<title>Three-dimensional shape measurement using digital shearography</title>

When phase-shifting digital holography with a continuous fringe-scanning scheme is implemented using a PC-based measurement system without any synchronous circuit, nonuniform phase-shifted interference fringes are captured because of the fluctuation in the image-capturing interval. To cope with the nonuniform phase shifts, a statistical generalized phase-shifting approach is employed. Because the algorithm is designed to use an arbitrary phase shift, the nonuniform phase shifts do not obstruct object-wave retrieval. Moreover, multiple interference fringes can be obtained in a short time owing to the continuous fringe-scanning scheme. However, the wavefront calculation method is not designed for sequentially recorded interference fringes. To use multiple interference fringes appropriately, we develop a least-squares wavefront calculation method combined with corrections for the initial phase and the direction of phase rotation. We verify the proposed method by numerical simulations and optical experiments. The results show that the object wave with the same initial phase can be correctly reconstructed by using both phase correction methods simultaneously.

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.

Projector nonlinearity is a common problem for digital structured light-based three-dimensional (3-D) shape measurement techniques. A temporal-spatial binary encoding method is presented for the purpose of eluding it. We build a 3-D shape measurement scheme by combining our proposed method with phase measurement profiling. A standard sinusoidal fringe pattern is divided into more than two binary fringe patterns using specially designed temporal and spatial binary encoding rule based on intensity hierarchic quantification, and then are in-focus projected onto the measured object at a time sequence to reconstruct a frame phase-shifting fringe image. On account of the projected binary fringe pattern strictly consisting of zeros and ones, the influence of the projector nonlinearity on the measurement result can be effectively ruled out and simultaneously enables high-quality sinusoidality. In-depth investigations by theoretical analysis and experiments are conducted to demonstrate the performance of this method.

Phase-shifting shadow moire is an efficient technique for three-dimensional (3-D) shape measurement, but the technique is always suffered from the error of phase shifter. A three-frame random phase-shifting algorithm is proposed to relax the requirements of repeatability and accuracy of the phase shifter in a shadow moire system. The proposed method can act as a bridge between the three phase-shifting analysis methods and two-shot phase-shifting methods. The value of introduced arbitrary and unknown phase shift can be calculated by a determinate formula. The algorithm has the advantages of being free from the disturbance of background elimination and less computation. In addition, the proposed method is immune to nonsinusoidal fringe patterns and can be implemented easily in many applications. Both simulation and optical experimental results verify the feasibility of this method.

In this work, we deduce explicit conditions to design phase shifting algorithms (PSAs) in order to suppress errors in phase measurement, the double-frequency ripple distortion and the spurious piston. These errors arise when non-uniform phase-shifting interferograms are processed with conventional PSAs which assume uniform phase shifts. By modeling the non-uniform phase shifts as a polynomial of the unperturbed phase shift value !0, we show that the conditions for eliminating those errors are associated to the m-th derivative of the PSA's frequency transfer function (FTF). Thus, we propose an approach to design robust algorithms based on the FTF formalism, and we present a ready-to-apply eight-frame PSA. Finally, our conclusions are supported by computer simulations.

When phase-shifting (PS) and temporal-phase-unwrapping algorithms are employed for three-dimensional (3D) shape measurement, the measured object must be kept static during the projection and acquisition. If the measured object is moving, deviation among multiple fringe images will inevitably occur, which results in the 3D measurement error. In this paper, a novel 3D measurement technique for rigid moving objects obtained by using the PS algorithm and the three-pitch heterodyne-unwrapping (TPHU) algorithm is proposed, which consists of six steps. First, the pixel offset is estimated based on a centroid deviation calculation. Second, the phase offsets among multiple fringes are calculated. Third, a novel set of fringes for the projection is generated. Fourth, we offset the captured fringe images according to the pixel offsets to generate a new set of captured fringe images for the phase calculation. Fifth, the wrapped phase is calculated by the PS algorithm. Finally, the unwrapped phase is calculated by the TPHU algorithm. The proposed method can be applied to the 3D shape reconstruction of a rigid movement object with a determined movement trajectory and constant movement speed. This approach not only greatly improves the measurement efficiency but also inherits the high accuracy and robustness of the PS and TPHU algorithms.

A comprehensive theoretical study on the head-on collision between two solitary waves in a thin elastic plate floating on an inviscid fluid of finite depth is investigated analytically by means of a singular perturbation method. The effects of plate compression are also taken into account. The Poincare–Lighthill–Kuo method has been used to derive the solution up to the fourth order of the resulting nonlinear differential equation, which in principle gives the asymptotic series solution. It is found that after collision, both the hydroelastic solitary waves preserves their original shape and positions. However a collision does have imprints on colliding waves with non-uniform phase shift up to the third order which creates tilting in the wave profile. Maximum run-up amplitude, wave speed, phase shift and distortion profile have also been calculated and plotted for two colliding solitary waves.

Color-coded fringe projection profilometry (FPP) is one of the most important single-frame three-dimensional (3D) shape measurement techniques. Transverse chromatic aberration (TCA) in a color-coded FPP system yields imaging deviations, which will result in 3D shape measurement error. In this work, a TCA compensation method for color-coded FPP is proposed. The proposed method establishes the relationship among phase, pixel deviation, and imaging position for decreasing the 3D shape measurement error introduced by TCA. Compared to existing methods that use the relationship among depth, pixel deviation, and imaging position or polynomial fitting for each pixel, the proposed method offers the merits of precise and simple implementation, since it voids the cumulative error in-depth calculation and removes the time-consuming process of polynomial fitting for each pixel. For validation, a color-coded FPP system is established and calibrated. Two verification experiments, including a translation test and 3D shape measurement of a regular spherical object, are conducted with the calibrated system. Experimental results demonstrate that the proposed TCA compensation method is capable of effectively reducing the 3D shape measurement error caused by TCA.

Binary coding methods have been widely used for three-dimensional (3D) shape measurement. However, traditional binary coding methods always require a sequence of binary patterns to encode the fringe order. This paper presents a spatial-temporal binary coding (STBC) method for 3D shape measurement with three binary patterns, including a spatial pattern and two temporal patterns. Specifically, the spatial pattern is used to encode a 4-bit spatial code, and the temporal patterns are used to encode the 2-bit temporal code. By combining the spatial code and the temporal code, 6-bit code words can be obtained that distinguish 26=64 fringe periods of sinusoidal phase-shifting patterns. A robust dual-region labeling decoding scheme is also developed to extract the 6-bit code word for fringe order determination, and then absolute-phase unwrapping can be carried out. Experiments have been conducted, and their results confirm that the STBC method can reliably reconstruct the 3D shapes of measured objects with only three binary patterns.

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 (3D) shape measurement of fringe projection profilometry (FPP) with high speed and accuracy is required in many applications. The measurement speed of FPP system is mainly limited by the low frame rate of pattern projection. Although defocusing binary patterns can be used to increase the measurement speed greatly, it will decrease the measurement depth. To this end, a novel triangular pattern generation method is proposed in this paper. By designing and projecting binary strip patterns with focused projector, images of triangular patterns can be then acquired with high frame rate. Fast and accurate 3D shape measurement can be finally obtained by using phase-shifting triangular patterns. Compared with the traditional sinusoidal patterns, the proposed method can improve the projection rate more than 4 times. Moreover, the proposed method will not decrease the measurement depth. Experimental results are given to confirm the validity of the proposed method.

Light-in-Flight recording by holography makes it possible to perlorm accurate three-dimensional shape measurements by single-line contouring. Because ultrashort light pulses are used, both stationary and moving objects may be recorded, e.g., fast-rotating turbine blades, mobile scale models, active human beings, etc. The evaluation is accomplished by an image processing system that reads the contouring line that varies along the hologram and transforms it into spatial coordinates, thereby measuring the three-dimensional shape. There are a number of possible application areas of the method, ranging from practical engineering to medicine.

Blade is the key component of the aero-engine. In generally, it requires precise size and accurate shape, so the three-dimensional shape measurement of the blade is very important. Fringe projection profilometry is generally used to measure the 3D shape of an object, because of the advantages of non-contact operation, full-field acquisition, high resolution, and fast data processing. In this paper, 3D shape measurement method based on surface structured light that combination of gray-code and phase-shift projection is proposed. The wrapped phase of the blade is got by the four-step phase-shift method. In the projection process of the gray-code, due to the ambient light, different surface reflectance and surface physical discontinuous, the edge of gray-code becomes smooth, leading to decode error, so that the gray-code pattern must be binaried before decode. Through the accurate binarization of the blade gray-code pattern, the decoding cycle of the blade is achieved, and the unwrapped phase is achieved by phase unwrapping. Then the unwrapped phase difference between the blade and reference plane is got, the height of the blade can be obtained by the relationship between the phase difference and the height. The experimental and simulation results show that the proposed method can achieve a high precision, high speed and low cost 3D shape measurement of the blade. The measurement accuracy reaches 0.03 mm. The proposed method extends the reliability and practicality of the fringe projection profilometry.

With the development of science and technology and industrial production, the application of three-dimensional shape measurement technology in the fields of machine vision, industrial monitoring, mechanical engineering and medical testing has become increasingly widespread. The optical three-dimensional shape measurement technology is a method that uses a series of optical methods to obtain the three-dimensional information of the measured object, and the structured light measurement method is a widely used measurement method. In the structured light measurement method, the generation of fringe patterns is a very important link. Here, we use holographic method to generate a fringe pattern whose period and phase can be easily modulated. First, a black and white fringe pattern with a certain spatial frequency is generated according to the cosine structured light period and phase. The spatial frequency of the black and white fringe is determined by the spatial frequency of the structured light to be generated and the magnification of the projection system. A prism phase that can cause lateral movement is applied to the black part of the black and white stripes, so that the light incident on the black area deviates from the optical axis to the first order of diffraction, while the light incident on other parts (without being modulated) continues to travel along the optical axis. In this way, the deviated beam is bright and dark, and finally structured illumination is obtained. Theories and experiments verify the effectiveness of the method. The method to generate fringe patterns is simple, fast, accurate and can be conveniently controlled, and can be well applied to three-dimensional shape measurement technology.

In the measurement of three-dimensional structure beyond the diffraction limit of the objective lens of the optical system using the speckle interferometry, it was reported that the shape of a measured object can be measured by detecting the change of phase distribution generated by a lateral shift instead of analyzing the focal image of the measured object. In this paper, it is investigated that the three-dimensional shape measurement can be realized by detecting the phase change of only the zeroth-order diffraction light by the lateral shift of the object using experimental results. As a result, it is confirmed that the shape information of the object is included in the zeroth-order diffraction light and that the three-dimensional shape can be measured by using only zeroth-order light without any higher order diffraction lights.