scholarly journals GARLM: Greedy Autocorrelation Retrieval Levenberg–Marquardt Algorithm for Improving Sparse Phase Retrieval

2018 ◽  
Vol 8 (10) ◽  
pp. 1797 ◽  
Author(s):  
Zhuolei Xiao ◽  
Yerong Zhang ◽  
Kaixuan Zhang ◽  
Dongxu Zhao ◽  
Guan Gui
Keyword(s):  
Fourier Transform ◽  
Local Search ◽  
Phase Retrieval ◽  
Greedy Algorithms ◽  
Optimal Estimation ◽  
Retrieval Algorithm ◽  
Marquardt Algorithm ◽  
Levenberg Marquardt ◽  
Local Search Procedure ◽  
Unknown Signal

The goal of phase retrieval is to recover an unknown signal from the random measurements consisting of the magnitude of its Fourier transform. Due to the loss of the phase information, phase retrieval is considered as an ill-posed problem. Conventional greedy algorithms, e.g., greedy spare phase retrieval (GESPAR), were developed to solve this problem by using prior knowledge of the unknown signal. However, due to the defect of the Gauss–Newton method in the local convergence problem, especially when the residual is large, it is very difficult to use this method in GESPAR to efficiently solve the non-convex optimization problem. In order to improve the performance of the greedy algorithm, we propose an improved phase retrieval algorithm, which is called the greedy autocorrelation retrieval Levenberg–Marquardt (GARLM) algorithm. Specifically, the proposed GARLM algorithm is a local search iterative algorithm to recover the sparse signal from its Fourier transform magnitude. The proposed algorithm is preferred to existing greedy methods of phase retrieval, since at each iteration the problem of minimizing the objective function over a given support is solved by using the improved Levenberg–Marquardt (ILM) method and matrix transform. A local search procedure such as the 2-opt method is then invoked to get the optimal estimation. Simulation results are given to show that the proposed algorithm performs better than the conventional GESPAR algorithm.

scholarly journals Five years of CO, HCN, C<sub>2</sub>H<sub>6</sub>, C<sub>2</sub>H<sub>2</sub>, CH<sub>3</sub>OH, HCOOH and H<sub>2</sub>CO total columns measured in the Canadian high Arctic

2014 ◽  
Vol 7 (6) ◽  
pp. 1547-1570 ◽  
Author(s):  
C. Viatte ◽  
K. Strong ◽  
K. A. Walker ◽  
J. R. Drummond
Keyword(s):  
Time Series ◽  
Fourier Transform ◽  
Biomass Burning ◽  
Atmospheric Chemistry ◽  
A Priori ◽  
Estimation Method ◽  
Optimal Estimation ◽  
High Arctic ◽  
Retrieval Algorithm ◽  
Data Set

Abstract. We present a five-year time series of seven tropospheric species measured using a ground-based Fourier transform infrared (FTIR) spectrometer at the Polar Environment Atmospheric Research Laboratory (PEARL; Eureka, Nunavut, Canada; 80°05' N, 86°42' W) from 2007 to 2011. Total columns and temporal variabilities of carbon monoxide (CO), hydrogen cyanide (HCN) and ethane (C2H6) as well as the first derived total columns at Eureka of acetylene (C2H2), methanol (CH3OH), formic acid (HCOOH) and formaldehyde (H2CO) are investigated, providing a new data set in the sparsely sampled high latitudes. Total columns are obtained using the SFIT2 retrieval algorithm based on the optimal estimation method. The microwindows as well as the a priori profiles and variabilities are selected to optimize the information content of the retrievals, and error analyses are performed for all seven species. Our retrievals show good sensitivities in the troposphere. The seasonal amplitudes of the time series, ranging from 34 to 104%, are captured while using a single a priori profile for each species. The time series of the CO, C2H6 and C2H2 total columns at PEARL exhibit strong seasonal cycles with maxima in winter and minima in summer, in opposite phase to the HCN, CH3OH, HCOOH and H2CO time series. These cycles result from the relative contributions of the photochemistry, oxidation and transport as well as biogenic and biomass burning emissions. Comparisons of the FTIR partial columns with coincident satellite measurements by the Atmospheric Chemistry Experiment Fourier Transform Spectrometer (ACE-FTS) show good agreement. The correlation coefficients and the slopes range from 0.56 to 0.97 and 0.50 to 3.35, respectively, for the seven target species. Our new data set is compared to previous measurements found in the literature to assess atmospheric budgets of these tropospheric species in the high Arctic. The CO and C2H6concentrations are consistent with negative trends observed over the Northern Hemisphere, attributed to fossil fuel emission decrease. The importance of poleward transport for the atmospheric budgets of HCN and C2H2 is highlighted. Columns and variabilities of CH3OH and HCOOH at PEARL are comparable to previous measurements performed at other remote sites. However, the small columns of H2CO in early May might reflect its large atmospheric variability and/or the effect of the updated spectroscopic parameters used in our retrievals. Overall, emissions from biomass burning contribute to the day-to-day variabilities of the seven tropospheric species observed at Eureka.

scholarly journals Five years of CO, HCN, C<sub>2</sub>H<sub>6</sub>, C<sub>2</sub>H<sub>2</sub>, CH<sub>3</sub>OH, HCOOH, and H<sub>2</sub>CO total columns measured in the Canadian High Arctic

2013 ◽  
Vol 6 (6) ◽  
pp. 11345-11403
Author(s):  
C. Viatte ◽  
K. Strong ◽  
K. A. Walker ◽  
J. R. Drummond
Keyword(s):  
Fourier Transform ◽  
Biomass Burning ◽  
Atmospheric Chemistry ◽  
A Priori ◽  
Estimation Method ◽  
Correlation Coefficients ◽  
Optimal Estimation ◽  
High Arctic ◽  
Retrieval Algorithm ◽  
Atmospheric Variability

Abstract. We present a five-year timeseries of seven tropospheric species measured using a ground-based Fourier Transform InfraRed (FTIR) spectrometer at the Polar Environment Atmospheric Research Laboratory (PEARL, Eureka, Nunavut, Canada, 80°05' N, 86°42' W) from 2007 to 2011. Total columns and temporal variabilities of carbon monoxide (CO), hydrogen cyanide (HCN), and ethane (C2H6), as well as the first derived total columns at Eureka of acetylene (C2H2), methanol (CH3OH), formic acid (HCOOH), and formaldehyde (H2CO) are investigated, providing a new dataset in the sparsely sampled high latitudes. Total columns are obtained using the SFIT2 retrieval algorithm based on the Optimal Estimation Method. The microwindows, as well as the a priori profiles and variabilities are selected to optimize the information content of the retrievals, and error analyses are performed for all seven species. Our retrievals show good sensitivities in the troposphere. The seasonal amplitudes of the timeseries, ranging from 34 to 104%, are captured while using a single a priori profile for each species. The timeseries of the CO, C2H6 and C2H2 total columns at PEARL exhibit strong seasonal cycles with maxima in winter and minima in summer, in opposite phase to the HCN, CH3OH, HCOOH and H2CO timeseries. These cycles result from the relative contributions of the photochemistry, oxidation, and transport, as well as biogenic and biomass burning emissions. Comparisons of the FTIR partial columns with coincident satellite measurements by the Atmospheric Chemistry Experiment Fourier Transform Spectrometer (ACE-FTS) show good agreement. The correlation coefficients and the slopes range from 0.56 to 0.97, and 0.50 to 3.35, respectively, for the seven target species. Our new dataset is compared with previous measurements found in the literature to assess atmospheric budgets of these tropospheric species in the high Arctic. The CO and C2H6 concentrations are consistent with negative trends observed over the Northern Hemisphere, attributed to fossil fuel emission decrease. The importance of poleward transport on the atmospheric budgets of HCN and C2H2 is highlighted. Columns and variabilities of CH3OH, and HCOOH at PEARL are comparable to previous measurements performed at other remote sites. However, the small columns of H2CO in early May might reflect its large atmospheric variability, and/or the effect of the updated spectroscopic parameters used in our retrievals. Overall, emissions from biomass burning contribute to the day-to-day variabilities of the seven tropospheric species observed at Eureka.

scholarly journals An improved magnetometer calibration and compensation method based on Levenberg–Marquardt algorithm for multi-rotor unmanned aerial vehicle

2020 ◽  
Vol 53 (3-4) ◽  
pp. 276-286
Author(s):  
Helong Wu ◽  
Xinbiao Pei ◽  
Jihui Li ◽  
Huibin Gao ◽  
Yue Bai
Keyword(s):  
Least Squares ◽  
Unmanned Aerial Vehicle ◽  
Optimal Estimation ◽  
Calibration Model ◽  
Marquardt Algorithm ◽  
Levenberg Marquardt ◽  
Magnetometer Data ◽  
Aerial Vehicle ◽  
Magnetometer Calibration ◽  
Yaw Angle

In order to improvethe yaw angle accuracy of multi-rotor unmanned aerial vehicle and meet the requirement of autonomous flight, a new calibration and compensation method for magnetometer based on Levenberg–Marquardt algorithm is proposed in this paper. A novel mathematical calibration model with clear physical meaning is established. “Hard iron” error and “Soft iron” error of magnetometer which affect the yaw accuracy of unmanned aerial vehicle are compensated. Initially, Levenberg–Marquardt algorithm is applied to the process of sphere fitting for the original magnetometer data; the optimal estimation of sphere radius and initial “Hard iron” error are obtained. Then, the ellipsoid fitting is performed, and the optimal estimation of “Hard iron” error and “Soft iron” error are obtained. Finally, the calibration parameters are used to compensate for the magnetometer’s output during unmanned aerial vehicle flight. Traditional ellipsoid fitting based on least squares algorithm is taken as reference to prove the effectiveness of the proposed algorithm. Semi-physical simulation experiment proves that the proposed magnetometer calibration method significantly enhances the accuracy of magnetometer. Static test shows that the yaw angle error is reduced from 1.2° to 0.4° when using the proposed calibration model to calibrate magnetometers. In dynamic tests, the sensor MTi’s output is used as reference. The data fusion of magnetometer compensated by the proposed new calibration model based on Levenberg–Marquardt algorithm can accurately track the desired attitude angle. Experimental results indicate that the accuracy of magnetometer in the yaw angle estimation has been greatly enhanced. In the process of attitude estimated, the compensation magnetometer data given by this new method have faster convergence speed, higher accuracy, and better performance than the compensation magnetometer data given by traditional ellipsoid fitting based on least squares algorithm.

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