Multiple-wavelength phase-shifting interferometry

Geometric-feature-based approach to human face reconstruction with high measurement speed

This paper presents a method based on geometry for three-dimensional (3D) face reconstruction without the need for additional images, hardware components, or objects. In our proposed method, we consider part of the nose as the feature region because its shape remains almost constant during the measurement. The geometry of this region was used to provide cues for phase unwrapping. We first spatially unwrap the phase and determine the integer multiple of 2π to be added by comparing the recovered result of the feature region and its actual shape. Then, the face can be reconstructed with the acquired absolute phase. Experimental results demonstrated that our method is capable of reconstructing a dynamic face with high measurement speed, and only three phase-shifted fringes are required per frame.

Adaptive horizontal scaling method for speckle-assisted fringe projection profilometry

Phase-shifting method is widely used in fringe projection profilometry to obtain high-precision wrapped phase maps. The wrapped phase map needs to be converted to an absolute phase map to recover 3D information. The speckle pattern based phase unwrapping method requires only one additional auxiliary pattern, showing great potential for fast 3D measurements. In this paper, a speckle assisted four-steps phase-shifting method was proposed for 3D measurements. This method requires five structured light patterns to complete 3D measurements, including four-steps phase-shifting fringe patterns and a speckle pattern which is used to remove phase ambiguity. The main challenge of speckle based phase unwrapping method is to overcome the mismatch problem which often occurs in some very steep surfaces. In order to improve the speckle matching accuracy, an adaptive horizontal scaling method was proposed. A voting strategy based on phase-connected regions was proposed to reduce the computational overhead. The experiments demonstrate its superior performance, and an accuracy of 0.21 mm was achieved.

Semi-Global Matching Assisted Absolute Phase Unwrapping

Measuring speed is a critical factor to reduce motion artifacts for dynamic scene capture. Phase-shifting methods have the advantage of providing high-accuracy and dense 3D point clouds, but the phase unwrapping process affects the measurement speed. This paper presents an absolute phase unwrapping method capable of using only three speckle-embedded phase-shifted patterns for high-speed three-dimensional (3D) shape measurement on a single-camera, single-projector structured light system. The proposed method obtains the wrapped phase of the object from the speckle-embedded three-step phase-shifted patterns. Next, it utilizes the Semi-Global Matching (SGM) algorithm to establish the coarse correspondence between the image of the object with the embedded speckle pattern and the pre-obtained image of a flat surface with the same embedded speckle pattern. Then, a computational framework uses the coarse correspondence information to determine the fringe order pixel by pixel. The experimental results demonstrated that the proposed method can achieve high-speed and high-quality 3D measurements of complex scenes.

Ultrahigh-dynamic-range wavefront sensor based on absolute double-slit interferometry

This Letter reports a new, to the best of our knowledge, technique for the quality testing of steep optical samples by introducing an absolute interferometry method based on a double-slit interference experiment. We determine the quality of the sample with an ultrahigh-dynamic-range wavefront sensor by determining the deformation of the central fringe of the double-slit interferometer recorded for two different separations of the slits. The transmission function of the double slit is implemented on an amplitude spatial light modulator. Therefore, the slits' location can be easily displaced over the entire area of the sample's wavefront. We applied the proposed method on two samples: a microscope slide and a conventional ophthalmic lens, and maximum absolute phase variations of 0.33 and 26.7 rad were measured, respectively. Our estimation shows that an absolute phase variation of about 700 rad can be measured with this method.

Digital image correlation assisted absolute phase unwrapping

This paper presents an absolute phase unwrapping method for high-speed three-dimensional (3D) shape measurement. This method uses three phase-shifted patterns and one binary random pattern on a single-camera, single-projector structured light system. We calculate the wrapped phase from phase-shifted images and determine the coarse correspondence through the digital image correlation (DIC) between the captured binary random pattern of the object and the pre-captured binary random pattern of a flat surface. We then developed a computational framework to determine fringe order number pixel by pixel using the coarse correspondence information. Since only one additional pattern is used, the proposed method can be used for high-speed 3D shape measurement. Experimental results successfully demonstrated that the proposed method can achieve high-speed and high-quality measurement of complex scenes.

Pixel MTF response effect on non-null interferometry

The pixel modulation transfer function response degrades the contrast of non-null interferometric surface figure measurements. We experimentally quantify this effect for spatial frequencies ranging from 0 to 363 lp/mm (≈3.33 times the Nyquist limit). Our results show a low SNR spatial frequency band that behaves like a low-pass filter for sub-Nyquist interferometry and a stop-band filter for multiple-wavelength phase-shifting interferometry. We also introduce a multiple-mode, multiple-wavelength interferometry approach to measure optical surfaces with slope departure angles mapping to spatial frequencies in this low SNR band. The extended measurement range of this approach is achieved without using a sparse-array detector.

Parameter optimization of frequency sweeping digital holography for the measurement of ground optical surfaces

This paper describes the dependence of the precision of digital holographic methods on measurement parameters. The predominantly discussed parameters are illumination intensity and its homogeneity, surface microroughness, the influence of measurement geometry, as well as object shape, since most of them can be optimized by experimental arrangement. Frequency sweeping digital holography as well as dual-wavelength digital holography in the Fourier arrangement are tested and the results are discussed. It transpires that the methods are not very sensitive to object microroughness or overall reflectivity. Instead, it is the similarity of signal and reference waves that has the highest impact on measurement. After parameter optimization, the holographic methods can be advantageously used for ground surface measurements in optical workshops.

Fast phase extraction of a synthetic wavelength from a dual-wavelength interferogram through Hilbert transformation

Phase imaging is an important measurement method in optical metrology, and phase extraction is one of its essential components. A method using one-frame dual-wavelength interferometry (DWI) for fast phase extraction is proposed. Using this method, the Hilbert transform (HT) is applied twice to a wavelength-multiplexed interferogram to obtain two images. The HT is then applied twice to the sum of the squares of these two images. The synthetic wavelength phase can then be obtained. Compared with other common multiple-frame DWI methods, the proposed approach requires only one multiplexed interferogram and does not have to consider parameter changes in interferograms of different frames. Furthermore, the proposed method was determined to require the least calculation time, and the simulation and experimental results confirmed its high accuracy and efficiency. It suggests that the proposed method is suitable for real-time measurements.

Unambiguous measurement range and error tolerance in dual-wavelength interferometry

In dual or multiwavelength interferometry, the traditional equivalent wavelength method is widely used for phase recovery to enlarge the unambiguous measurement range (UMR). In fact, however, this method ignores information of size and sign (positive or negative) of single wavelength wrapped phases, and the extension of the UMR is not sufficient. For the reflective measurement, the largest UMR of the dual or multiwavelength interferometry is half of the least-common multiple (LCM) of single wavelengths, called the LCM effective wavelength, which is often several times the equivalent wavelength. But why do we often use the equivalent wavelength and seldom use the larger UMR in practice? Existing research reveals that the actual UMR is related to the measurement error of single-wavelength-wrapped phases, and half of the LCM effective wavelength is only the theoretical value. But how do errors affect the UMR? We think the quantitative analysis and description are lacking. In this paper, we continue to study this problem, analyze it in a graphical method, and give quantitative descriptions. The simulation experiments are carried out and verify our analysis.

State-of-the-art active optical techniques for three-dimensional surface metrology: a review [Invited]

This paper reviews recent developments of non-contact three-dimensional (3D) surface metrology using an active structured optical probe. We focus primarily on those active non-contact 3D surface measurement techniques that could be applicable to the manufacturing industry. We discuss principles of each technology, and its advantageous characteristics as well as limitations. Towards the end, we discuss our perspectives on the current technological challenges in designing and implementing these methods in practical applications.

Absolute phase measurement with four patterns based on variant shifting phases

Fringe projection profilometry has been proverbially utilized for measuring the shapes of objects. A common challenge in those systems is to accurately obtain a smooth absolute phase. Many new methods have been proposed to address this challenge. In this paper, we discuss a technique based on variant shifting phases. This approach embeds codewords into the shifting phase and only needs four patterns. However, reliable measurement results are difficult to achieve with a large number of codewords because of the phase errors. To address this shortcoming, we present a robust coding method that embeds a specific code sequence into the shifting phase and can generate more than 36 periods. The fringe order is determined using unique three-adjacent-codes combining the current period and its neighbors. An error correction algorithm is also proposed to optimize the codewords. The proposed method is experimentally verified using an established measurement system. The result shows that the proposed method is robust and efficient.

Fringe harmonics elimination in multi-frequency phase-shifting fringe projection profilometry

In fringe projection profilometry, the purpose of using two- or multi-frequency fringe patterns is to unwrap the measured phase maps temporally. Using the same patterns, this paper presents a least squares algorithm for, simultaneously with phase-unwrapping, eliminating the influences of fringe harmonics induced by various adverse factors. It is demonstrated that, for most of the points over the measured surface, projecting two sequences of phase-shifting fringe patterns having different frequencies enables providing sufficiently many equations for determining the coefficient of a high order fringe harmonic. As a result, solving these equations in the least squares sense results in a phase map having higher accuracy than that depending only on the fringe patterns of a single frequency. For the other few points which have special phases related to the two frequencies, this system of equations becomes under-determined. For coping with this case, this paper suggests an interpolation-based solution which has a low sensitivity to the variations of reflectivity and slope of the measured surface. Simulation and experimental results verify that the proposed method significantly suppresses the ripple-like artifacts in phase maps induced by fringe harmonics without capturing extra many fringe patterns or correcting the non-sinusoidal profiles of fringes. In addition, this method involves a quasi-pointwise operation, enabling correcting position-dependent phase errors and being helpful for protecting the edges and details of the measurement results from being blurred.

Single-Shot Imaging of Two-Wavelength Spatial Phase-Shifting Interferometry

In this investigation, we propose an effective method to measure 3D surface profiles of specimens with single-shot imaging. Based on the two-wavelength interferometric principle and spatial phase-shifting technique using a polarization pixelated camera, the proposed system can not only rapidly measure the phase, but also overcome the 2π-ambiguity problem of typical phase-shifting interferometry. The rough surface profile can be calculated by the visibility of the interference fringe and can compensate for the height discontinuity by phase jumps occurring in a fine height map. An inclined plane mirror and a step height specimen with 9 μm were used for the validation of capability of measuring continuously smooth surface and large step heights. The measurement results were in good agreement with the results of typical two-wavelength interferometry.

Compact snapshot multiwavelength interferometer

We propose a snapshot and compact multiwavelength interferometer for real-time testing of optical surfaces with large slopes. A color camera with a micropolarizer array simultaneously captures the phase-shifted interferograms of multiple wavelengths, from which four interferograms with π/2 phase shift at each wavelength are extracted for multiwavelength phase-shifting interferometric measurement. The accuracy of the proposed system is validated experimentally, and the testing of freeform surface with large slopes is also demonstrated. The proposed system provides a feasible way to obtain the instantaneous online measurement of freeform surfaces while minimizing environmental disturbance.

Fourier spectra for nonuniform phase-shifting algorithms based on principal component analysis

We develop an error-free nonuniform phase-stepping algorithm (nPSA) based on principal component analysis (PCA). PCA-based algorithms typically give phase-demodulation errors when applied to nonuniform phase-shifted interferograms. We present a straightforward way to correct those PCA phase-demodulation errors. We give mathematical formulas to fully analyze PCA-based nPSA (PCA-nPSA). These formulas give a) the PCA-nPSA frequency transfer function (FTF), b) its corrected Lissajous figure, c) the corrected PCA-nPSA formula, d) its harmonic robustness (R_H), and e) its signal-to-noise-ratio (SNR). We show that the PCA-nPSA can be seen as a linear quadrature filter and, as consequence, one can find its FTF. Using the FTF, we show why plain PCA often fails to demodulate nonuniform phase-shifted fringes. Previous works on PCA-nPSA (without FTF), give specific numerical/experimental fringe data to "visually demonstrate" that their new nPSA works better than its competitors. This often leads to biased/favorable fringe pattern selections which "visually demonstrate" the superior performance of their new nPSA. This biasing is herein totally avoided because we provide figures-of-merit formulas based on linear systems and stochastic process theories. However, and for illustrative purposes only, we provide specific fringe data phase-demodulation, including comprehensive analysis and comparisons.

Simultaneous dual-wavelength digital holographic microscopy with compensation of chromatic aberration for accurate surface characterization

In this paper, a simultaneous dual-wavelength digital holographic microscope with compensation of chromatic aberration is presented. The microscope has been used to calibrate two standard step heights of 1340 and 1000 nm. The extracted results have been compared with those extracted by the single-shot, dual-wavelength digital holographic microscope, and the comparison has shown that measurements by the proposed microscope are more accurate by around 40 nm. We claim that the high accuracy in measurement is due to the fact that the proposed system works in real time with very low chromatic aberration. Judging the performance of the measurement process over time is explained by the control charts. Evidence from the control measurements sets a probable limit on precision for this process at about 3σ=3.5  nm.

High-speed three-dimensional shape measurement based on shifting Gray-code light

The measuring technique combining a phase-shifting algorithm and Gray-code light has been widely used in three-dimensional (3D) shape measurement for static scenes because of its high robustness and anti-noise ability. However, in the high-speed measurement, phase unwrapping errors occur easily on the boundaries of adjacent Gray-code words because of the defocus of the projector, the motion of the objects and the non-uniform reflectivity of the surface. To overcome this challenge, a high-speed 3D shape measurement method based on shifting Gray-code light has been proposed in this paper. Firstly, the average intensity of three captured phase-shifting fringe images are used as a pixel-wise threshold to binarize the Gray codes and to eliminate most phase unwrapping errors caused by the non-uniform reflectivity, ambient light variations, and the defocus of projector. Then, the shifting Gray-code (SGC) coding strategy is proposed to avoid the remaining errors of phase unwrapping on the edge of the code words. In this strategy, no additional patterns are projected, and two sets of decoding words with staggered boundaries are built in the temporal sequences for one wrapped phase. Finally, the simple, efficient, and robust phase unwrapping can be achieved in the high-speed dynamic measurement. This proposed method has been applied to reconstruct 3D shape of randomly collapsing objects in a large depth range, and the experimental results demonstrate that it can reliably obtain high-quality shape and texture information at 310 frames per second.

Multi-Wavelength Digital-Phase-Shifting Moiré Based on Moiré Wavelength

Multi-wavelength digital-phase-shifting moiré was demonstrated using multiple moiré wavelengths determined by system calibration over the full working depth. The method uses the extended noisy phase map as a reference to unwrap the phase map with a shorter wavelength, and thus achieve a less noisy and more accurate continuous phase map. The moiré wavelength calibration determines a moiré-wavelength to height relationship that permits pixelwise refinement of the moiré wavelength and height during 3D reconstruction. Only a single pattern has to be projected and, thus, a single image captured to compute each phase map with a different wavelength to perform digital-phase-shifting moiré temporal phase unwrapping. Only two captured images are required for two-wavelength phase unwrapping and three captured images for three-wavelength phase unwrapping. The method has been demonstrated in the 3D surface-shape measurement of an object with surface discontinuities and spatially isolated objects.

Pixel-by-pixel absolute phase retrieval assisted by an additional three-dimensional scanner

This paper presents a novel absolute phase unwrapping method assisted by a low-cost three-dimensional (3D) scanner. The proposed absolute phase unwrapping method leverages a low-cost 3D scanner to capture rough 3D data of the scene and transforms the rough 3D data to the world coordinate system to generate an artificial reference phase map ϕ_ref. By referring to ϕ_ref, we can do absolute phase unwrapping directly without projecting any additional patterns, such that the digital fringe projection (DFP) system can achieve higher measurement speed. We develop a multiresolution system consisting of a DFP system and a Kinect V2 to validate our method. Experiments demonstrate that our method works for a large depth range, and the speed of the low-cost 3D scanner is not necessarily the maximum speed of our proposed method. Assisted by the Kinect V2, whose maximum speed is only 30 Hz, our DFP system achieves 53 Hz with a resolution 1600×1000 pixels when we measure dynamic objects that are moving in a large depth range of 400 mm.

Absolute Phase Retrieval Using One Coded Pattern and Geometric Constraints of Fringe Projection System

Fringe projection technologies have been widely used for three-dimensional (3D) shape measurement. One of the critical issues is absolute phase recovery, especially for measuring multiple isolated objects. This paper proposes a method for absolute phase retrieval using only one coded pattern. A total of four patterns including one coded pattern and three phase-shift patterns are projected, captured, and processed. The wrapped phase, as well as average intensity and intensity modulation, are calculated from three phase-shift patterns. A code word encrypted into the coded pattern can be calculated using the average intensity and intensity modulation. Based on geometric constraints of fringe projection system, the minimum fringe order map can be created, upon which the fringe order can be calculated from the code word. Compared with the conventional method, the measurement depth range is significantly improved. Finally, the wrapped phase can be unwrapped for absolute phase map. Since only four patterns are required, the proposed method is suitable for real-time measurement. Simulations and experiments have been conducted, and their results have verified the proposed method.

Depth-driven variable-frequency sinusoidal fringe pattern for accuracy improvement in fringe projection profilometry

Sinusoidal fringe pattern is widely used in optical profilometry; however, the traditional constant-frequency sinusoidal fringe pattern reduces 3D measurement accuracy in the defocus region. To this end, this paper presents a variable-frequency sinusoidal fringe pattern method that is optimized by the measurement depth. The proposed method improves the pixel matching accuracy and thus increases measurement accuracy. This paper theoretically determines the optimal frequency by analyzing the pixel matching error caused by intense noise in a captured image; presents the online frequency optimization along abscissa and ordinate axes in the sinusoidal fringe patterns; and details the encoding and decoding to use variable-frequency fringe patterns for 3D profilometry. Simulations and experiments demonstrate that our proposed method can improve the 3D measurement accuracy and increase measurement robustness.

Snapshot multi-wavelength interference microscope

A snapshot multi-wavelength interference microscope is proposed for high-speed measurement of large vertical range discontinuous microstructures and surface roughness. A polarization CMOS camera with a linear micro-polarizer array and Bayer filter accomplishes snapshot multi-wavelength phase-shifting measurement. Four interferograms with 𝜋/2 phase shift are captured at each wavelength for phase measurement, the 2𝜋 ambiguities are removed by using two or three wavelengths.

High-speed and high-accuracy 3D surface measurement using a mechanical projector

This paper presents a method to achieve high-speed and high-accuracy 3D surface measurement using a custom-designed mechanical projector and two high-speed cameras. We developed a computational framework that can achieve absolute shape measurement in sub-pixel accuracy through: 1) capturing precisely phase-shifted fringe patterns by synchronizing the cameras with the projector; 2) generating a rough disparity map between two cameras by employing a standard stereo-vision method using texture images with encoded statistical patterns; and 3) utilizing the wrapped phase as a constraint to refine the disparity map. The projector can project binary patterns at a speed of up to 10,000 Hz, and the camera can capture the required number of phase-shifted fringe patterns with 1/10,000 second, and thus 3D shape measurement can be realized as high as 10,000 Hz regardless the number of phase-shifted fringe patterns required for one 3D reconstruction. Experimental results demonstrated the success of our proposed method.

Absolute three-dimensional shape measurement with two-frequency square binary patterns

This paper presents a novel method to achieve absolute three-dimensional shape measurement solely using square binary patterns. This method uses six patterns: three low-frequency phase-shifted patterns and three phase-shifted high-frequency patterns. The phase obtained from the low-frequency phase temporally unwraps the phase obtained from high-frequency patterns. The projector is defocused such that the high-frequency patterns produce a high-quality phase, but the phase retrieved from low-frequency patterns has a large harmonic error that fails the two-frequency temporal phase unwrapping process. In this paper, we develop a computational framework to address the challenge. The proposed computational framework includes four major approaches to alleviate the harmonic error problem: (i) use more than one period of low-frequency patterns enabled by a geometric constraint-based phase unwrapping method; (ii) artificially apply a large Gaussian filter to low-frequency patterns before phase computation; (iii) create an error lookup table to compensate for harmonic error; and (iv) develop a boundary error correction method to alleviate problems associated with filtering. Both simulation and experimental results demonstrated the success of the proposed method.

Superfast high-resolution absolute 3D recovery of a stabilized flapping flight process

Scientific research of a stabilized flapping flight process (e.g. hovering) has been of great interest to a variety of fields including biology, aerodynamics, and bio-inspired robotics. Different from the current passive photogrammetry based methods, the digital fringe projection (DFP) technique has the capability of performing dense superfast (e.g. kHz) 3D topological reconstructions with the projection of defocused binary patterns, yet it is still a challenge to measure a flapping flight process with the presence of rapid flapping wings. This paper presents a novel absolute 3D reconstruction method for a stabilized flapping flight process. Essentially, the slow motion parts (e.g. body) and the fast-motion parts (e.g. wings) are segmented and separately reconstructed with phase shifting techniques and the Fourier transform, respectively. The topological relations between the wings and the body are utilized to ensure absolute 3D reconstruction. Experiments demonstrate the success of our computational framework by testing a flapping wing robot at different flapping speeds.

Modified Gray-Level Coding Method for Absolute Phase Retrieval

Fringe projection systems have been widely applied in three-dimensional (3D) shape measurements. One of the important issues is how to retrieve the absolute phase. This paper presents a modified gray-level coding method for absolute phase retrieval. Specifically, two groups of fringe patterns are projected onto the measured objects, including three phase-shift patterns for the wrapped phase, and three n-ary gray-level (nGL) patterns for the fringe order. Compared with the binary gray-level (bGL) method which just uses two intensity values, the nGL method can generate many more unique codewords with multiple intensity values. With assistance from the average intensity and modulation of phase-shift patterns, the intensities of nGL patterns are normalized to deal with ambient light and surface contrast. To reduce the codeword detection errors caused by camera/projector defocus, nGL patterns are designed as n-ary gray-code (nGC) patterns to ensure that at most, one code changes at each point. Experiments verify the robustness and effectiveness of the proposed method to measure isolated objects with complex surfaces.

Computer-aided-design-model-assisted absolute three-dimensional shape measurement

Conventional three-dimensional (3D) shape measurement methods are typically generic to all types of objects. Yet, for many measurement conditions, such a level of generality is inessential when having the preknowledge of the object geometry. This paper introduces a novel adaptive algorithm for absolute 3D shape measurement with the assistance of the object computer-aided-design (CAD) model. The proposed algorithm includes the following major steps: (1) export the 3D point cloud data from the CAD model; (2) transform the CAD model into the camera perspective; (3) obtain a wrapped phase map from three phase-shifted fringe images; and (4) retrieve absolute phase and 3D geometry assisted by the CAD model. We demonstrate that if object CAD models are available, such an algorithm is efficient in recovering absolute 3D geometries of both simple and complex objects with only three phase-shifted fringe images.

Three-dimensional absolute shape measurement by combining binary statistical pattern matching with phase-shifting methods

This paper presents a novel method that leverages the stereo geometric relationship between projector and camera for absolute phase unwrapping on a standard one-projector and one-camera structured light system. Specifically, we use only one additional binary random image and the epipolar geometric constraint to generate a coarse correspondence map between projector and camera images. The coarse correspondence map is further refined by using the wrapped phase as a constraint. We then use the refined correspondence map to determine a fringe order for absolute phase unwrapping. Experimental results demonstrated the success of our proposed method.

Simultaneous phase-shifting dual-wavelength interferometry based on independent component analysis

An independent component analysis-based simultaneous phase-shifting dual-wavelength interferometry approach is proposed. By using a one-time phase-shifting procedure, the simultaneous phase-shifting operation of two illumination wavelengths can be implemented, and then the background intensity and two orthogonal independent components of each single wavelength can be separated from a sequence of simultaneous phase-shifting dual-wavelength interferograms with random phase shifts. Subsequently, the wrapped phases of single wavelength can be calculated by above two orthogonal independent components; thus the unambiguous phase of synthetic wavelength can be achieved. Both the simulation and experimental results show that the proposed approach reveals the advantages of high accuracy, rapid speed, high stability, and good adaptability for arbitrary phase shifts.

Absolute three-dimensional shape measurement with a known object

This paper presents a novel method for absolute three-dimensional (3D) shape measurement that does not require conventional temporal phase unwrapping. Our proposed method uses a known object (i.e., a ping-pong ball) to provide cues for absolute phase unwrapping. During the measurement, the ping-pong ball is positioned to be close to the nearest point from the scene to the camera. We first segment ping-pong ball and spatially unwrap its phase, and then determine the integer multiple of 2π to be added such that the recovered shape matches its actual geometry. The nearest point of the ball provides zmin to generate the minimum phase Φmin that is then used to unwrap phase of the entire scene pixel by pixel. Experiments demonstrated that only three phase-shifted fringe patterns are required to measure absolute shapes of objects moving along depth z direction.

Simple, flexible, and accurate phase retrieval method for generalized phase-shifting interferometry

This paper presents a non-iterative phase retrieval method from randomly phase-shifted fringe images. By combining the hyperaccurate least squares ellipse fitting method with the subspace method (usually called the principal component analysis), a fast and accurate phase retrieval algorithm is realized. The proposed method is simple, flexible, and accurate. It can be easily coded without iteration, initial guess, or tuning parameter. Its flexibility comes from the fact that totally random phase-shifting steps and any number of fringe images greater than two are acceptable without any specific treatment. Finally, it is accurate because the hyperaccurate least squares method and the modified subspace method enable phase retrieval with a small error as shown by the simulations. A MATLAB code, which is used in the experimental section, is provided within the paper to demonstrate its simplicity and easiness.

Quantized phase coding and connected region labeling for absolute phase retrieval

This paper proposes an absolute phase retrieval method for complex object measurement based on quantized phase-coding and connected region labeling. A specific code sequence is embedded into quantized phase of three coded fringes. Connected regions of different codes are labeled and assigned with 3-digit-codes combining the current period and its neighbors. Wrapped phase, more than 36 periods, can be restored with reference to the code sequence. Experimental results verify the capability of the proposed method to measure multiple isolated objects.

Motion-induced error reduction by combining Fourier transform profilometry with phase-shifting profilometry

We propose a hybrid computational framework to reduce motion-induced measurement error by combining the Fourier transform profilometry (FTP) and phase-shifting profilometry (PSP). The proposed method is composed of three major steps: Step 1 is to extract continuous relative phase maps for each isolated object with single-shot FTP method and spatial phase unwrapping; Step 2 is to obtain an absolute phase map of the entire scene using PSP method, albeit motion-induced errors exist on the extracted absolute phase map; and Step 3 is to shift the continuous relative phase maps from Step 1 to generate final absolute phase maps for each isolated object by referring to the absolute phase map with error from Step 2. Experiments demonstrate the success of the proposed computational framework for measuring multiple isolated rapidly moving objects.

Pixel-wise absolute phase unwrapping using geometric constraints of structured light system

This paper presents a method to unwrap phase pixel by pixel by solely using geometric constraints of the structured light system without requiring additional image acquisition or another camera. Specifically, an artificial absolute phase map, Φmin, at a given virtual depth plane z = zmin, is created from geometric constraints of the calibrated structured light system; the wrapped phase is pixel-by-pixel unwrapped by referring to Φmin. Since Φmin is defined in the projector space, the unwrapped phase obtained from this method is absolute for each pixel. Experimental results demonstrate the success of this proposed novel absolute phase unwrapping method.

Single-shot absolute 3D shape measurement with Fourier transform profilometry

Fourier transform profilometry (FTP) is one of the frequently adopted three-dimensional (3D) shape measurement methods due to its ability to recover single-shot 3D shapes, yet it is challenging to retrieve the absolute phase map solely from one single grayscale fringe image. This paper presents a computational framework that overcomes this limitation of FTP with digital fringe projection (DFP). By using geometric constraints, an absolute phase map can be retrieved point-by-point from one single grayscale fringe image. Experiments demonstrate the success of our proposed framework with single-shot absolute 3D shape measurement capability.

Phase retrieval based on temporal and spatial hybrid matching in simultaneous phase-shifting dual-wavelength interferometry

In simultaneous phase-shifting dual-wavelength interferometry, by matching both the phase-shifting period number and the fringe number in interferogram of two wavelengths to the integers, the phase with high accuracy can be retrieved through combining the principle component analysis (PCA) and least-squares iterative algorithm (LSIA). First, by using the approximate ratio of two wavelengths, we can match both the temporal phase-shifting period number and the spatial fringe number in interferogram of two wavelengths to the integers. Second, using above temporal and spatial hybrid matching condition, we can achieve accurate phase shifts of single-wavelength of phase-shifting interferograms through using PCA algorithm. Third, using above phase shifts to perform the iterative calculation with the LSIA method, the wrapped phases of single-wavelength can be determined. Both simulation calculation and experimental research demonstrate that by using the temporal and spatial hybrid matching condition, the PCA + LSIA based phase retrieval method possesses significant advantages in accuracy, stability and processing time.

Enhanced two-frequency phase-shifting method

One of the major challenges of employing a two-frequency (or two-wavelength) phase-shifting algorithm for absolute three-dimensional shape measurement is its sensitivity to noise. Therefore, three- or more-frequency phase-shifting algorithms are often used in lieu of a two-frequency phase-shifting algorithm for applications where the noise is severe. This paper proposes a method to use geometric constraints of digital fringe projection system to substantially reduce the noise impact by allowing the use of more than one period of equivalent phase map for temporal phase unwrapping. Experiments successfully verified the enhanced performance of the proposed method without increasing the number of patterns.

Synthesis of multi-wavelength temporal phase-shifting algorithms optimized for high signal-to-noise ratio and high detuning robustness using the frequency transfer function

Synthesis of single-wavelength temporal phase-shifting algorithms (PSA) for interferometry is well-known and firmly based on the frequency transfer function (FTF) paradigm. Here we extend the single-wavelength FTF-theory to dual and multi-wavelength PSA-synthesis when several simultaneous laser-colors are present. The FTF-based synthesis for dual-wavelength (DW) PSA is optimized for high signal-to-noise ratio and minimum number of temporal phase-shifted interferograms. The DW-PSA synthesis herein presented may be used for interferometric contouring of discontinuous industrial objects. Also DW-PSA may be useful for DW shop-testing of deep free-form aspheres. As shown here, using the FTF-based synthesis one may easily find explicit DW-PSA formulae optimized for high signal-to-noise and high detuning robustness. To this date, no general synthesis and analysis for temporal DW-PSAs has been given; only ad hoc DW-PSAs formulas have been reported. Consequently, no explicit formulae for their spectra, their signal-to-noise, their detuning and harmonic robustness has been given. Here for the first time a fully general procedure for designing DW-PSAs (or triple-wavelengths PSAs) with desire spectrum, signal-to-noise ratio and detuning robustness is given. We finally generalize DW-PSA to higher number of wavelength temporal PSAs.

GPC light shaping a supercontinuum source

Generalized Phase Contrast (GPC) is a versatile tool for efficiently rerouting and managing photon energy into speckle-free contiguous spatial light distributions. We have previously shown theoretically and numerically that a GPC Light Shaper shows robustness to shift in wavelength and can maintain both projection length scale and high efficiency over a range [0.75λ(0); 1.5λ(0)] with λ(0) as the characteristic design wavelength. With this performance across multiple wavelengths and the recent availability of tabletop supercontinuum lasers, GPC light shaping opens the possibility for creatively incorporating various multi-wavelength approaches into spatially shaped excitations that can enable new broadband light applications. We verify this new approach using a supercontinuum light source, interfaced with a compact GPC light shaper. Our experiments give ~70% efficiency, ~3x intensity gain, and ~85% energy savings, limited, however, by the illumination equipment, but still in very good agreement with theoretical and numerical predictions.

Single-wavelength phase retrieval method from simultaneous multi-wavelength in-line phase-shifting interferograms

From a sequence of simultaneous multi-wavelength phase-shifting interferograms (SMWPSIs), a novel single-wavelength phase retrieval method based on the least-squares iterative algorithm is proposed and utilized in dual-wavelength interferometry. Firstly, only one time phase-shifting procedure implements the phase shifts of all illumination wavelengths simultaneously, and then the accurate wrapped phases of each single-wavelength can be respectively retrieved from SMWPSIs by the least-squares iterative operation, so the phase of synthetic wavelength can be obtained by the subtraction easily. Using the proposed method, both the simulation and the experimental results demonstrate that the optical setup is simpler; the requirements for the displacement of the phase-shifting device and the number of the captured interferograms are smaller compared to the traditional phase-shifting multi-wavelength interferometry or off-axis multi-wavelength interferometry. Even in the case that the phase-shifts are unknown, the wrapped phases and the phase-shifts of each single-wavelength can be obtained by the proposed method.

Measurement of large discontinuities using single white light interferogram

White light interferometry is a widely used tool to extend the unambiguous measurement range of a monochromatic interferometer. In this work, we discuss Hilbert transformation analysis of a single white light interferogram acquired with a single-chip color CCD camera for step height measurement which lies beyond the unambiguous range of the monochromatic interferometry. The color interferogram is decomposed and phase maps for red, green, and blue components are calculated independently using Hilbert transformation. This procedure makes the measurement faster, simpler, and cost-effective. The usefulness of the technique is demonstrated on micro-sample.

Absolute three-dimensional shape measurement using coded fringe patterns without phase unwrapping or projector calibration

This paper presents a novel stereo-phase-based absolute three-dimensional (3D) shape measurement that requires neither phase unwrapping nor projector calibration. This proposed method can be divided into two steps: (1) obtain a coarse disparity map from the quality map; and (2) refine the disparity map using wrapped phase. Fringe patterns are modified to encode the quality map for efficient and accurate stereo matching. Experiments demonstrated that the proposed method could achieve high-quality 3D measurement even with extremely low-quality fringe patterns.

Influence of roughness in the phase-shifting speckle method: an experimental study with applications

In this paper, a quantification of the influence of roughness in the phase-shifting speckle technique (phase-difference) is presented. For this purpose, a series of experiments were carried out on calibrated samples of different roughness varying the equivalent wavelength Λ, by combining the wavelengths of an argon-ion laser. In order to use this method correctly, on the basis of the phase fringe contrast, a criterion for the minimal Λ (limit) needed for a specific roughness is given. Besides, considering the investigated relation between roughness and effective wavelength as an inverse problem, a procedure to measure the roughness of a surface is proposed. Laboratory experiments to prove the validity of the method are shown.

Method of excess fractions with application to absolute distance metrology: analytical solution

Multiwavelength interferometry provides a solution to a number of applications in metrology for the measurement of optical path differences longer than the source wavelength. To this day, the method of excess fractions (EF) has proved to provide very long, unambiguous measurement ranges with the highest reliability for a given set of wavelengths and level of phase noise. This is achieved because EF combines the individual phase values in an equivalent least-square problem and evaluates the correspondence for all possible solutions. However, this procedure can be slow for a number of applications. In this paper, an analytical solution for EF is presented that allows the direct calculation of the unknown integer fringe order. It is shown that this solution is consistent with the other phase unwrapping approaches as beat wavelength or Chinese remainder theorem-based solutions, but moreover, it can be understood as a unified representation and solution of the fringe order problem.

Improving accuracy of distance measurements based on an optoelectronic oscillator by measuring variation of fiber delay

The ultra-long optical fiber of an optoelectronic oscillator (OEO) and the high spectral purity of its high frequency oscillation signal open the possibility of high-accuracy distance measurements at a long range. However, the longer the fiber length in an OEO, the more prone the system is to surrounding disturbance, which in turn leads to fluctuation of the loop delay and a reduction in distance measurement accuracy. In this paper, an intensity modulated light signal is combined with the light signal of an OEO in terms of wavelength division multiplexing (WDM) and is propagated through the fiber. The phase shift has been measured in real time to compensate for variations in fiber delay. With this method, experimental results showed a standard deviation of 14.8 μm.

Characterization of surface modifications by white light interferometry: applications in ion sputtering, laser ablation, and tribology experiments

In materials science and engineering it is often necessary to obtain quantitative measurements of surface topography with micrometer lateral resolution. From the measured surface, 3D topographic maps can be subsequently analyzed using a variety of software packages to extract the information that is needed. In this article we describe how white light interferometry, and optical profilometry (OP) in general, combined with generic surface analysis software, can be used for materials science and engineering tasks. In this article, a number of applications of white light interferometry for investigation of surface modifications in mass spectrometry, and wear phenomena in tribology and lubrication are demonstrated. We characterize the products of the interaction of semiconductors and metals with energetic ions (sputtering), and laser irradiation (ablation), as well as ex situ measurements of wear of tribological test specimens. Specifically, we will discuss: i. Aspects of traditional ion sputtering-based mass spectrometry such as sputtering rates/yields measurements on Si and Cu and subsequent time-to-depth conversion. ii. Results of quantitative characterization of the interaction of femtosecond laser irradiation with a semiconductor surface. These results are important for applications such as ablation mass spectrometry, where the quantities of evaporated material can be studied and controlled via pulse duration and energy per pulse. Thus, by determining the crater geometry one can define depth and lateral resolution versus experimental setup conditions. iii. Measurements of surface roughness parameters in two dimensions, and quantitative measurements of the surface wear that occur as a result of friction and wear tests. Some inherent drawbacks, possible artifacts, and uncertainty assessments of the white light interferometry approach will be discussed and explained.

Computerized interferometric surface measurements [Invited]

The addition of electronics, computers, and software to interferometry has enabled enormous improvements in optical metrology. This paper discusses four areas in which computerized interferometric measurement improvements have been made in the measurement of surface shape and surface roughness: (a) The use of computer-generated holograms for the testing of aspheric optics, (b) phase-shifting interferometry for getting interferometric data into a computer so the data can be analyzed, (c) computerized interference microscopes, including multiple-wavelength and coherence scanning, for the precision measurement of surface microstructure, and (d) vibration-insensitive dynamic interferometers for enabling precise measurements in noncontrolled environments.

Advanced Nanomeasuring Techniques for Surface Characterization

Advanced precise and accurate nanomeasurement techniques play an important role to improve the function and quality of surface characterization. There are two basic approaches, the hard measuring techniques and the soft computing measuring techniques. The advanced soft measuring techniques include coordinate measuring machines, roundness testing facilities, surface roughness, interferometric methods, confocal optical microscopy, scanning probe microscopy, and computed tomography at the level of nanometer scale. On the other hand, a new technical committee in ISO is established to address characterization issues posed by the areal surface texture and measurement methods. This paper reviews the major advanced soft metrology techniques obtained by optical, tactile, and other means using instruments, classification schemes of them, and their applications in the engineering surfaces. Furthermore, future trends under development in this area are presented and discussed to display proposed solutions for the important issues that need to be addressed scientifically.