Phase Probability Density Function Research Articles

Estimation of the Phase Probability Density Function by Solving the Inverse Problem

Estimation of the Phase Probability Density Function by Solving the Inverse Problem

System nonlinearity correction based on a multi-output support vector regression machine

In a fringe projection profilometry system, the phase error introduced by the projector's gamma distortion is the main source of errors. To overcome this problem, we present a phase compensation scheme for multi-dimensional harmonic coefficient prediction based on a multi-output support vector regression machine(M-SVR), The scheme first constructs a significant characteristic relationship between phase probability density function (PDF) and phase multi-harmonic coefficients, creates simulation data without a priori knowledge, constructs a data set with a certain sample size, and then trains the M-SVR model. The trained M-SVR model is used to capture the potential features of the experimental distorted phase and output the multi-dimensional harmonic parameters with nonlinear relationships, followed by error compensation of the distorted phase using an immobile point iteration algorithm for the purpose of correcting the system nonlinearity. We demonstrate the validity and stability of the model through simulation and experimental trials. Most importantly, the preprocessed M-SVR model also has the potential to participate in error correction of other measurement experiments with reasonable sample and hyperparameter settings, which greatly saves the time and cost of multiple experiments.

Regularization of Fourier Series with Approximate Coefficients for the Problem of Phase Probability Density Function Estimation

Regularization of Fourier Series with Approximate Coefficients for the Problem of Phase Probability Density Function Estimation

Regularization of Fourier series with approximate coefficients for the problem of phase probability density function estimation

Regularization of Fourier series with approximate coefficients for the problem of phase probability density function estimation

A New Look at the $\eta$-$\mu$ Fading Model

In this paper, the $\eta$ - $\mu$ fading model is looked at with a unifying perspective, through which correlation and power imbalance between its quadrature processes are simultaneously taken into account. It is formally shown that these two phenomena, quantified by two respective parameters, do not alter the resulting envelope statistics but have a great impact on the corresponding phase. To this end, an exact, simple, closed-form formulation is found that maps these parameters into a resulting $\eta$ parameter that keeps the envelope probability density function (PDF) unaltered. An exact formulation is also found for the phase PDF, given in easily computable infinite series. Closed-form approximation for such a PDF is also found that yields excellent results as compared with the exact one. As a particular case of the general phase formulation, the phase of the Hoyt process is extended to include power imbalance and correlation, and this is given in simple closed-form. Additionally, and as a by-product, some useful mathematical identities are obtained.

Trajectory Planning of a Moving Robot Empowers 3D Localization of RFID Tags With a Single Antenna

In this work, we present a method for 3D localization of RFID tags by a reader-equipped robot with a single antenna. The robot carries a set of sensors, which enable it to create a map of the environment and locate itself in it (Simultaneous Localization and Mapping - SLAM). Then we exploit the collected phase measurements to localize large tag populations in real-time. We show that by forcing the robot to move along non-straight trajectories, thus creating non-linear synthetic apertures, the circular ambiguity of the possible tag's locations is eliminated and 3D localization is accomplished. A reliability metric is introduced, suitable for real-time assessment of the localization error. We investigate how the curvature of the robot's trajectory affects the accuracy under varying multipath conditions. It is found that increasing the trajectory's slope and number of turns improves the accuracy of the method. We introduce a phase model that accounts for the effects of multipath and derive the closed form expression of the resultant's phase probability density function. Finally, the proposed method is extended when multiple antennas are available. Experimental results in a “multipath-rich” indoor environment demonstrate a mean 3D error of 35cm, achieved in a few seconds.

The $\alpha$ -$\eta$ -$\kappa$ -$\mu$ Fading Model: New Fundamental Results

A number of new results aiming at facilitating the use of the $\alpha $ - $\eta $ - $\kappa $ - $\mu $ fading model are presented. These results include: 1) fast convergent, with no recursions, series representations for the envelope probability density function (PDF) and cumulative distribution function (CDF); 2) higher order moments of the envelope; 3) moment generating function (MGF) of the envelope; 4) asymptotic behavior of the envelope PDF and also of the envelope CDF; 5) a procedure for parameter estimation; 6) new closed-form expressions for particular cases of the envelope distribution; 7) new mathematical identity for the generalized Laguerre polynomial and the generalized hypergeometric function; and 8) approximate, but tight, series representation for the phase PDF. As application examples, the following are given: 1) outage probability; 2) outage capacity; 3) amount of fading; and 4) bit error rate, for which the asymptotic behavior is also presented.

A Two-Layer Secure Quantization Algorithm for Secret Key Generation With Correlated Eavesdropping Channel

When the eavesdropping channel has a certain correlation with the legitimate channel, the physical layer key is susceptible to eavesdroppers. To ensure the security of the secret key, we propose a two-layer secure (TLS) quantization algorithm in this paper. First, we model the correlation between the distance and the channel correlation and derive the joint probability density function of channel phase and the key capacity. Then, we describe the detail of the TLS algorithm and prove the validity of our algorithm. Finally, we evaluate the performances of the TLS quantization algorithm using two parameters-key rate and bit disagreement rate. The Simulation results verify the effectiveness of the algorithm.

Unwrapped Phase Estimation via Normalized Probability Density Function for Multibaseline InSAR

Interferometric synthetic aperture radar (InSAR) is a powerful technique for obtaining terrain information based on the interferometric phase. Multibaseline (MB) InSAR is an extension of the conventional InSAR and is used to improve the estimation accuracy and reliability of the unwrapped phase. Based on a newly defined normalized phase probability density function (pdf), a novel wrapped-to-unwrapped phase (W2UP) estimation method is proposed for MB-InSAR. First, the concept of the normalized pdf is introduced to overcome the limitation of the fixed $2\pi $ period for different baseline cases. Then, a new maximum likelihood estimation is established using the MB normalized pdfs, which has a much steeper peak around the true phase value than the single baseline case and leads to higher estimation accuracy. The proposed W2UP method estimates the unwrapped phase from multiple filtered interferograms, so it is less influenced by the phase noise. Both the theoretical analysis and results using the simulated and real MB data are provided to verify the effectiveness of the proposed method.

Fast and Accurate Computation of the Multilook Interferometric Phase Probability Density Function

This letter reports a method to compute the multilook interferometric phase probability density function (pdf) used in interferometric synthetic aperture radar (InSAR). The method is fast and accurate for at least 10 000 looks and the full range of interferometric correlations. This is accomplished by computing the products of large and small numbers in the underlying hypergeometric series logarithmically and by deriving a single-term recursion relation to accelerate the computation. This letter extends the usable range of inputs to the multilook interferometric phase pdf for radar systems with a large number of looks and the entire practical range of interferometric correlation.

Backplane aberration calibration of spatial light modulators using a phase-retrieval algorithm.

The calibration and correction of backplane aberration of a liquid-crystal spatial light modulator (LCSLM) are important for its proper functioning. To simplify the calibration procedurally, we study a random-illumination-phase-retrieval-based method. Our method improves the convergence of the phase-retrieval-based calibration and reduces the calibration complexity. However, the cross talk of the LCSLM deteriorates the calibration performance. We determined the relationship between the probability density function of the random phases and the light-intensity pattern and proposed an algorithm to compensate for the cross talk. We conducted a series of simulations to test the performance of the above-mentioned algorithms. The results show that our algorithms are effective and outperform other methods.

The $\alpha $ -$\eta $ -$\kappa $ -$\mu $ Fading Model

A fading model—the $\alpha $ - $\eta $ - $\kappa $ - $\mu $ model—is proposed that accounts for virtually all relevant short-term propagation phenomena described in the literature. These phenomena include nonlinearity of the medium, power of the scattered waves, power of the dominant components, and number of multipath clusters. They are mapped onto physical parameters, and apart from the first one, which is assumed to influence only the resulting signal envelope, the others affect independently its in-phase and quadrature components. Imbalance parameters are then introduced so as to better assess the impact of these phenomena on the entire process. The signal is described by means of its envelope and phase probability density functions (pdfs). To this end, complex-based and envelope-based models are proposed and explored. An exact joint envelope-phase pdf is found in the closed form. The marginal statistics in these cases remain in the integral forms. Interestingly, the envelope density function obtained through the envelope-based model is found to compute more efficiently than that obtained through the complex model. In addition, capitalizing on the results in the literature, exact series-expansion formulations are found for the envelope pdf as well as for the envelope cumulative distribution function. To the best of the author’s knowledge, this is probably the most general and unifying, physically based, complex fading model proposed in the literature. Because of its comprehensiveness, a number of issues remain open and constitute a fertile field for investigation.

SU‐F‐T‐191: 4D Dose Reconstruction of Intensity Modulated Proton Therapy (IMPT) Based On Breathing Probability Density Function (PDF) From 4D Cone Beam Projection Images: A Study for Lung Treatment

Purpose:With energy repainting in lung IMPT, the dose delivered is approximate to the convolution of dose in each phase with corresponding breathing PDF. This study is to compute breathing PDF weighted 4D dose in lung IMPT treatment and compare to its initial robust plan.Methods:Six lung patients were evaluated in this study. Amsterdam shroud image were generated from pre‐treatment 4D cone‐beam projections. Diaphragm motion curve was extract from the shroud image and the breathing PDF was generated. Each patient was planned to 60 Gy (12GyX5). In initial plans, ITV density on average CT was overridden with its maximum value for planning, using two IMPT beams with robust optimization (5mm uncertainty in patient position and 3.5% range uncertainty). The plan was applied to all 4D CT phases. The dose in each phase was deformed to a reference phase. 4D dose is reconstructed by summing all these doses based on corresponding weighting from the PDF. Plan parameters, including maximum dose (Dmax), ITV V100, homogeneity index (HI=D2/D98), R50 (50%IDL/ITV), and the lung‐GTV's V12.5 and V5 were compared between the reconstructed 4D dose to initial plans.Results:The Dmax is significantly less dose in the reconstructed 4D dose, 68.12±3.5Gy, vs. 70.1±4.3Gy in the initial plans (p=0.015). No significant difference is found for the ITV V100, HI, and R50, 92.2%±15.4% vs. 96.3%±2.5% (p=0.565), 1.033±0.016 vs. 1.038±0.017 (p=0.548), 19.2±12.1 vs. 18.1±11.6 (p=0.265), for the 4D dose and initial plans, respectively. The lung‐GTV V12.5 and V5 are significantly high in the 4D dose, 13.9%±4.8% vs. 13.0%±4.6% (p=0.021) and 17.6%±5.4% vs. 16.9%±5.2% (p=0.011), respectively.Conclusion:4D dose reconstruction based on phase PDF can be used to evaluate the dose received by the patient. A robust optimization based on the phase PDF may even further improve patient care.

On the Phase Statistics of the κ-μ Process

A comprehensive study on the $\kappa $ - $\mu $ phase statistics is conducted. A substantial number of new formulas, exact and approximate, closed form, and integral form, are presented. In particular, a tight close-form approximation for the phase probability density function is found that yields results almost indistinguishable from the exact integral formulation. Most strikingly, the approximate formulation comprises the exact Nakagami- $m$ phase as well as the exact Von Mises (Tikhonov) densities. Joint statistics involving combinations of the envelope, phase, and their time derivatives are derived in an exact manner. The exact phase crossing rate is then obtained in an integral form. A closed-form approximation is proposed that yields very good results as compared to the exact formulation. A Monte Carlo simulation plot is used to validate the formula of the exact phase crossing rate. The formulations presented here drastically facilitate the use of the phase statistics of the $\kappa $ - $\mu $ fading model.

Robust Ambiguous Clutter Suppression for the Near-Shore Water Areas With Spaceborne Multichannel SAR Systems

In practice, the ambiguous image of the strong ground clutter may dominate in the adjacent water areas in the case of high-azimuth ambiguity level for the spaceborne synthetic aperture radar. In this letter, a robust ambiguous clutter suppression method is proposed for the ground moving target indication. The subspace projection approach is used to suppress the ambiguous clutter, and the notch broadening technique is used to widen the ambiguous clutter notches for the robustness. The width of the notch is determined by the probability density function of the interferogram's phase. The validity and robustness of the proposed method are verified by the simulation results.

Stochastic Analysis of Cycle Slips in Injection-Locked Oscillators and Analog Frequency Dividers

A detailed investigation of cycle slips in injection-locked oscillators (ILOs) and analog frequency dividers is presented. This nonlinear phenomenon gives rise to a temporal desynchronization between the injected oscillator and the input source due to noise perturbations. It involves very different time scales so even envelope-transient-based Monte Carlo analyses may suffer from high computational cost. The analysis method is based on an initial extraction of a reduced-order nonlinear model of the injected oscillator based on harmonic-balance simulations. This model has been improved with a more accurate description of oscillation dependence on the input source either at the fundamental frequency or, in the case of a frequency divider, at a given harmonic frequency. The reduced-order model enables an efficient stochastic analysis of the system based on the use of the associated Fokker-Planck equation in the phase probability density function. Several methods for the solution of the associated Fokker-Planck equation are compared with one of them being applicable under a wider range of system specifications. The analysis enables the prediction of the parameter-space regions that are best protected against cycle slips. The technique has been applied to two microwave ILOs and has been validated through commercial software envelope simulations in situations where the computational cost of the envelope simulations was acceptable, and through measurements. The measurement procedure of the cycle slipping phenomenon has been significantly improved with respect to previous work.

English

Amplitude and phase statistics of SAR complex interferogram are significant in the study of interferometry and polarimetry. To reduce statistical variations, multi-look processing is adopted by averaging spatially the complex interferogram. In this study, we derive and validate three kinds of probability density functions (PDFs) of multi-look interferogram for different surface feature scenes. For simple homogeneous areas with the gamma distribution intensity, a concise product-form interferometry phase PDF is derived, which is equivalent to a conventional Gauss hypergeometric PDF. For complicated areas with the K and G0 distributions intensity, two new interferometry amplitude PDFs named as Gamma-K and Gamma-G are proposed, and their phase PDFs are approximately preserved. Finally three typical areas including grass, mountain, and city are picked out from a pair of RADARSAT-2 SAR images and studied. Experimental results indicate good agreement between the computed histograms and the theoretical distributions. The results obtained can be applied to the feature classification of polarisation SAR data and the estimation of decorrelation effect of interferometric SAR. Science Journal, Vol. 64, No. 6, November 2014, pp.564-570, DOI:http://dx.doi.org/10.14429/dsj.64.4747

An Analytical Formula Approximating the Multilook Interferometric-Phase Variance for InSAR

For interferometric synthetic aperture radar (InSAR), the second moment of interferometric phase, i.e., its variance, is a useful metric for InSAR applications (e.g., configuration design). Due to the high degree of nonlinearity in the probability density function of the multilook interferometric phase, it is extremely difficult to derive the analytical expression of the interferometric-phase variance as a function of the number of looks. However, in this letter, we will show that, if we follow an indirect route, then an analytical formula for approximating the interferometric-phase variance for any number of looks can be obtained. The key step of deriving this analytical formula is to transform the interferometric phase into the Euler domain. Simulation results show an excellent agreement between the measured variance curve and the curve obtained by the newly proposed formula except for some small coherence values.

Nakagami-m Phase Model: Further Results and Validation

This paper contributes to and expands on the Nakagami-m phase model. It derives exact, closed-form expressions for both the phase cumulative distribution function and its inverse. In addition, empirical first- and second-order statistics obtained from measurements conducted in a body-area network scenario were used to fit the phase probability density function, the phase cumulative distribution function, and the phase crossing rate expressions. Remarkably, the unlikely shapes of the phase statistics, as predicted by the theoretical formulations, are actually encountered in practice.

The effects of time delays on synchronization properties in a network of neural mass models

Patterns of (de)synchronization are key to neural functioning. We asked what general mechanisms are responsible for the emergence and disappearance of these (de)synchronization patterns. To answer this question we analytically revealed potential effects of physiologically motivated time delays in neural populations. We investigated a network of Neural Mass Models (NMM), here Freeman's seminal model that has successfully been used to model alpha-oscillations [1] and EEG activity in the olfactory system [2] and visual cortex [3]. In contrast to these local applications we were interested in distributed activities over larger networks and hence coupled arbitrarily many NMMs involving the aforementioned time delays. That is, we represented activity in distinct brain areas by means of oscillatory excitatory/inhibitory pairs and consider network behavior as representative for global brain activity. We characterized synchronization as the appearance of clustered stationary phase probability density functions (PDFs). In general, this requires a phase dynamics description of the network, which can be established by applying a slowly varying amplitude approximation (see also [4]), together with a continuity equation covering the PDF's evolution. In fact we succeeded to show the existence of synchronized phase PDFs even for small inter area coupling strengths in the case of both zero and constant finite delays, whereas synchronization disappeared when delays were distributed throughout the network. Homogeneity of the effective coupling matrices (D) appearing in the phase dynamics equation is crucial in this respect as it is required for the existence of clustered phase PDFs. Loss of homogeneity in D can be caused by distributed delays as well as heterogeneity in the structural coupling matrix, leading to the conclusion that the mechanism modulating desynchronization is equal for both cases. The effect of distributed delays is consistent with analytic results in phase oscillator systems [5,6]. This also holds for the drop in oscillation frequency for increased time delay values [5,7] that we observed in simulation results. These two findings validate the description of network behavior in terms of its phase dynamics. In biologically more realistic networks time delays have been shown to determine network synchronization properties in a similar way [8]. Furthermore delays have turned out to be crucial in the behavior of resting state networks [9]. Our results suggest that structural heterogeneity may be compensated for by an appropriate distribution of time delays, thereby explaining the necessity of time delays in RSNs to establish biological plausible synchronization patterns.