Temporal phase unwrapping of digital hologram sequences

Phase unwrapping algorithm based on phase diversity wavefront reconstruction and virtual Hartmann-Shack technology

Phase unwrapping (PU) algorithms play a crucial role in various phase measurement techniques. Traditional algorithms cannot work well in strong noise environments, which makes it very difficult to obtain the accurate absolute phase from the noisy wrapped phase. In this Letter, we introduce a novel, to the best of our knowledge, phase unwrapping algorithm named PD-VHS. This algorithm innovatively employs point spread function (PSF) filtering to eliminate noise from the wrapped phase. Furthermore, it combines a phase diversity (PD) wavefront reconstruction technology with a virtual Hartmann-Shack (VHS) technology for phase reconstruction and phase unwrapping of the filtered PSFs. In simulations, hundreds of random noise wrapped phases, containing the first 45 Zernike polynomials (excluding piston and the two tilt terms) and the wavefront RMS = 0.5λ and 1λ, are used to compare the classical quality-map guided algorithm, the VHS algorithm with decent noise immunity, with our PD-VHS algorithm. When signal-to-noise ratio (SNR) drops to just 2 dB, the mean root mean square errors (RMSEs) of the residual wavefront between the unwrapped result and the absolute phase of the quality-map guided algorithm and the VHS algorithm are up to 3.99λ, 0.44λ, 4.29λ, and 0.85λ, respectively; however, our algorithm RMSEs are low: 0.11λ and 0.17λ. Simulation results demonstrated that the PD-VHS algorithm significantly outperforms the quality-map guided algorithm and the VHS algorithm under large-scale noise conditions.

Unifying temporal phase unwrapping framework using deep learning

Temporal phase unwrapping (TPU) is significant for recovering an unambiguous phase of discontinuous surfaces or spatially isolated objects in fringe projection profilometry. Generally, temporal phase unwrapping algorithms can be classified into three groups: the multi-frequency (hierarchical) approach, the multi-wavelength (heterodyne) approach, and the number-theoretic approach. For all of them, extra fringe patterns of different spatial frequencies are required for retrieving the absolute phase. Due to the influence of image noise, people have to use many auxiliary patterns for high-accuracy phase unwrapping. Consequently, image noise limits the efficiency and the measurement speed greatly. Further, these three groups of TPU algorithms have their own theories and are usually applied in different ways. In this work, for the first time to our knowledge, we show that a generalized framework using deep learning can be developed to perform the TPU task for different groups of TPU algorithms. Experimental results show that benefiting from the assistance of deep learning the proposed framework can mitigate the impact of noise effectively and enhance the phase unwrapping reliability significantly without increasing the number of auxiliary patterns for different TPU approaches. We believe that the proposed method demonstrates great potential for developing powerful and reliable phase retrieval techniques.

Rapid deformation analysis in digital holographic interferometry using graphics processing unit accelerated Wigner-Ville distribution

High-speed measurement of temporally varying displacement derivatives of a deformed object is an important and challenging problem in precision metrology. This paper proposes an approach for temporal deformation analysis using digital holographic interferometry, where a series of digital holograms is recorded corresponding to a test object subjected to variable load. Numerical reconstruction followed by superposition of these holograms provides a huge stack of interference or fringe pattern data, which needs to be processed to extract the displacement derivative information. This operation involves great computational complexity, and we propose a highly efficient Wigner-Ville distribution method based on a graphics processing unit to address the challenge. Experimental results clearly demonstrate the viability of the proposed approach for high-speed deformation analysis in digital holographic interferometry.

Weighted least-squares phase-unwrapping algorithm based on the orientation coherence for discontinuous optical phase patterns

Phase unwrapping is one of the key steps of optical interferogram analysis, among which phase discontinuity is still a challenge. In this paper, we propose a new weighted least-squares phase-unwrapping algorithm for discontinuous optical phase patterns. In the proposed algorithm, the orientation coherence is introduced to define the new weighting coefficient, which can accurately show the wrapped phase quality. According to our proposed algorithm, the new weighting coefficient has a good performance on distinguishing the continuous regions and the discontinuous regions in wrapped phase patterns. This advantage of our algorithm can ensure a more reliable unwrapped result for discontinuous optical phase patterns. We test the proposed algorithm on the computer-simulated speckle phase images and two experimentally obtained phase images, respectively, and compare them with the other five widely used methods. The experimental results demonstrate the performance of our new weighted least-squares phase-unwrapping algorithm.

Time-directional filtering of wrapped phase for observing transient phenomena with parallel phase-shifting interferometry

Recent development of parallel phase-shifting interferometry (PPSI) enables accurate measurement of time-varying phase maps. By combining a high-speed camera with PPSI, it became possible to observe not only time-varying but also fast phenomena including fluid flow and sound in air. In such observation, one has to remove static phase (time-invariant or slowly-varying phase unrelated to the phenomena of interest) from the observed phase maps. Ordinarily, a signal processing method for eliminating the static phase is utilized after phase unwrapping to avoid the 2π discontinuity which can be a source of error. In this paper, it is shown that such phase unwrapping is not necessary for the high-speed observation, and a time-directional filtering method is proposed for removing the static phase directly from the wrapped phase without performing phase unwrapping. In addition, experimental results of simultaneously visualizing flow and sound with 42 000 fps are shown to illustrate how the time-directional filtering changes the appearance. A MATLAB code is included within the paper (also in https://goo.gl/N4wzdp) for aiding the understanding of the proposed method.

High-precision real-time 3D shape measurement using a bi-frequency scheme and multi-view system

High-speed and high-precision 3D shape measurement plays a central role in diverse applications such as automatic online inspection, robotics control, and human-computer interaction. Conventional multi-frame phase-shifting-based fringe projection profilometry techniques face inherent trade-offs between the speed and measurement precision, which are fundamentally limited by the fringe density and extra pattern projections used for de-ambiguity of fringe orders. Increasing the frequency of the projection fringes can obviously improve the measurement precision; however, it creates difficulties in the subsequent phase unwrapping. For this reason, to date, the frequency of the fringes in typical real-time 3D shape measurement techniques is generally less than 30 to guarantee a reasonable reliability of phase unwrapping. To overcome this limitation, a bi-frequency phase-shifting technique based on a multi-view fringe projection system is proposed, which significantly enhances the measurement precision without compromising the measurement speed. Based on the geometric constraints in a multi-view system, the unwrapped phase of the low-frequency (10-period) fringes can be obtained directly, which serves as a reference to unwrap the high-frequency phase map with a total number of periods of up to 160. Besides, the proposed scheme with 10-period and 160-period fringes is suitable for slightly defocusing projection, allowing a higher projection rate and measurement speed. Experiments on both static and dynamic scenes are performed, verifying that our method can achieve high-speed and high-precision 3D measurement at 300 frames per second with a precision of about 50 μm.

Flexible cross-correlated (C2) imaging method for the modal content characterization in a broad range of wavelengths

We demonstrate a flexible cross-correlated (C²) imaging method in the time domain by application of a tunable and highly flexible light source. An advantage of the flexible C² method is shown by characterization of the step-index fiber (SMF28) over a broad range of wavelengths from 870nm to 1090nm and by the modal analysis of the distributed modal filtering (DMF) rod fiber within a wavelength range from 1050nm to 1090nm. Also, the influence of the spectral shape and bandwidth on the imaging trace is investigated by deliberately adjusting the input spectrum of the light source. The modal intensity as well as the phase distribution are extracted by the alternative method of 2D FT filtering. Being exceptionally tunable the flexible C² method gives an ability to adapt the system's parameters in a desired manner satisfying even measurements of very specific fiber designs opening up new possibilities for advanced modal characterization of fibers over broad range of wavelengths.

Flexible calibration of phase-to-height conversion in fringe projection profilometry

A flexible new technique is presented to calibrate the monocular system of phase-based fringe projection profilometry, which is made of a camera, a projector, and a computer. The proposed algorithm mainly consists of a more flexible phase-to-height conversion model and a minimum norm solution, followed by a nonlinear optimization based on the maximum likelihood criterion. In the whole calibration procedure, this method only requires the camera to capture a few two-dimensional checkerboard target images and several deformed fringe images with at least three different orientations. The proposed technique neither subjects to certain limitations nor measures additional geometry parameters in advance. Also, no highly precise gauge blocks or extra reference phases are involved. In contrast with the existing methods, the proposed technique is easier to use and more flexible. Experiments have been performed to validate the performance of this technique.

Invalid phase values removal method for absolute phase recovery

A novel approach is presented for more effectively removing invalid phase values in absolute phase recovery. The approach is based on a detailed study involving the types and cases of invalid phase values. Meanwhile, some commonalities of the existing removal algorithms also are thoroughly analyzed. It is well known that rough absolute phase and fringe order maps can very easily be obtained by temporal phase unwrapping techniques. After carefully analyzing the components and fringe order distribution of the rough fringe order map, the proposed method chiefly adopts an entirely new strategy to refine a pure fringe order map. The strategy consists of three parts: (1) the square of an image gradient, (2) subregion areas of the binary image, and (3) image decomposition and composition. In combination with the pure fringe order map and a removal criterion, the invalid phase values can be identified and filtered out from the rough absolute phase map. This new strategy not only gets rid of the limitations of traditional removal methods but also has a two-fold function. The paper also offers different metrics from the experiment to evaluate the quality of the final absolute phase. In contrast with other removal methods, experimental results have verified the feasibility, effectiveness, and superiority of the proposed method.

Temporal phase-unwrapping of static surfaces with 2-sensitivity fringe-patterns

Here we describe a 2-step temporal phase unwrapping formula that uses 2-sensitivity demodulated phases for measuring static surfaces. The first phase demodulation has at most 1-wavelength sensitivity and the second one is G-times (G>>1.0) more sensitive. Measuring static surfaces with 2-sensitivity fringe patterns is well known and recent published methods combine 2-sensitivities measurements mostly by triangulation. Two important applications for our 2-step unwrapping algorithm is profilometry and synthetic aperture radar (SAR) interferometry. In these two applications the object or surface being analyzed is static and highly discontinuous; so temporal unwrapping is the best strategy to follow. Phase-demodulation in profilometry and SAR interferometry is very similar because both share similar mathematical models.

Quality-guided phase unwrapping implementation: an improved indexed interwoven linked list

Quality-guided phase unwrapping (QGPU) is a widely used technique, and an adjoin list plays a very important role in the QGPU process. Indexed interwoven linked list (I2L2) is a data structure for implementing the adjoin list. In this paper, we propose three improvements on the I2L2. The first improvement is resumed searching, which records the highest nonempty level in the I2L2 and reduces the computational redundancy; the second is an adaptive mapping between the quality values and the I2L2 levels, which reduces the effect of concentrated quality value distribution. Last, I2L2-H, a new variant of the I2L2 combining the advantages of both the I2L2 and heap, is developed. With these three improvements, the improved I2L2 is over 6 times faster than the original one in the best cases, and it can process large phase maps in almost real time.

From speckle pattern photography to digital holographic interferometry [Invited]

Speckles are inherently an interference phenomenon produced when an optically rough surface or a turbulent medium introduces some degree of randomness to a reflected or a transmitted electromagnetic field. Speckles are often nuisance in coherent image formation. Speckle patterns are however a useful tool for displacement and deformation as well as vibration and stress analysis. The development of speckle photography to speckle interferometry and digital holographic interferometry is described in this paper.

Quality-guided phase unwrapping technique: comparison of quality maps and guiding strategies

Quality-guided phase unwrapping is a widely used technique with different quality definitions and guiding strategies reported. It is thus necessary to do a detailed comparison of these approaches to choose the optimal quality map and guiding strategy. For quality maps, in the presence of noise, transform-based methods are found to be the best choice. However in the presence of discontinuities, phase unwrapping is itself unresolved and hence quality-guided phase unwrapping is not sufficient. For guiding strategies, classical, two-section, and stack-chain guiding strategies are chosen for comparison. If accuracy is the foremost criterion then the classical guiding strategy with a data structure of indexed interwoven linked list is best. If speed is of essence then the stack-chain guiding strategy is the one to use.

Dual-frequency pattern scheme for high-speed 3-D shape measurement

A novel dual-frequency pattern is developed which combines a high-frequency sinusoid component with a unit-frequency sinusoid component, where the high-frequency component is used to generate robust phase information, and the unit-frequency component is used to reduce phase unwrapping ambiguities. With our proposed pattern scheme, phase unwrapping can overcome the major shortcomings of conventional spatial phase unwrapping: phase jumping and discontinuities. Compared with conventional temporal phase unwrapped approaches, the proposed pattern scheme can achieve higher quality phase data using a less number of patterns. To process data in real time, we also propose and develop look-up table based fast and accurate algorithms for phase generation and 3-D reconstruction. Those fast algorithms can be applied to our pattern scheme as well as traditional phase measuring profilometry. For a 640 x 480 video stream, we can generate phase data at 1063.8 frames per second and full 3-D coordinate point clouds at 8.3 frames per second. These achievements are 25 and 10 times faster than previously reported studies.

Design considerations in projection phase-shift moiré topography based on theoretical analysis of fringe formation

Moiré topography is a well-established optical technique to measure the shape of three-dimensional surfaces, based on the geometric interference between an optical grid and its image deformed by an object surface. The technique produces fringes that represent contours of equal height, and from the recordings of several phase-shifted topograms surface height coordinates can be calculated. To perform these calculations, it is assumed that object height variation is small in comparison with the measurement setup dimensions, and this approximation leads to systematic errors in measurement accuracy. We present the mathematical description of the fringe formation process in projection moiré topography, and on the basis of these equations we establish the relation between setup geometry and upper limits of the systematic measurement errors. We derive the equations that determine design specifications needed to reduce the effects of approximations to be below the measurement resolution of the setup. It is shown that setup geometry should be adapted to the gray-scale measurement resolution of the imaging system. We show that, using an iterative correction from one fringe order to the next, measurement accuracy can be maintained over the entire object depth.

High-speed digital holographic interferometry for vibration measurement

A system based on digital holographic interferometry for the measurement of vibrations is presented. A high-power continuous laser (10 W) and a high-speed CCD camera are used. Hundreds of holograms of an object that has been subjected to dynamic deformation are recorded. The acquisition speed and the time of exposure of the detector are determined by the vibration frequency. Two methods are presented for triggering the camera in order to acquire at a given phase of the vibration. The phase of the wavefront is calculated from the recorded holograms by use of a two-dimensional digital Fourier-transform method. The deformation of the object is obtained from the phase. By combination of the deformations recorded at different times it is possible to reconstruct the vibration of the object.

Polarization microscopy by use of digital holography: application to optical-fiber birefringence measurements

We present a digital holographic microscope that permits one to image polarization state. This technique results from the coupling of digital holographic microscopy and polarization digital holography. The interference between two orthogonally polarized reference waves and the wave transmitted by a microscopic sample, magnified by a microscope objective, is recorded on a CCD camera. The off-axis geometry permits one to reconstruct separately from this single hologram two wavefronts that are used to image the object-wave Jones vector. We applied this technique to image the birefringence of a bent fiber. To evaluate the precision of the phase-difference measurement, the birefringence induced by internal stress in an optical fiber is measured and compared to the birefringence profile captured by a standard method, which had been developed to obtain high-resolution birefringence profiles of optical fibers.