Absolute Phase Retrieval Research Articles

Two-plus-two fringe projection profilometry based on phase-shifted coding

In fringe projection profilometry based on temporal phase unwrapping, determining a fringe order map commonly requires a large number of fringes. To reduce the fringe number, this paper proposes a concise absolute phase retrieval algorithm just by projecting four fringes. The first two orthogonal fringes with relatively large frequency can collect reliable height information. The second two fringes are designed the same as the first two, but the only difference is that each 2π-phase of them is shifted by a unique amount, which can robustly label a large number of fringe orders. For decoding the fringes, we develop an average intensity one-time extraction algorithm, which allows for the rapid acquisition of the two pairs of alternating current components. From this, the wrapped phase containing height information and the stair-coded phase providing fringe orders can be directly extracted by arctangent operation in a point-to-point manner. Furthermore, we also develop a universal fringe order correction algorithm that can simultaneously correct the common errors and the misalignment between the wrapped phase and fringe orders. Experiment results demonstrate that this method achieves comparable accuracy and adaptability to the phase-coding method, while utilizing two fewer fringes.

Deep learning-based frequency-multiplexing composite-fringe projection profilometry technique for one-shot 3D shape measurement

The pursuit of precise and efficient 3D shape measurement has long been a focal point within the fringe projection profilometry (FPP) domain. However, achieving precise 3D reconstruction for isolated objects from a single fringe image remains a challenging task in the field. In this paper, a deep learning-based frequency-multiplexing (FM) composite-fringe projection profilometry (DFCP) is proposed, where an end-to-end absolute phase retrieval network (APR-Net) is trained to directly recover the absolute phase map from a FM composite fringe pattern. The obtained absolute phase map exhibits exceptional precision and is devoid of spectrum crosstalk or leakage disturbance typically encountered in traditional FM techniques. APR-Net is intricately crafted, incorporating a nested strategy along with the concept of centralized information interaction. A diverse dataset encompassing various scenarios and free from spectrum aliasing is assembled to guide the training process. A seven-map loss calculation scheme is employed to guide the training process, of which the efficacy is proved through ablation experiments. In the first qualitative experiment, DFCP demonstrates comparable phase accuracy to ground truths with 47 fewer projected images, outperforming other three methods with mean absolute phase errors of 0.0052 rad, 0.1761 rad, 0.0169 rad, and 0.0139 rad. The second qualitative experiment and the quantitative evaluation respectively prove DFCP’s capability in high dynamic range 3D measurement and in precise 3D measurement, with sphere diameter errors of 0.0780 mm and 0.0726 mm, and a spherical centroid distance error of 0.0555 mm.

Absolute phase retrieval based on spatial ternary phase coding with circular fringe projection

Fringe projection profilometry (FPP) based on temporal phase unwrapping (TPU) plays a vital role due to its high precision and fast speed in three-dimensional (3D) measurements. However, a fundamental contradiction arises between the necessity for numerous auxiliary patterns and the goal of enhancing measurement efficiency, particularly when ensuring a big codeword space for fringe order. Thus, a spatial ternary phase coding (STPC) method is proposed, which achieves a big codeword space by projecting just one auxiliary coding pattern while ensuring a robust decoding. By equally dividing each fringe period into several segments, ternary coding of fringe order is accomplished using an arccosine operation to code three distinct phase values within each segment. Meanwhile, circular grating is adopted due to its excellent performance in phase sensitivity, which offers a higher measuring accuracy in 3D reconstruction. Accordingly, a robust TPU is achieved by combining the introduced phase unwrapping scheme and the circle center-cosine theorem (CCCT) strategy. Furthermore, high-efficiency measurements are attained through projecting three phase-shifting circular gratings and one STPC grating, rendering it suitable for real-time measurements. Experimental results with various static objects and real-time scene demonstrate the feasibility and validity of the proposed method.

Absolute phase retrieval based on fringe amplitude encoding without any additional auxiliary pattern

An absolute phase retrieval method based on fringe amplitude encoding is proposed. Different from the conventional intensity coding methods which are based on time division multiplexing with multiple additional auxiliary patterns, the proposed fringe order encoding strategy is codeword overlapping interaction based on space division multiplexing. It just directly encodes different fringe amplitudes for different periods in corresponding sinusoidal phase-shifting patterns to generate space division multiplexing composite sinusoidal phase-shifting patterns and quantifies the fringe amplitudes into four levels as encoding strategy, so it can retrieve absolute phase without any additional auxiliary patterns. To improve the anti-interference capability of the proposed method, a codeword extraction method based on image morphological processing is proposed to segment the grayscale. Consequently, both the phase-shifting sinusoidal deformed patterns and the single frame space division multiplexing four gray-level codewords for fringe order recognition can be extracted respectively from the captured composite deformed patterns. Then, a half-period single-connected domain correction method is also proposed to correct the codewords. Moreover, in order to suppress the effect of jump errors, the phase zero points are constructed to segment the positive and negative ranges of the phase, making the phase unwrapping process segmented. The experimental results demonstrate the feasibility and effectivity of the proposed method.

Anti-crosstalk absolute phase retrieval method for microscopic fringe projection profilometry using temporal frequency-division multiplexing.

In microscopic fringe projection profilometry (MFPP), the traditional absolute phase retrieval method using composite frequency fringe has the shortcomings of low accuracy and poor robustness due to mutual crosstalk of harmonic from the different channels of frequency-division multiplexing. In this study, an absolute phase retrieval method that avoids the inter-channel crosstalk is proposed. By introducing guard bands to accommodate the frequency channels corresponding to the second harmonic that dominate the high order harmonics, the aliasing between the second harmonic and the fundamental is eliminated. Consequently, phase maps without crosstalk can be demodulated using appropriate phase-shifting algorithms. The proposed method is well-suited for high-precision three-dimensional shape measurement scenarios in many fields such as integrated circuit manufacturing process control and micro-electro-mechanical system quality inspection. The experiment results demonstrate that the anti-crosstalk method is effective and can realize three-dimensional reconstruction for discontinuous planar surface and spherical surface.

A novel phase-shifting segmented phase coding method for absolute phase retrieval

The phase-coding method is widely used during 3D measurement for attaining absolute phase retrieval. However, traditional the phase-coding method still suffers from low accuracy and insufficient reliability. To address these issues, a phase-shifting segmented phase coding method was proposed in this work for absolute phase retrieval in complex object measurements, which specifically improved upon the segmented method of phase coding. Coding patterns with different phase-shifting were obtained by embedding the codeword into the phase-shifting phase. These patterns were also combined and arranged to create novel segmented coding patterns. In addition, this segmented encoding can be self-connected based on its different phase-shifting information to obtain the complete fringe order, achieving thus absolute phase retrieval of over 36 codewords. To solve the problem of the periodic misalignment of the wrapped phase and fringe order, the complementary phase coding method was encoded by the bidirectional coding method, and the separation method of the bidirectional coding was improved. From experimental results, it was demonstrated that the proposed method has high measurement accuracy and reliability.

Generalized fringe order correction based on phase shift segmentation for absolute phase retrieval

We present a fringe order correction method for various absolute phase retrieval algorithms. In fringe projection profilometry based on absolute phase retrieval (FPP-APR), the mismatches tend to be easily generated around the 2π discontinuities or edges of the wrapped phase due to the defocus and noise of the system. In this paper, a staggered wrapped phase with a phase shift of 2π / N segments the calculated wrapped phase into a left-wrapped phase and a right-wrapped phase. Then, a statistical method independently determines the fringe order of each closed-wrapped phase without the 2π discontinuities. The experiments demonstrate that the proposed fringe order correction has the generality and simplicity for the existing absolute phase retrieval and performs high robustness to measure the colorfully complex objects of daily life.

Single-Shot Multi-Frequency 3D Shape Measurement for Discontinuous Surface Object Based on Deep Learning

It is challenging to stably and rapidly achieve accurate absolute phase retrieval for isolated objects with a single-shot pattern in fringe projection profilometry (FPP). In this context, a single-shot multi-frequency absolute phase retrieval (SAPR-DL) method based on deep learning is proposed, which only needs to capture one fringe image to obtain the full-field precise absolute phase. Specifically, a low-frequency deformed fringe image is loaded into the trained one-to-two deep learning framework (DLFT) to predict unit-frequency and high-frequency deformed fringe images. Then, three fringe images with different frequencies are loaded into the trained deep learning phase retrieval framework (DLPR) to calculate the corresponding absolute phase. The experimental results prove that the proposed SAPR-DL method can obtain the three-dimensional (3D) shape measurement of multiple complex objects by collecting a single-shot fringe image, showing great prospects in advancing scientific and engineering applications.

Hue-indexing-based absolute phase retrieval method using a discrete hue sequence

To achieve an effective phase unwrapping for hue-based fringe projection profilometry, this paper proposes a hue-indexing-based absolute phase retrieval method using a discrete hue sequence. First, the hue component is extracted as the wrapped phase for 3D reconstruction by projecting a programmed hue fringe pattern. Afterward, a hue-indexing sequence with a random combination of six unique hues from the hue map is designed for hue unwrapping. By assigning a hue index of the fringe geometric center, the defocusing effect in the hue unwrapping is corrected, where the fringe order of the wrapped hue is then uniquely identified. The simulations show that the root mean square (RMS) of the residual error is 2.2185×10-4 r a d, and the effectiveness of the proposed method is further verified through experiments using a plaster statue and a compressor blade. The measurement comparison of the proposed method and the coordinate measuring machine is provided, where the RMS error of the measured deviation is 4.09×10-2 m m.

Absolute phase retrieval using one coding pattern for the dynamic 3-D measurement

This paper proposes an absolute phase retrieval using one additional coding pattern to measure the complex object. Compared with traditional temporal phase unwrapping (TPU) methods, the proposed coding pattern is generated from phase-shifting patterns and the phase difference rather than stair stripes containing different quantified gray-level. In this paper, N-step phase-shifting profilometry (PSP) can provides 2 N codewords, which can be used to label the large number of fringe orders. To ensure measuring accuracy, the errors closed to turning points can be identified by quantified phase difference, and then compensated. Experimental results demonstrate that the proposed method can achieve the same level of measuring accuracy as the classical PSP, and has strong robustness on the measurement of the object with sharp and colorful surfaces. The dynamic experiment shows that the proposed method only requiring one additional pattern can achieve the 3-D reconstructed rate of 30 fps.

Half-Period Gray-Level Coding Strategy for Absolute Phase Retrieval

N-ary gray-level (nGL) coding strategy is an effective method for absolute phase retrieval in the fringe projection technique. However, the conventional nGL method contains many unwrapping errors at the boundaries of codewords. In addition, the number of codewords is limited in only one pattern. Consequently, this paper proposes a new gray-level coding method based on half-period coding, which can improve both these two deficiencies. Specifically, we embed every period with a 2-bit codeword, instead of a 1-bit codeword. Then, special correction and decoding methods are proposed to correct the codewords and calculate the fringe orders, respectively. The proposed method can generate n2 codewords with n gray levels in one pattern. Moreover, this method is insensitive to moderate image blurring. Various experiments demonstrate the robustness and effectiveness of the proposed strategy.

Absolute phase retrieval of shiny objects using fringe projection and deep learning with computer-graphics-based images

Fringe projection profilometry is a high-precision method used to measure the 3D shape of an object by projecting sinusoidal fringes onto an object. However, fringes projected onto a metallic or shiny object are distorted nonlinearly, which causes significant measurement errors. A high-precision measurement method for shiny objects that employs computer graphics (CG) and deep learning is proposed. We trained a deep neural network by projecting fringes on a shiny object in CG space. Our results show that the method can reduce the nonlinear fringe distortion caused by gloss in real space.

Improved dual-frequency phase-coding fringe projection method for 3D shape measurement

The phase-coding method is useful in 3D shape measurement due to its low sensitivity to surface contrast and camera noise. However, as the number of fringe periods increases, the wrapped phase recovered from the sinusoidal phase-shifting patterns is always misaligned with the fringe order calculated by the stair-coding patterns. To solve this, an improved dual-frequency phase-coding fringe projection method is designed to realize absolute phase retrieval. Specifically, the auxiliary fringe order determined by phase-shifting is used to get the absolute phase, which can eliminate fringe-order errors. Moreover, to increase speed and the number of codewords, this paper presents a scheme that only requires seven patterns, including a uniform gray image, two high-frequency sinusoidal fringe patterns, two low-frequency sinusoidal fringe patterns and two phase-coding fringe patterns. Therefore, this method requires fewer patterns to realize high-resolution measurement than existing methods. The success of the proposed method can be proved by experiments.

A Universal Self-Correcting Approach for Abnormal Jump Errors in Absolute Phase Retrieval

Codeword-assisted temporal phase unwrapping (TPU) is extensively applied in absolute phase retrieval, in which the fringe order embedded in the temporal sequence identifies the interval of the wrapped phase. However, owing to inevitable image blurring and random noise, the fringe order cannot be aligned with the interval of the wrapped phase at the pixel level and tends to fail easily under high-frequency measurements, thereby yielding abnormal jump errors. Existing approaches suffer from the limitations of universality, efficiency, and fringe frequency. To this end, this work proposes a universal self-correcting approach to remove abnormal jump errors efficiently and accurately. Specifically, the virtual fringe order is introduced by analysing the intensity changes of the phase-shifted patterns, and then it is combined with the real fringe order to form a complementary fringe order pair. Additionally, to enhance the anti-interference ability of the virtual fringe order, the fringe order histogram check technique (FOHCT) is proposed to eliminate fringe order errors. The proposed approach can efficiently conduct various codeword-assisted absolute phase retrieval without introducing additional patterns. Furthermore, benefitting from no error accumulation, our approach is applicable for high-frequency measurements, thereby improving the measurement capability. The measurement results validated the universality and efficiency of the proposed approach. High-frequency measurements demonstrated that the proposed approach outperformed the median filtering and the phase coding approach in terms of error rate (0.009% vs. 3.13% vs. 1.76%).

The Absolute Phase Retrieval Based on the Rotation of Phase-Shifting Sequence

The absolute phase retrieval has been the focus of developments in three-dimensional (3D) shape measurement. The traditional absolute phase retrievals based on phase-shifting algorithm mainly utilize extra patterns to generate fringe order map, which decreases the measuring speed and income more phase shift error. In this paper, the cue of fringe order is directly embedded into phase-shifting patterns to minimize the number of projecting patterns. Fringe order is encoded by rotating the sequence of phase shift amounts, causing the 2&#x03C0; discontinuity to be extended into &#x00B1;2&#x03C0;/<i>N</i>, &#x00B1;4&#x03C0;/<i>N</i>,..., &#x00B1;2(<i>N</i>-1)&#x03C0;/<i>N</i>, &#x00B1;2&#x03C0;. Due to the coding periodicity, the regional wrapped phase without 2&#x03C0; ambiguity can be extracted from different wrapped phase maps. When decoding, the coding phase jump points can be located and identified from one or multiple wrapped phase maps, and then used to determine fringe order by the bidirectional data processing. Experimental results have proved that the proposed method without extra coding patterns not only provides high robustness to shadow and considerable accuracy closed to that of traditional phase-coded method, but also inherits the performance to measure the complex and colorful object using traditional sinusoidal phase-shifting patterns.

Self-Unwrapping Phase-Shifting for Fast and Accurate 3-D Shape Measurement

Phase unwrapping is one of the most important procedures in three-dimensional phase measurement profilometry and has attracted great research interest in the past few decades. However, existing phase unwrapping methods require additional patterns or pre-assumptions to determine the fringe orders for further absolute phase retrieval, which dramatically affect the measurement efficiency. To address this problem, we propose a generic self-unwrapping phase-shifting algorithm that retrieves the absolute phase without external information or priors. To this end, we firstly embed a novel space-varying phase shift that uniquely determines the fringe order information into sinusoidal patterns, and then extract it to retrieve the absolute phase by pixel-wise calculation. Our method achieves higher efficiency over previous methods while preserving the measurement precision, and the minimum number of patterns required is only four. All theoretical, simulation and extensive experimental results demonstrate its superiority in fast and accurate 3D shape measurement.

Absolute phase retrieval for a single-shot fringe projection profilometry based on deep learning

A deep learning-based method is proposed to recover the absolute phase value from a single fringe pattern. We propose a deep neural network architecture that includes two subnetworks used for wrapping phase calculation and phase unwrapping, respectively. The training set is generated with the absolute phase obtained by the combination of phase shifting and gray coding. In addition, a reference plane is adopted to provide periodic range information for phase unwrapping. Then according to the output of the well-trained network, a high-quality absolute phase is obtained through only a single fringe pattern of the measured object. Experiments on the test set verify that high accuracy for complex texture objects is acquired using the proposed method, which indicates its potential in high-speed measurement.

Optimized phase-coding method for absolute phase retrieval based on K-means algorithm

The phase-coding method for absolute phase measurement possesses a strong robustness because its codewords are encoded into phase domain rather than intensity. However, a misalignment problem arises between fringe order and wrapped phase in this method owing to random noises and image blurring, especially in the high-frequency fringes situation. Therefore, the measurement result is unavailable. To overcome this problem, an optimized phase-coding algorithm is proposed in the study. We use the sinusoidal patterns to generate two opposite binary masks, which can locate half of each codeword. Then, the K-means algorithm is used to correct the codewords errors. The accurate fringe order is obtained by merging two halves of codewords and the absolute phase is recovered afterward. The used method is experimentally verified by an established measurement system, and the results show the proposed method is robustness and efficiency.

Joint Coding Strategy of the Phase Domain and Intensity Domain for Absolute Phase Retrieval

Codewords used to determine fringe orders are significant for absolute phase retrieval based on the phase-coding method. However, because of inevitable interference from image blurring, image noise, and system nonlinearity, it is challenging to significantly increase the number of codewords for high-accuracy phase retrieval. To address this challenge, this article proposes a joint coding strategy to perform the coding of high-density stripes. More specifically, codewords are simultaneously embedded into the phase domain and the intensity domain of composite fringe patterns, which greatly extends the encoding range of valid codewords. Moreover, to improve the anti-interference capability of the proposed coding strategy, a regional correction technique (RCT) is presented to remove jump phase errors induced by the inevitable interference. The results of experiments indicate that the proposed method is effective and accurate for both complex objects and a standard sphere.

Absolute phase retrieval for colored objects based on three phase-shifting amount codes.

We propose an absolute phase retrieval method based on three phase-shifting amount codes (3-PSA-codes) to measure the colored object with one additional pattern. 3-PSA-codes adopt the coding concept of 3-digit-codes, in which the code elements of three consecutive periods are treated as a unique code word for one period. However, to measure the colored object more effectively in the proposed method, each code element is embedded into the PSA domain and retrieved from the phase difference. Fringe patterns for the wrapped phase are artfully employed in the code element retrieval. Hence, for the first time, to the best of our knowledge, the code element related to the phase can be determined by one additional pattern. It breaks the constraint that temporal methods require multiple additional patterns to overcome the adverse effect of the surface color of objects on absolute phase retrieval. Experimental results demonstrate that the proposed 3-PSA-codes have strong robustness in the measurement of the colored object.