Phase-coding Method 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.

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.

Self-correction of fringe order jump error induced by system nonlinearity based on phase-coding method

Self-correction of fringe order jump error induced by system nonlinearity based on phase-coding method

Pixel-Wise Phase Unwrapping With Adaptive Reference Phase Estimation for 3-D Shape Measurement

Phase unwrapping plays an important role in phase-based three-dimensional (3D) shape measurement. This paper proposes an effective pixel-wise phase unwrapping method without requiring additional fringe patterns. Firstly, compared with the geometric constraint-based (GCB) method which needs calibrate the fringe projection system and create a reference phase at the closest plane <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">z_min</i> within the measurement volume, the proposed method can adaptively and directly estimate the reference phase from the wrapped phase without system calibration and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">z_min</i> estimation. Secondly, based on the finding that the slope angle of the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">z_min</i> plane ignored by the GCB method has significant impact on the measurement range, thus the phase gradients that highly correlates with the optimal slope angle are involved into reference phase estimation. The wrapped phase can be pixel-by-pixel unwrapped by comparing with the reference phase. Both simulations and experiments have been conducted to demonstrate the efficiency of the proposed method. Compared with traditional two-frequency method and phase-coding method, the proposed method can improve the measuring speed by 2 times. Experiment results show that the proposed method can effectively remove the phase unwrapping errors and achieve the mean absolute phase error of 0.0137 rad, which has obvious advantages relative to traditional methods.

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.

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.

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.

Waveform-Coded Steady-State Visual Evoked Potentials for Brain-Computer Interfaces

This study presents a novel waveform-coding method for multi-target steady-state visual evoked potential (SSVEP)-based brain–computer interfaces (BCIs). Three periodic waveforms including square, sawtooth, and sinusoidal waves at various frequencies and initial phases were employed to elicit discriminable SSVEPs. A virtual keyboard was first designed using 36 visual stimuli modulated by the combinations of different frequencies, phases, and waveforms. With the virtual keyboard, 13 healthy participants performed offline and online BCI experiments with a cue-guided spelling task. The task-related component analysis (TRCA)-based algorithm was used to identify a target visual stimulus. The offline results showed that the visual stimuli tagged with different properties could accurately be identified by analyzing the elicited SSVEPs. Moreover, the online spelling task achieved promising performance with an averaged information transfer rate (ITR) of 62.6 ± 32.5 bits/min. This study validated the feasibility of implementing a multi-command SSVEP-based BCI using the hybrid waveform-, frequency- and phase-coding method. The proposed waveform-coding method provides a completely new channel for multi-target stimulus coding, expanding the research fields of an SSVEP-based BCI.

Two-Digit Phase-Coding Strategy for Fringe Projection Profilometry

Phase-coding methods have been widely used for three-dimensional (3-D) shape measurement, which use sinusoidal phase-shifting patterns to recover wrapped phase and the stair phase-coding patterns to determine the fringe order. However, due to the system nonlinearity, random noises, and image blurring, the total number of codewords that can be used is limited. In order to increase the number of codewords, this article presents a two-digit phase-coding (TDPC) strategy, in which each fringe of the sinusoidal phase-shifting patterns corresponds to a two-digit codeword encoded in the phase-coding patterns. In this way, more codewords can be generated without increasing additional patterns. Experiments of the cup and sculptures have been carried out, and the results confirm that the TDPC method can recover the 3-D shape of objects more robustly. Meanwhile, the precision experiment of a standard ball proves the high accuracy of the proposed method.

Dynamic three-dimensional shape measurement with a complementary phase-coding method

The traditional phase-coding (TPC) method has been widely used for three-dimensional (3D) shape measurement. However, the fringe order determined from the coding patterns cannot be precisely aligned with the wrapped phase calculated from the sinusoidal patterns when measuring dynamic objects. Consequently, phase unwrapping errors will occur around the 2π phase jumps of the wrapped phase. To address this problem, the present work proposes a complementary phase-coding (CPC) method for dynamic 3D shape measurement. Specifically, the projection of the sinusoidal and coding patterns forms a cycle and the se patterns are shifted by a half-period between even and odd cycles. Combining two fringe orders of adjacent even and odd cycles, the absolute phase map can be correctly recovered. Moreover, to improve the measurement speed, all 8-bit gray patterns are converted into 1-bit binary patterns by a binary dithering technique. Both simulations and experiments demonstrate the robustness and efficiency of the proposed method.

Enhanced phase-coding method for three-dimensional shape measurement with half-period codeword.

The phase-coding method has been widely used for 3D shape measurement, which uses sinusoidal phase-shifting patterns to recover the wrapped phase and the stair phase-coding patterns to determine the fringe order. However, due to random noises and image blurring, the fringe order is always misaligned with the wrapped phase, which will lead to fringe order errors. This paper presents an enhanced phase-coding method to address this misalignment problem by using half-period codewords, in which each codeword is aligned to the half-period of the sinusoidal patterns. Then, two complementary fringe orders with half-period dislocation can be calculated, which can effectively eliminate the fringe order errors. To extend the coding range of stair phase, this paper further develops a computational scheme based on the geometric constraint method. Simulations and experiments have been carried out, and their results confirm that the enhanced method can reliably recover the 3D shape of the measured objects.

An improved phase-coding method for absolute phase retrieval based on the path-following algorithm

Although the traditional phase-coding method for absolute phase retrieval possesses strong robustness, the method leads to unwrapping artifacts as a mass of codewords required. To solve this problem, a novel improved method is proposed in this paper. The incorrect points in the recovered phase via the traditional method are located using a computational framework, and the path-following algorithm is adopted as an aid to correct the unwrapping artifacts. The effectiveness of the proposed method is experimentally verified by an established measurement system. The reconstructed geometry of standard sphere has sufficient accuracy and the absolute phase retrieval of a complex star object has high quality.

Fast phase-measuring profilometry through composite color-coding method

Phase-measuring profilometry (PMP) has become one of the most effective means of three-dimensional (3D) shape measurement. Owing to the low robustness of the phase-unwrapping algorithm and the large number of projection patterns, the off-the-shelf PMP methods have a limited speed and accuracy for the measurement of discontinuous objects. To improve the measurement speed and accuracy, a composite color-coding method is presented in this paper. The main idea of the coding is embedding sinusoidal stripes, phase-coding stripes, and Gray code into the red, blue, and green channels of a color pattern, respectively. The Gray code is used for identifying the order of the sub-stair phase, which is quantized into m levels with a stair height of 2π/m. Each stair corresponds to the 2π phase-changed period of the sinusoidal fringes. The complexity of the phase unwrapping is low, and the proposed method needs fewer patterns for measurement than conventional methods. The experimental results indicate that the proposed method can achieve more accurate and efficient 3D shape measurement than the traditional phase-coding methods and is suitable for measuring discontinuous objects with complex shapes.

The Adaptive Multi-Level Phase Coding Method in Audio Steganography

Audio steganography allows and inspires many researchers to design methods for secure communication. Based on the evaluation on the existing methods, it was found that most methods focused on one or two requirements while disregarding others, causing imbalanced performance. Moreover, most methods lack adaptivity and dynamic allocation. Therefore, in this research, a method called Adaptive Multi-level Phase Coding (AMPC) was proposed to optimize the above issues. The reverse logic of the main tradeoffs was used to empirically design several embedding levels that that simultaneously attained good performance for all aspects as much as possible. Then, an adaptive component was added by selecting the embedding level that provided the best performance for each embedding process. Moreover, the error spreading factor was introduced to achieve a fair payload distribution. The performance balance objective requires a new formulation that will enable the accurate selection of the degree of modification, multiple-bit embedding per modification, and reduced retrieval errors. As a result, the interval centering quantization (ICQ) was formulated and implemented in the proposed method. The experimental results show that AMPC successfully fulfilled the research objectives. Also, AMPC surpassed other phase coding methods in all aspects while time-domain methods achieved the highest transparency and capacity with the lowest robustness. Moreover, experiments show that the implementation of adaptive multi-level concept is able to improve the existing method's performance significantly. In summary, AMPC was able to achieve a stable embedding rate of 33 Kbps at 35 dB of SNR, which is higher than the recorded embedding rate of other phase coding methods.

Optimal pulse width modulation technique combined with stair phase-coding method for absolute phase retrieval with projector defocusing

The accurate three-dimensional shape measurement of a complex surface is significantly important in industrial testing. However, three-dimensional profilometry by conventional sinusoidal fringe projection using a phase-shifting algorithm performs suboptimally because of the nonlinear intensity response of projectors. To overcome this problem, the present paper proposes using a combined optimizing sinusoidal pulse width modulation (SPWM) technique and stair phase-coding approach to obtain the unwrapped phase. Properly optimizing SPWM with a small degree of a defocusing projector to generate sinusoidal fringe patterns can overcome the undesired harmonics and nonlinear gamma effect. Two groups of four-step phase-shifting fringe patterns are used. One group of the stripes contains four sinusoidal patterns generated by the SPWM technique, which is used to determine the wrapped phase. The other group of stripes contains four sinusoidal patterns with the codeword embedded into the stair phase, whose stair changes are perfectly aligned with the 2π discontinuities of the sinusoidal fringe phase, which is used to determine the fringe order for the phase unwrapping. Moreover, for the defocusing projection, because the frequency of the phase-coding fringe increases, the fringe order decision becomes less reliable. Thus, a self-correction phase unwrapping method is applied for phase retrieval. Experiments were conducted to verify the performance of the proposed method.

Regulation of multiple off-axis acoustic vortices with a centered quasi-plane wave

In treating a circular point-source array using the phase-coded method, a composite acoustic field of multiple off-axis acoustic vortices (AVs) with a centered quasi-plane wave is proposed which is superimposed by an AV beam and a co-axial non-AV beam generated by the even- and odd-numbered source arrays, respectively. The acoustic pressure and the phase of the composite acoustic field are derived using explicit formulae and demonstrated by numerical simulations. Off-axis sub-AVs (SAVs) are shown to be formed at the intersections of the radial pressure distributions of the AV and non-AV beams for the same pressure amplitude and opposite phases. Off-axis SAVs can be generated on a circumference centered with a regular polygon of quasi-plane waves. The radii and azimuthal angles of the off-axis AVs and the radius of the centered quasi-plane wave are determined by the topological charge and the initial phase difference of the acoustic beams. With the established 16-source experimental system, the generation of multiple off-axis SAVs with a centered quasi-plane wave is demonstrated by scanning measurements of the composite field with a topological charge of 1, 2, or 3, which is also proved by the qualitatively consistent simulations for N = 16. The proposed model may be applied to multi-point vortex trapping with homogeneous particle dispersion within a finite area around the central axis. These favorable results provide the feasibility of controllable manipulation for drug particles or cells and suggest wide potential applications in the fields of sonochemistry and biomedical engineering.

Improved phase-coding methods with fewer patterns for 3D shape measurement

Absolute phase information is required when multiple separated objects are being measured. The recent proposed phase-coding method can effectively achieve absolute phase retrieval by embedding the fringe order information in phase domain. However, it suffers from the limitation of codewords and the requirement of more coding patterns. To overcome these problems, this paper proposed several improved phase-coding methods to solve the absolute phase with fewer patterns. The four-frame and five-frame method are suitable when the fringes are sparse, since they can respectively encode N∕2 and N fringe periods(generally N≤16). If really dense fringe patterns with more than N periods are required, the improved two-step phase-coding algorithm can be adopted since it can encode about N2∕2 fringe periods. Experiments were conducted and the results verify the success of the proposed methods.

Absolute phase retrieval via color phase-coding

Phase-coding method that embeds codewords into phase rather than intensity performs well for absolute phase retrieval. However, this method becomes instable when dealing with a large number of codewords. This paper designs three color phase-coding fringes to determine the fringe order pixel by pixel. Specifically, six groups of N-stairs are encoded into three coding fringes, and each group is endowed with a different color to eliminate ambiguity between groups. Such that three additional fringes can generate more than 6N codewords by combining the color and phase information. Experiment results confirmed the validity of the proposed method.

A Flexible Fringe Projection Vision System with Extended Mathematical Model for Accurate Three-Dimensional Measurement.

In order to acquire an accurate three-dimensional (3D) measurement, the traditional fringe projection technique applies complex and laborious procedures to compensate for the errors that exist in the vision system. However, the error sources in the vision system are very complex, such as lens distortion, lens defocus, and fringe pattern nonsinusoidality. Some errors cannot even be explained or rendered with clear expressions and are difficult to compensate directly as a result. In this paper, an approach is proposed that avoids the complex and laborious compensation procedure for error sources but still promises accurate 3D measurement. It is realized by the mathematical model extension technique. The parameters of the extended mathematical model for the ’phase to 3D coordinates transformation’ are derived using the least-squares parameter estimation algorithm. In addition, a phase-coding method based on a frequency analysis is proposed for the absolute phase map retrieval to spatially isolated objects. The results demonstrate the validity and the accuracy of the proposed flexible fringe projection vision system on spatially continuous and discontinuous objects for 3D measurement.