise.

Statistical study and experimental verification of high-resolution methods in phase-shifting interferometry

The introduction of high-resolution phase-shifting interferometry methods such as annihilation filter, state space, multiple-signal classification, minimum norm, estimation of signal parameter via rotational invariance, and maximum-likelihood estimator have enabled the estimation of phase in an interferogram in the presence of harmonics and noise. These methods are also effective in holographic moiré where incorporating two piezoelectric transducers (PZTs) yields two orthogonal displacement components simultaneously. Typically, when these methods are used, the first step involves estimating the phase steps pixelwise; then the interference phase distribution is computed by designing a Vandermonde system of equations. In this context, we present a statistical study of these methods for the case of single and dual PZTs. The performance of these methods is also compared with other conventional benchmarking algorithms involving the single PZT. The paper also discusses the significant issue of an allowable pair of phase steps in the presence of noise using a robust statistical tool such as the Cramér-Rao bound. Furthermore, experimental validations of these high-resolution methods are presented for the estimation of single phase in holographic interferometry and for the estimation of multiple phases in holographic moiré.

An integral approach to phase shifting interferometry using a super-resolution frequency estimation method

The objective of this paper is to describe an integral approach - based on the use of a super-resolution frequency estimation method - to phase shifting interferometry, starting from phase step estimation to phase evaluation at each point on the object surface. Denoising is also taken into consideration for the case of a signal contaminated with white Gaussian noise. The other significant features of the proposal are that it caters to the presence of multiple PZTs in an optical configuration, is capable of determining An integral approach to phase shifting the harmonic content in the signal and effectively eliminating their influence on measurement, is insensitive to errors arising from PZT miscalibration, is applicable to spherical beams, and is a robust performer even in the presence of white Gaussian intensity noise.

Ultrasound image reconstruction using the finite-rate-of-innovation principle

Recently, a method of finding the spectral samples of non-periodic-finite-rate-of-innovation (NP-FRI) signals using a sum-of-sincs (SoS) sampling kernel was proposed in the literature. In the SoS approach, the kernel is repeated at a rate dependent on the delays of the FRI signal. The number of repetitions depends on both the duration and the delays of pulses constituting the FRI signal. In this paper, we show that the kernel repetition can be avoided and perfect reconstruction can be obtained by working with the SoS kernel directly provided that certain sampling criteria are satisfied. We place a lower bound on the sampling rate to ensure that exact signal reconstruction is achieved using filtered samples. To suppress the effect of noise, we use Cadzow denoising technique. Reconstruction is achieved using the annihilating filter method. We report results on data simulated using Field II software as well as real cardiac ultrasound data. The experimental results show that, with nearly 10 times less data than that required by the standard technique, the proposed method gives a comparable quality of reconstruction. The reconstruction accuracy can be controlled by choosing the model order of the NP-FRI signal appropriately.

High-resolution frequency estimation technique for recovering phase distribution in interferometers

An integral approach to phase measurement is presented. First, the use of a high-resolution technique for the pixelwise detection of phase steps is proposed. Next, the robustness of the algorithm that is developed is improved by incorporation of a denoising procedure during spectral estimation. The pixelwise knowledge of phase steps is then applied to the Vandermonde system of equations for retrieval of phase values at each pixel point. Conceptually, our proposal involves the design of an annihilating filter that has zeros at the frequencies associated with the polynomial that describes the fringe intensity. The parametric estimation of this annihilating filter yields the desired spectral information embedded in the signal, which in our case represents the phase steps. The proposed method offers the advantage of extracting the interference phase of nonsinusoidal waveforms in the presence of miscalibration error of the piezoelectric transducer. In addition, in contrast to previously reported methods, this method does not require the application of selective phase steps between data frames for nonsinusoidal waveforms.

Super-resolution Fourier transform method in phase shifting interferometry.

The paper proposes a super-resolution Fourier transform method for phase estimation in phase shifting interferometry. Incorporation of a super-resolution technique before the application of Fourier transform significantly enhances the resolution capability of the proposed method. The other salient features of the method lie in its ability to handle multiple harmonics, PZT miscalibration, and arbitrary phase steps in the optical configuration. The method does not need addition of any carrier fringes to separate the spectral contents in the intensity fringes. The proposed concept thus overcomes the limitations of other methods based on Fourier transform techniques. The robustness of the proposed method is studied in the presence of noise.

Resolution-enhanced Fourier transform method for the estimation of multiple phases in interferometry.

A phase shifting method based on high-resolution frequency estimation and Fourier transform technique is introduced. This method, also referred to as the eigenvector method, draws on the complementary strengths of both these methods. The salient feature of the method lies in its ability to handle nonsinusoidal wave-forms, multiple piezoelectric transducers, and arbitrary phase steps in an optical configuration. The method does not need the addition of carrier fringes to separate the spectral contents in the intensity fringes. The proposed concept thus overcomes the limitations of methods based on Fourier transform techniques. The robustness of the proposed method is studied in the presence of noise.

Model-based processing of a holographic moiré.

A state space model for the determination of dual phase distributions in a holographic moiré in the presence of nonsinusoidal waveforms, random noise, and miscalibration of the piezoelectric (PZT) devices is proposed. The extraction of these phase terms requires incorporating two PZTs into the moiré setup. A Toeplitz approximation method (TAM) is applied for phase determination, and modification to the Toeplitz covariance matrix formed from the phase-shifted moiré fringes by application of a denoising step in the state-feedback matrix is proposed. This step ensures that the phase terms can even be estimated at a signal-to-noise ratio much lower than that of the original TAM or by our previously suggested polynomial based method.

Estimation of multiple phases in interferometry in the presence of nonlinear arbitrary phase steps

A phase shifting device (PZT) which is commonly applied in interferometry for phase measurement, unfortunately, has a nonlinear response to the applied voltage. In certain configurations such as, holographic moiré, where incorporation of two PZTs yields multiple phase information regarding the two orthogonal displacement components, the nonlinear response of the two PZTs yields highly erroneous result. In this context, we present for the first time a method for compensating multiple nonlinearities in the PZTs. Experimental results show the feasibility of the proposed method. The statistical performance of this method is also verified by comparing with the Cramér-Rao lower bound.