Phase Measurements Research Articles

MOSARIX: Multi-crystal spectrometer in the tender x-ray range at SOLEIL synchrotron.

We have built and commissioned a novel standalone multi-crystal x-ray spectrometer (MOSARIX) in the von Hamos configuration based on highly annealed pyrolytic graphite crystals. The spectrometer is optimized for the energy range of 2-5keV, but this range can be extended up to 20keV by using higher reflection orders. With its nine crystals and a Pilatus detector, MOSARIX achieves exceptional detection efficiency with good resolving power (better than 4000), opening the door to study small cross section phenomena and perform fast in situ measurements. The spectrometer operates under a He atmosphere, which provides a flexible sample environment for measurements in gas, liquid, and solid phases.

Advancements in laser-based spatiotemporal measurements of flow boiling

Recent advancements in laser and imaging systems, as well as in computational processing capabilities, have made quantitative optical imaging, which is often combined with laser illumination, highly adaptable, robust and reliable. Laser-based diagnostic techniques, such as planar laser-induced fluorescence (PLIF), offer the possibility of simultaneous spatiotemporally resolved measurements of temperature fields in the liquid phase at boiling conditions. In this paper, we examine the applicability of two-colour PLIF (2cPLIF), where the ratio between individual fluorescent emissions from uniformly dispersed dyes is used to take temperature measurements in the liquid phase in the presence of moving vapour-liquid interfaces typical of boiling flow. The implementation of 2cPLIF necessitates uniformity in the concentration of different dyes across the flow field. However, in the case of a multiphase flow such as boiling, thermophoresis can lead to inhomogeneous dye distributions. To overcome this challenge, a single-dye multispectral planar laser-induced fluorescence (SDMS-PLIF) method has been developed, which employs fluorescent emissions in different spectral bands of the same dye (Nile Red). The spectral characteristics of Nile Red were measured using a spectrometer to identify its temperature-sensitive bands over a wide range of dye concentrations, from 0.3 to 30 mg/L. Following this, we demonstrate the measurement capabilities of SDMS-PLIF thermography as applied to a boiling flow in a miniaturised vertical square channel, gaining insight into the thermohydrodynamic interactions between vapour bubbles and a heated wall.

Analysis of Multipath Changes in the Polish Permanent GNSS Stations Network

This study examines the influence of multipath errors on Global Navigation Satellite System (GNSS) measurements collected at ASG-EUPOS reference stations between 2010 and 2021. Multipath occurs when GNSS signals reflect off surrounding objects before reaching the receiver antenna, leading to positioning errors. In the case of reference stations, all available mitigation techniques were used to minimize the impact of multipath. However, it is still detectable and affects the measurement results. For carrier phase differential positioning, it increases the ambiguous search space, which results in a decrease in determining rover—reference station vector accuracy. The study employs two linear combinations (Code-Minus-Carrier and Multipath Pseudorange Observable) to quantify the multipath effect on both pseudorange and carrier phase measurements. Based on the research, it was found that the multipath values changed depending on the change of the receiver and the terrain around the reference stations. The study observed a gradual decrease in multipath errors from 2010 to 2021, likely due to technological advancements in receiver design. No significant increase in multipath errors was observed due to environmental changes around the stations, suggesting a minimal influence from new reflecting objects nearby. Based on the analyses conducted, it is also recommended to perform periodic tests to detect incorrect receiver configuration or operation.

A method to enhance the accuracy of radio frequency power measurements in particle accelerator high‐frequency systems

AbstractThis letter presents a method for accurately calculating radio frequency (RF) power in a standing wave transmission line system. The proposed approach utilizes two directional couplers in conjunction with a low‐level radio frequency control system. RF power is determined through the application of proposed equations derived from amplitude and phase measurements of coupling signals obtained from the two directional couplers. Experimental validation of the method is conducted using two directional couplers with directivities of 26 and 28 dB, respectively, operating at 162.5 MHz across the entire wavelength. Results indicate that employing a single directional coupler yields average power errors of approximately 6.6% and 5.8%, respectively. However, by implementing the proposed method, the measurement error can be reduced to approximately 1.0%.

Evaluating technology enhanced learning by using single‐case experimental design: A systematic review

AbstractSingle‐case experimental designs (SCEDs) may offer a reliable and internally valid way to evaluate technology‐enhanced learning (TEL). A systematic review was conducted to provide an overview of what, why and how SCEDs are used to evaluate TEL. Accordingly, 136 studies from nine databases fulfilling the inclusion criteria were included. The results showed that most of the studies were conducted in the field of special education focusing on evaluating the effectiveness of computer‐assisted instructions, video prompts and mobile devices to improve language and communication, socio‐emotional, skills and mental health. The research objective of most studies was to evaluate the effects of the intervention; often no specific justification for using SCED was provided. Additionally, multiple baseline and phase designs were the most common SCED types, with most measurements in the intervention phase. Frequent data collection methods were observation, tests, questionnaires and task analysis, whereas, visual and descriptive analysis were common methods for data analysis. Nearly half of the studies did not acknowledge any limitations, while a few mentioned generalization and small sample size as limitations. The review provides valuable insights into utilizing SCEDs to advance TEL evaluation methodology and concludes with a reflection on further opportunities that SCEDs can offer for evaluating TEL.Practitioner notesWhat is already known about this topic SCEDs use multiple measurements to study a single participant over multiple conditions, in the absence and presence of an intervention SCEDs can be rigorous designs for evaluating behaviour change caused by any intervention, including for testing technology‐based interventions. What this paper adds Reveals patterns, trends and gaps in the use of SCED for TEL. Identifies the study disciplines, EdTech tools and outcome variables studied using SCEDs. Provides a comprehensive understanding of how SCEDs are used to evaluate TEL by shedding light on methodological techniques. Enriches insights about justifications and limitations of using SCEDs for TEL. Implications for practice and/or policy Informs about the use of the rigorous method, SCED, for evaluation of technology‐driven interventions across various disciplines. Contributes therefore to the quality of an evidence base, which provides policymakers, and different stakeholders a consolidated resource to design, implement and decide about TEL.

Improved phase-to-height mapping method combine with device attitude

Phase-to-height mapping is one of the important processes in three dimensional phase measurement profilometry. But, in traditional phase-to-height mapping method, the measurement accuracy is affected by device attitude, so it needs saving a large amount of mapping equations to achieve high-quality phase-to-height mapping. In order to improve that, this paper proposes an improved phase-to-height mapping method combine with device attitude. Firstly, we get the unwrapped phase of the target. Then, using generalized regression neural network is used to reduce the offset of phase information at the same height due to the randomness of device attitude. Last, the phase-to-height mapping is completed by substituting the unwrapped phase (the difference between having detected object and no detected object) of eliminate the offset into improved phase-to-height mapping method. Experimental results show that the proposed method could achieve high-quality phase-to-height mapping with less mapping equation and less memory space. Compared with the nonlinear phase-to-height mapping method (probabilistic neural network to eliminate phase offset), its accuracy is improved by 44.30%. Compared with the nonlinear phase-to-height mapping method (radial basis function neural network to eliminate phase offset), the accuracy is improved by 39.58%.

Vortex shedding behind porous flat plates normal to the flow

This work examines the influence of body porosity on the wake past nominally two-dimensional rectangular plates of fixed width $D$ in the moderate range of Reynolds numbers $Re = UD/\nu$ (with $U$ the incoming velocity and $\nu$ the kinematic viscosity) between 15 000 and 70 000. With porosity $\beta$ defined as the ratio of open to total area of the plate, it is well established that as porosity increases, the wake shifts from the periodic von Kármán shedding behaviour to a regime where this vortex shedding is absent. This change impacts the fluid forces acting on the plate, especially the drag, which is significantly lower for a wake without vortex shedding. We analyse experimentally the transition between these two regimes using hot-wire anemometry, particle-image velocimetry and force measurements. Coherence and phase measurements are used to determine the existence of regular, periodic vortex shedding based on the velocity fluctuations in the two main shear layers on either side of the wake. Results show that, independent of $Re$ , the wake exhibits the classical Kármán vortex shedding pattern for $\beta <0.2$ but this is absent for $\beta >0.3$ . In the intermediate range, $0.2<\beta <0.3$ , there is a transitional regime that has not previously been identified; it is characterised by intermittent shedding. The flow alternates randomly between a vortex shedding and a non-shedding pattern and the total proportion of time during which vortex shedding is observed (the intermittency) decreases with increasing porosity.

Analyzing the added value of common data environments for organizational and project performance of BIM-based projects

Using common data environments (CDEs) adds value to BIM-based construction projects' organizational and project performance. However, CDEs are used very limitedly in the construction industry. One of the reasons for the limited implementation of CDEs can be associated with the need for construction companies' knowledge about the positive impacts of CDEs on performance management. A well-structured CDE can provide countless benefits and promote long-term improvements in construction projects and organizations, increasing their business success. Despite the acknowledged importance of CDEs, research needs to investigate the impacts of CDEs on project and organizational performance considering the construction KPIs. This study aims to reveal (1) how the CDEs facilitate performance measurement in the construction phase of BIM-based projects and (2) how the CDEs positively affect the project and organizational performance in the construction phase of BIM-based projects. This scope uses seven construction KPIs: time, cost, quality, safety, productivity, organizational sustainability, and client satisfaction. This study conducts a systematic literature review, semi-structured interviews with five subject-matter experts, and the two-rounded Delphi method to fulfill the research objective. The results show that implementing the CDEs in the construction phase of BIM-based projects positively affects productivity, quality, and time KPIs, followed by organizational sustainability, cost, client satisfaction, and safety, respectively. This research contributes to collating and uncovering the added value of CDEs for the organizational and performance management of BIM-based projects. Accordingly, this study would increase the awareness of construction companies about ‘how they can benefit from the data located in the CDEs from project management through knowledge management in the best way'.

Alteration of circadian sleep-wake rhythm and salivary melatonin secretion in idiopathic/isolated REM sleep behavior disorder: Preliminary evidence

Objective/BackgroundIdiopathic/isolated REM sleep behavior disorder (iRBD) is widely regarded as an early sign of neurodegeneration leading to synucleinopathies. While circadian rhythm alterations in iRBD have been preliminarily demonstrated, evidence on melatonin secretion patterns in this clinical condition is limited. To address this knowledge gap, this exploratory study aimed to integrate salivary melatonin measurement with actigraphic monitoring in individuals with iRBD and age-matched healthy controls (HC) under real-life conditions. MethodsParticipants diagnosed with iRBD and HC underwent clinical evaluation and wore an actigraph for seven days and nights. Salivary melatonin concentrations were measured at five time points during the last night of recording. Comparative analyses were conducted on clinical data, actigraphic parameters, and melatonin levels between the two groups. ResultsiRBD participants (n = 18) showed greater motor (p < 0.01) and non-motor symptoms (p < 0.001), alongside disruptions in circadian sleep-wake rhythm compared to HC (n = 10). Specifically, actigraphy revealed a delayed central phase measurement (p < 0.05), reduced activity during the most active hours (p < 0.001), and decreased relative amplitude (p < 0.05). Total salivary melatonin concentration was significantly lower in iRBD (p < 0.05), with a slight but non-significant phase delay in dim light melatonin onset. ConclusionsThis exploratory study highlights a dysregulation of circadian sleep-wake rhythm coupled with reduced melatonin secretion in iRBD. Future research could add to these preliminary findings to evaluate novel treatment approaches to regulate the sleep-wake cycle and elucidate the implications of circadian dysregulation in the conversion from iRBD to neurodegeneration.

Room-scale Location Trace Tracking via Continuous Acoustic Waves

The increasing prevalence of smart devices spurs the development of emerging indoor localization technologies for supporting diverse personalized applications at home. Given marked drawbacks of popular chirp signal-based approaches, we aim at developing a novel device-free localization system via the continuous wave of the inaudible frequency. To achieve this goal, solutions are developed for fine-grained analyses, able to precisely locate moving human traces in the room-scale environment. In particular, a smart speaker is controlled to emit continuous waves at inaudible 20kHz , with a co-located microphone array to record their Doppler reflections for localization. We first develop solutions to remove potential noises and then propose a novel idea by slicing signals into a set of narrowband signals, each of which is likely to include at most one body segment’s reflection. Different from previous studies, which take original signals themselves as the baseband, our solutions employ the Doppler frequency of a narrowband signal to estimate the velocity first and apply it to get the accurate baseband frequency, which permits a precise phase measurement after I-Q (i.e., in-phase and quadrature) decomposition. A signal model is then developed, able to formulate the phase with body segment’s velocity, range, and angle. We next develop novel solutions to estimate the motion state in each narrowband signal, cluster the motion states for different body segments corresponding to the same person, and locate the moving traces while mitigating multi-path effects. Our system is implemented with commodity devices in room environments for performance evaluation. The experimental results exhibit that our system can conduct effective localization for up to three persons in a room, with the average errors of 7.49 cm for a single person, with 24.06 cm for two persons, with 51.15 cm for three persons.

Spinet-QSM: model-based deep learning with schatten p-norm regularization for improved quantitative susceptibility mapping.

Quantitative susceptibility mapping (QSM) provides an estimate of the magnetic susceptibility of tissue using magnetic resonance (MR) phase measurements. The tissue magnetic susceptibility (source) from the measured magnetic field distribution/local tissue field (effect) inherent in the MR phase images is estimated by numerically solving the inverse source-effect problem. This study aims to develop an effective model-based deep-learning framework to solve the inverse problem of QSM. This work proposes a Schatten -norm-driven model-based deep learning framework for QSM with a learnable norm parameter to adapt to the data. In contrast to other model-based architectures that enforce the l -norm or l -norm for the denoiser, the proposed approach can enforce any -norm ( ) on a trainable regulariser. The proposed method was compared with deep learning-based approaches, such as QSMnet, and model-based deep learning approaches, such as learned proximal convolutional neural network (LPCNN). Reconstructions performed using 77 imaging volumes with different acquisition protocols and clinical conditions, such as hemorrhage and multiple sclerosis, showed that the proposed approach outperformed existing state-of-the-art methods by a significant margin in terms of quantitative merits. The proposed SpiNet-QSM showed a consistent improvement of at least 5% in terms of the high-frequency error norm (HFEN) and normalized root mean squared error (NRMSE) over other QSM reconstruction methods with limited training data.

Effects of cognitive behavioural therapy and bright light therapy for insomnia in youths with eveningness: study protocol for a randomised controlled trial

BackgroundInsomnia and eveningness are common and often comorbid conditions in youths. While cognitive behavioural therapy for insomnia (CBT-I) has been suggested as a promising intervention, it remains unclear whether it is sufficient to also address circadian issues in youths. In addition, despite that light has been shown to be effective in phase-shifting one’s circadian rhythm, there has been limited data on the effects of bright light therapy and its combination with CBT-I on sleep and circadian outcomes in youths. The current protocol outlines a randomised controlled trial that examines the efficacy of CBT-I and CBT-I plus bright light therapy (BLT) in reducing insomnia severity, improving mood symptoms and daytime functioning (e.g. sleepiness, fatigue, cognitive function), and improving subjective and objective sleep and circadian measures compared to a waitlist control group.MethodsWe will carry out a randomised controlled trial (RCT) with 150 youths aged 12–24 who meet the criteria of insomnia and eveningness. Participants will be randomised into one of three groups: CBT-I with bright light therapy, CBT-I with placebo light, and waitlist control. Six sessions of CBT-I will be delivered in a group format, while participants will be currently asked to use a portable light device for 30 min daily immediately after awakening throughout the intervention period for bright light therapy. The CBT-I with light therapy group will receive bright constant green light (506 lx) while the CBT-I with placebo light group will receive the modified light device with the LEDs emitting less than 10 lx. All participants will be assessed at baseline and post-treatment, while the two active treatment groups will be additionally followed up at 1 month and 6 months post-intervention. The primary outcome will be insomnia severity, as measured by the Insomnia Severity Index. Secondary outcomes include self-reported mood, circadian, daytime functioning, and quality of life measures, as well as sleep parameters derived from actigraphy and sleep diary and neurocognitive assessments. Objective measures of the circadian phase using dim-light melatonin onset assessment and sleep parameters using polysomnography will also be included as the secondary outcomes.DiscussionThis study will be the first RCT to directly compare the effects of CBT-I and BLT in youths with insomnia and eveningness. Findings from the study will provide evidence to inform the clinical management of insomnia problems and eveningness in youths.Trial registrationClinicalTrials.gov NCT04256915. Registered on 5 February 2020.

High-resolution cell imaging using white light phase shifting interferometry and iterative phase deconvolution.

An optimization algorithm is presented for the deconvolution of a complex field to improve the resolution and accuracy of quantitative phase imaging (QPI). A high-resolution phase map can be recovered by solving a constrained optimization problem of deconvolution using a complex gradient operator. The method is demonstrated on phase measurements of samples using a white light based phase shifting interferometry (WLPSI) method. The application of the algorithm on real and simulated objects shows a significant resolution and contrast improvement. Experiments performed on Escherichia coli bacterium have revealed its sub-cellular structures that were not visible in the raw WLPSI images obtained using a five phase shifting method. These features can give valuable insights into the structures and functioning of biological cells. The algorithm is simple in implementation and can be incorporated into other QPI modalities .

A fringe jump counting method for the phase measurement in the HCN laser interferometer on EAST and its FPGA-based implementation

Electron density in fusion plasma is usually diagnosed using laser-aided interferometers. The phase difference signal obtained after phase demodulation is wrapped, which is also called a fringe jump. A method has been developed to unwrap the phase difference signal in real time using FPGA, specifically designed to handle fringe jumps in the hydrogen cyanide (HCN) laser interferometer on the EAST superconducting tokamak. This method is designed for a phase demodulator using the fast Fourier transform (FFT) method at the front end. The method is better adapted for hardware implementation compared to complex mathematical analysis algorithms, such as field programmable gate array (FPGA). It has been applied to process the phase measurement results of the HCN laser interferometer on EAST in real time. Electron density results show good confidence in the fringe jump unwrapping method. Further possible application in other laser interferometers, such as the POlarimeter-INTerferometer (POINT) system on EAST tokamak is also discussed.

Algorithm for estimating the absolute total electron content of the ionosphere from dual-frequency phase and range satellite measurements

Objectives. The problem of developing an algorithm for estimating the absolute total electron content of the ionosphere from dual-frequency phase and range satellite measurements for a single receiving station of global navigation satellite systems is being solved.Methods. To obtain an estimate the phase measurement data are corrected using digital signal processing methods, well known total electron content formulas for phase and range measurements are applied and combined, and also the differential code bias of the receiving station is estimated using the least squares method.Results. It is shown that the total electron content calculated from phase measurements provides high accuracy, but up to an unknown constant, but the content calculated from range measurements allows one to obtain the absolute value, but with a large noise component and differential code bias of a satellite and receiver equipment. An algorithm for estimating the absolute total electron content of the ionosphere has been developed, its description and diagram are given. The algorithm was used to estimate the total electronic content within six months of observations, and the average error of the resulting estimate was calculated.Conclusion. The developed algorithm can be used to estimate the absolute total electron content of the ionosphere for a single receiving station of global navigation satellite systems. In contrast to theoretically known formulas for phase and range measurements, this article contains information about adjusting phase measurements and estimating the differential code delay of receiving station. Further research may be related to the adaptive selection of parameters and testing of the algorithm for working with nanosatellites of the CubeSat format.

High-Precision Time Difference of Arrival Estimation Method Based on Phase Measurement

In unmanned aerial vehicle (UAV)-based time difference of arrival (TDOA) positioning technique, baselines are limited due to communication constraints. In this case, the accuracy is highly sensitive to the TDOA measurements’ error. This article primarily addresses the problem of short-baseline high-precision time synchronization and TDOA measurement. We conducted a detailed analysis of error models in TDOA systems, considering both the time and phase measurement. We utilize the frequency division wireless phase synchronization technique in TDOA systems. Building upon this synchronization scheme, we propose a novel time delay estimation method that relies on phase measurements based on the integer least squares method. The performance of this method is demonstrated through Monte Carlo simulations and outdoor experiments. The standard deviations of synchronization and TDOA measurements in experiments are 1.12 ps and 1.66 ps, respectively. Furthermore, the circular error probable (CEP) accuracy is improved from 0.33%R to 0.02%R, offering support for the practical application of distributed short-baseline high-precision passive location techniques.

Evaluating an advanced system with flexible dielectric insole and inertial sensors for post-ACL reconstruction assessment

ObjectiveThis study aims to validate an integrated system combining capacitive pressure-sensing insoles with wearable inertial sensors to evaluate rehabilitation progress after ACL reconstruction. MethodsA total of 60 participants were recruited. The study proceeded in three stages: initial evaluation of the pressure sensors' sensitivity, then durability, accuracy assessment, and kinematics, and finally the return-to-play evaluation for post-ACL reconstruction patients. The study analyzed various gait parameters, jump tests (single-leg jumps, triple hops, and crossover hops), and the Limb Symmetry Index (LSI). ResultsThe study found comparable cadence rates (63.68 ± 3.72 steps/min for the integrated system vs. 59.73 ± 3.18 steps/min for Vicon), walking speeds (61.38±2.72 m/min vs. 64.92 ± 2.13 m/min), and step lengths (113.63 ± 9.12 cm vs. 114.26 ± 8.35 cm), the stance phase (61.18 ± 1.89% vs. 62.15 ± 2.25 %) and swing phase (33.14 ± 2.19% vs. 32.72 ± 1.92 %) measurements, along with peak joint angles for the ankle (18.2 ± 2.9° vs. 19.1 ± 2.3°), knee (63.3 ± 4.1° vs. 65.1 ± 4.4°), and hip (21.1 ± 3.4° vs. 22.5 ± 3.2°). Results from jump tests and the Limb Symmetry Index (LSI) were also analyzed. ConclusionThe integrated system offers a novel approach in evaluating lower extremity biomechanics post-ACL reconstruction, showing significant potential for improvement in sports medicine rehabilitation.

A method to build feature descriptor for GNSS spoofing detection by carrier phase double difference measurement

GNSS spoofing is a type of attack that aims to deceive GNSS receivers by transmitting fake signals that imitate the authentic ones. To detect such attacks, a possible solution is to calculate the double difference (DD) of carrier phase measurements between two antennas of two separated receivers. This DD measurement represents the angle of arrival (AoA) information of the signal. In the case of authentic signals, each signal comes from a different satellite, so the AoA values are different for each satellite. However, in the case of a spoofing attack, all the counterfeit signals from the same broadcaster come from the same direction, and thus the AoA values are the same for all DD measurements. Sum of Squares (SoS) detector and Dispersion of Double Difference (D3) detector are known as simple but effective methods for spoofing detection. However, these detectors are not robust in some special regions where cycle slips are large and vary rapidly. This paper proposes a new method for detecting GNSS spoofing by using carrier phase double difference measurements to generate a feature vector as a descriptor for each satellite in a set of observed satellites. The feature vectors are formed by statistical parameters of a series of consecutive double difference values computed for each reference satellite. Our proposed feature descriptor has been evaluated on a dataset containing both real and fake signals using an SVM classifier. The study shows that the proposed feature descriptor is highly effective in detecting spoofing satellite signals.

LC-MS/MS measurement of endogenous steroid hormones and phase II metabolites in blood volumetric absorptive microsampling (VAMS) for doping control purposes

BackgroundVolumetric Absorptive Microsampling (VAMS) is emerging as a valuable technique in the collection of dried biological specimens, offering a potential alternative to traditional sampling methods. The objective of this study was to assess the suitability of 30 μL VAMS for the measurement of endogenous steroid hormones. MethodsA novel LC-MS/MS method was developed for the quantification of 18 analytes in VAMS samples, including main endogenous free steroids and phase II metabolites of androgens. The method underwent validation in accordance with ISO/IEC 17025:2017 and World Anti-Doping Agency (WADA) requirements. Subsequently, it was applied to authentic VAMS samples obtained from 20 healthy volunteers to assess the stability of target analytes under varying storage conditions. ResultsThe validation protocol assessed method’s selectivity, matrix effect, extraction recovery, quantitative performance, carry-over and robustness. The analysis of authentic samples demonstrated the satisfactory stability of monitored steroids in VAMS stored at room temperature, 4 °C, −20 °C and −80 °C for up to 100 days and subjected to up to 3 freezing-thawing cycles. ConclusionsThe validated LC-MS/MS method demonstrated its suitability for the measurement of steroids in dried blood VAMS. The observed stability of steroidal compounds suggests promising prospects for future applications of VAMS, both in anti-doping contexts and clinical research.

The optimal algorithm for eliminating nonlinear error in phase measurement profilometry based on global statistical phase feature function

In a digital fringe projection structured light system, the nonlinear phase error is generated by the gamma effect of both the projector, camera, and other electronic devices. One of the existing nonlinear correction methods is active correction by projecting ideal fringes as far as possible, and the other is passive compensation after capturing aberrant fringes. The former has higher accuracy but needs to capture a large number of fringe patterns, while the latter does not need many fringe patterns, but is not only greatly affected by random noise and out-of-focus effects, but also has poor accuracy. In this paper, an optimal algorithm for eliminating nonlinear error based on global statistical phase feature function (GSPF) is proposed. The phase distribution can be estimated from the difference between the global cumulative distribution function (CDF) and the normalized (CDF). For an ideal fringe pattern without nonlinear error and a fringe pattern with nonlinear error, the region wrapped by the x-axis normalized CDF is much smaller than the region wrapped by the x-axis global CDF, and the larger the nonlinear error is, the larger the difference between the two is. Therefore, the GSPF can be used for nonlinear error correction. Then the optimal nonlinear error correction is performed based on the minimum difference between the compensated phase entropy and the ideal phase entropy. The method does not require too many steps of phase-shifting, and only three fringe patterns are needed to realize accurate and robust correction. Experimental results show that the method is fast, highly accurate and robust. Using this technique, high accuracy measurements can be achieved with the traditional three-step phase-shifting algorithm.