Riesz transform-based digital four-step phase-shifting interferometer

A hybrid algorithm is proposed in this study for demodulating a single spatial carrier fringe pattern (FP) of interferometric measurement, which essentially combines the spatial carrier phase shift (SCPS) method and Fourier transform (FT) method. It firstly extracts three phase-shifted FPs from a single spatial carrier FP, then employs the FT method and a subtraction operation to determine the phase shift of three phase-shifted FPs, and finally retrieves the phase map using a least-square phase shift algorithm. The subtraction operation could considerably mitigate the inherent edge error of the FT method, resulting in an increase of accuracy compared with the FT method. It also does not require the background and modulation amplitude of the spatial carrier FP to be constant; thus it is robust and quite suitable for engineering. The factors that may influence the performance of the proposed algorithm are analyzed, including the random and speckle noise, carrier frequency, shape of the background, and modulation amplitude. The feasibility of the proposed algorithm is validated by two experiments, comparing them with the temporal phase-shifted method. The proposed algorithm is expected to be used in interferometric measurement under adverse environments.

In this paper, we propose a novel technique for quantitative phase measurements of vibrating structures based on additive-subtractive speckle interferometry. Additive fringe patterns (of poor visibility) corresponding to the object deformation are obtained by pulsing the laser illumination at times when the deformation is at its maximum and minimum amplitudes. A relative inter-pulse phase shift is introduced within each repetitive deformation cycle (e.g. +(pi) /2 for the maximum and -(pi) /2 for the minimum amplitudes respectively) to obtain two consecutive conjugate pairs of phase-shifted additive fringe patterns. Additionally, and inter-frame phase shift is introduced between the conjugate pairs of additive fringe patterns, which are then subtracted and rectified to provide a phase-shifted additive-subtractive fringe pattern of good visibility. Typically four such phase-shifted additive-subtractive fringe patterns with relative phase shift of (pi) /2 between each other are obtained, and these are used to calculate the deformation phase map from which the vibration measurement of a circular membrane. The ability of the technique to perform in relatively severe noisy environments is demonstrated.

A novel phase-shifting profilometry to realize temporal phase unwrapping simultaneously with the least fringe patterns

Effective camera calibration by using phase-shifting fringe patterns

The intensity distribution of fringe patterns becomes nonsinusoidal in real testing environments. Thus, the performance of existing phase shift extraction algorithms, which usually compute the desired phase shift by arccosine function or arcsine function, may be affected. In the presented paper, we report an arctangent-function-based technique to solve this disturbance. First, two orthogonal fringe patterns are constructed through subtraction and addition of two background-removed images. Second, the unequal amplitude between two new fringe patterns is eliminated using a normalization process. Third, the phase shift is determined by computing the norms of the two new images. The proposed method is fast and can be implemented easily in many applications. We verify the algorithm performance and robustness using both simulated and experimental data, indicating the high accuracy of the presented method.

The conventional phase-shifting techniques commonly suffer from frequency aliasing because of their number of phase shifts below the critical sampling rate. As a result, fringe harmonics induce ripple-like artifacts in their reconstructed phase maps. For solving this issue, this paper presents an anti-aliasing phase-measuring technique. Theoretical analysis shows that, with phase-shifting, the harmonics aliased with the fundamental frequency component of a fringe signal depend on the greatest common divisor (GCD) of the used phase shifts. This fact implies a possibility of removing such aliasing effects by selecting non-uniform phase shifts that together with 2π have no common divisors. However, even if we do so, it remains challenging to separate harmonics from the fundamental fringe signals, because the systems of equations available from the captured fringe patterns are generally under-determined, especially when the number of phase shifts is very few. To overcome this difficulty, we practically presume that all the points over the fringe patterns have an identical characteristic of harmonics. Under this constraint, using an alternate iterative least-squares fitting procedure allows us to estimate the fringe phases and the harmonic coefficients accurately. Simulation and experimental results demonstrate that this proposed method enables separating high order harmonics from as few as 4 fringe patterns having non-uniform phase shifts, thus significantly suppressing the ripple-like phase errors caused by the frequency aliasing.

In the phase measuring deflectometry, two groups of fringe patterns in orthogonal directions are usually applied to establish the correspondences between the pixel pairs on the screen and camera. Usually, 16 phase-shifting fringe patterns with different spatial frequencies are required in order to calculate the absolute phases in the conventional temporal phase unwrapping algorithms. This requirement makes the measurement inefficient and not robust against environmental noise. In this paper, an efficient phase retrieval strategy is developed, which requires only six fringe patterns. The modulating information in one-direction is obtained by first using four fringe patterns, and then it is applied to assist the phase calculation in the other direction, so that only two extra fringe patterns are needed. Subsequently the phases are unwrapped by using the geometric constraints of the software configurable optical test system without additional image acquisition. The measurement time is saved by 5/8, compared to the conventional methods. In this way, the influence of the low-frequency disturbances can be suppressed in the workshop condition. Experiments demonstrate that the proposed method can reliably retrieve the absolute phases, and it is of significance to improve the measuring efficiency and stability of in situ deflectometry.

Carrier squeezing interferometry with π/2 phase shift at the synthetic wavelength: Phase extraction in simultaneous dual-wavelength interferometry

The phase shift interferometer can be used to detect the wavefront phase of optical system and the piezoelectric transducer is widely used as a phase shifter in phase shift interferometer. However, the displacement of piezoelectric transducer presents hysteresis nonlinearity. Hence, when the equal-step phase shift algorithm (e.g. the Schwider-Hariharan phase-shifting algorithm which obtains five images and sets the phase-shifting between images to 90 degrees) is used in the phase shift interferometer to calculate the wavefront phase, the nonlinearity of piezoelectric transducer will introduce error to the 90 degrees phase-shifting between images. Then the calculated wavefront phase will contain the error. Hence, it is necessary to propose a method to diminish the phase error. The method by choosing an optimized phase-shifting algorithm has been used to mitigate the phase error. However, the method does not directly focus on piezoelectric transducer itself. In the paper, we propose a method by compensating for the hysteresis nonlinearity displacement of piezoelectric transducer to mitigate the phase error. We analyze the relationship between the nonlinearity displacement of piezoelectric transducer and phase shift error and establish the error models to analyze the relationship between the phase shift error and phase error. Furtherly, we propose the Proportion Integration Differentiation (PID) closed-loop feedback control method to compensate for the hysteresis nonlinearity displacement of piezoelectric transducer. The results indicate that, with compensated, the maximum tracking error is reduced from 18.89 % to 5.5 %. And in the ascending stage, the PV error and RMS error with compensated are less 12.35nm and 4.34nm than the PV error and RMS error without compensated. In the descending stage, the PV error and RMS error with compensated are less 19.15nm and 6.76nm than the PV error and RMS error without compensated. Hence, the proposed compensation method can improve the detection accuracy of the phase shift interferometer.

Phase measuring deflectometry (PMD) is a well-established method for determining the topography of specular freeform surfaces. A disadvantage of the classical PMD method, however, is the sequential measurement process - it requires at least six camera images of phase-shifted sinusoidal fringe patterns for one measurement. Therefore, for moving objects in industrial production, as well as for non-fixable objects such as the human cornea, the classical PMD evaluation is not suitable anymore. To overcome this problem, single-shot methods using single-side-band demodulation have been presented, which allow for a deflectometric measurement based on just one single image capture. However, this kind of evaluation does not work for complex surface geometries that result in broadband fringe patterns, since the phase is only considered globally in the Fourier space. A new single-shot evaluation method for the phase determination using the Continuous Wavelet Transform (CWT) is presented. The advantage of the wavelet transform is that the signal can be evaluated locally in both spatial and frequency space, making it possible to measure even complex reflective surfaces in motion. First measurement results are shown and compared to the classic phase-shifting evaluation for a non-moving object. Furthermore, limits and possible enhancements of this new method are discussed. Phase measuring deflectometry (PMD) is a well-established method for determining the topography of specular freeform surfaces. A disadvantage of the classical PMD method, however, is the sequential measurement process - it requires at least six camera images of phase-shifted sinusoidal fringe patterns for one measurement. Therefore, for moving objects in industrial production, as well as for non-fixable objects such as the human cornea, the classical PMD evaluation is not suitable anymore. To overcome this problem, single-shot methods using single-side-band demodulation have been presented, which allow for a deflectometric measurement based on just one single image capture. However, this kind of evaluation does not work for complex surface geometries that result in broadband fringe patterns, since the phase is only considered globally in the Fourier space. A new single-shot evaluation method for the phase determination using the Continuous Wavelet Transform (CWT) is presented. The advantage of the wavelet transform is that the signal can be evaluated locally in both spatial and frequency space, making it possible to measure even complex reflective surfaces in motion. First measurement results are shown and compared to the classic phase-shifting evaluation for a non-moving object. Furthermore, limits and possible enhancements of this new method are discussed.

Fringe projection profilometry (FPP) is extensively utilized for the 3D measurement of various specimens. However, traditional FPP typically requires at least three phase-shifted fringe patterns to achieve a high-quality phase map. In this study, we introduce a single-shot FPP method based on common path polarization interferometry. In our method, the projected fringe pattern is created through the interference of two orthogonal circularly polarized light beams modulated by a liquid crystal spatial light modulator (LC-SLM). A polarization camera is employed to capture the reflected fringe pattern, enabling the simultaneous acquisition of four-step phase-shifting fringe patterns. The system benefits from advanced anti-vibration capabilities attributable to the common path self-interference optical path design. Furthermore, the utilization of a low-coherence LED light source results in reduced noise levels compared to a laser light source. The experimental results demonstrate that our proposed method can yield 3D measurement outcomes with high accuracy and efficiency.

Digital speckle pattern interferometry (DSPI) is used to produce a fringe pattern representing speckle correlation changes. This fringe pattern has the same interpretation as that obtained with holographic interferometry. Remarkable features of this technique are the possibilities of real- time processing and software treatment. Due to these characteristics, DSPI becomes an attractive option to the non-destructive testing an evaluation. Through different analysis of the fringes, it is possible to get quantitative information about the correlation changes that gave rise to the pattern. We are going to introduce three alternatives of digital speckle to precision measurements: focal length (fringe positional changes), angle measurements (fringe visibility changes), and topography (digital fringe generation by moire). Theoretical demonstrations will be outlined and experimental results will be presented.

Fringe projective 3D profile sensor can obtain dense coordinates map of object outline fast and quantitatively. Phase unwrapping technique plays an important role in the performance of the sensor when phase-shifting fringe patterns are used. The multi-period phase shift method can considerably increase the accuracy of measuring phase value. However, due to error of intensity approximation arisen from digital grating and the spatial resolution limits on the projector and cameras of the senor, the result of unwrapped phase value is still within noise only by means of temporal phase-unwrapping. A new algorithm combining the spatial and temporal phase-unwrapping methods is presented. The algorithm, which is a good compromise between the number of needed gratings and the unwrapping reliability, is especially designed for using two groups of digital phase-shifting gratings with different period. Experiments have been carried out by selecting about thirty pairs of periods to illustrate the efficiency of temporal and spatial criterions, and the results offer the tolerance of phase calculation which is unwrapped correctly. The algorithm has been implemented on a test system, and the ratio of unwrapping successfully reaches to 80% with the repeated error 0.05 rad.

Photostress analysis system was used to study the stress distribution on a circular disk. A reflection polariscope was used to observe the surface strains from the recorded fringe patterns by digital camera. The shape of sample is a circular disk was pressed on the top and bottom and the polariscope produced fringe patterns from sample on the model being stressed. The results were analyzed, the fringe pattern looked different in various magnitude of force because the stress distribution in sample was changed with magnitude of force. Deformation can be determined from the fringe patterns.

We propose a 2D generalization to the midpoint-based empirical mode decomposition algorithm (MBEMD). Unlike with the regular bidimensional empirical mode decomposition algorithm (BEMD), we do not interpolate the upper and lower envelopes, but rather directly find the mean envelope, utilizing well defined points between two extrema of different kind (midpoints). This approach has several advantages such as improved spectral selectivity and better time performance over the regular BEMD process. The MBEMD algorithm is then applied to the task of the interferometric fringe pattern analysis, to identify its distinct components. This allows to separate the oscillatory pattern component, which is of interest, from the background, noise and possibly other spurious interferometric patterns. In result, the phase demodulation error is reduced. Flexibility of the adaptive method allows for processing correlation fringe patterns met in the digital speckle pattern interferometry as well as the regular interferometric fringe patterns without any special tuning of the algorithm.