nonlinear least squares
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2022 ◽  
Vol 14 (2) ◽  
pp. 402
Author(s):  
Xinchao Xu ◽  
Mingyue Liu ◽  
Song Peng ◽  
Youqing Ma ◽  
Hongxi Zhao ◽  
...  

In order to complete the high-precision calibration of the planetary rover navigation camera using limited initial data in-orbit, we proposed a joint adjustment model with additional multiple constraints. Specifically, a base model was first established based on the bundle adjustment model, second-order radial and tangential distortion parameters. Then, combining the constraints of collinearity, coplanarity, known distance and relative pose invariance, a joint adjustment model was constructed to realize the in orbit self-calibration of the navigation camera. Given the problem of directionality in line extraction of the solar panel due to large differences in the gradient amplitude, an adaptive brightness-weighted line extraction method was proposed. Lastly, the Levenberg-Marquardt algorithm for nonlinear least squares was used to obtain the optimal results. To verify the proposed method, field experiments and in-orbit experiments were carried out. The results suggested that the proposed method was more accurate than the self-calibration bundle adjustment method, CAHVOR method (a camera model used in machine vision for three-dimensional measurements), and vanishing points method. The average error for the flag of China and the optical solar reflector was only 1 mm and 0.7 mm, respectively. In addition, the proposed method has been implemented in China’s deep space exploration missions.


2022 ◽  
Vol 10 (1) ◽  
pp. 102
Author(s):  
Zhiyao Zhu ◽  
Huilong Ren ◽  
Xiuhuan Wang ◽  
Nan Zhao ◽  
Chenfeng Li

The limit state function is important for the assessment of the longitudinal strength of damaged ships under combined bending moments in severe waves. As the limit state function cannot be obtained directly, the common approach is to calculate the results for the residual strength and approximate the limit state function by fitting, for which various methods have been proposed. In this study, four commonly used fitting methods are investigated: namely, the least-squares method, the moving least-squares method, the radial basis function neural network method, and the weighted piecewise fitting method. These fitting methods are adopted to fit the limit state functions of four typically sample distribution models as well as a damaged tanker and damaged bulk carrier. The residual strength of a damaged ship is obtained by an improved Smith method that accounts for the rotation of the neutral axis. Analysis of the results shows the accuracy of the linear least-squares method and nonlinear least-squares method, which are most commonly used by researchers, is relatively poor, while the weighted piecewise fitting method is the better choice for all investigated combined-bending conditions.


2022 ◽  
Vol 12 (2) ◽  
pp. 591
Author(s):  
Ahmed Yahia Kallel ◽  
Zheng Hu ◽  
Olfa Kanoun

For embedded impedance spectroscopy, a suitable method for analyzing AC signals needs to be carefully chosen to overcome limited processing capability and memory availability. This paper compares various methods, including the fast Fourier transform (FFT), the FFT with barycenter correction, the FFT with windowing, the Goertzel filter, the discrete-time Fourier transform (DTFT), and sine fitting using linear or nonlinear least squares, and cross-correlation, for analyzing AC signals in terms of speed, memory requirements, amplitude measurement accuracy, and phase measurement accuracy. These methods are implemented in reference systems with and without hardware acceleration for validation. The investigation results show that the Goertzel algorithm has the best overall performance when hardware acceleration is excluded or in the case of memory constraints. In implementations with hardware acceleration, the FFT with barycentre correction stands out. The linear sine fitting method provides the most accurate amplitude and phase determinations at the expense of speed and memory requirements.


2021 ◽  
Author(s):  
Morteza Kimiaei ◽  
Arnold Neumaier

AbstractThis paper suggests a new limited memory trust region algorithm for large unconstrained black box least squares problems, called LMLS. Main features of LMLS are a new non-monotone technique, a new adaptive radius strategy, a new Broyden-like algorithm based on the previous good points, and a heuristic estimation for the Jacobian matrix in a subspace with random basis indices. Our numerical results show that LMLS is robust and efficient, especially in comparison with solvers using traditional limited memory and standard quasi-Newton approximations.


Author(s):  
Yichun Hu ◽  
Siran Li ◽  
Jun Wang ◽  
Deqiang Zheng ◽  
Huidi Zhang ◽  
...  

Background: The aim of this study was to assess the relationship between serum 25-hydroxyvitamin D [25(OH)D] and serum intact parathyroid hormone (PTH) in Chinese childbearing women, and to estimate the optimum threshold of 25(OH)D that maximally inhibits the PTH, which is considered to be the optimal status for vitamin D sufficiency. Methods: Serum samples were selected from the biological samples’ bank of the Chinese Chronic Diseases and Nutrition Survey (CCDNS) 2015. The serum 25(OH)D concentration was determined by liquid chromatography tandem mass spectrometry and the serum PTH was determined by electronic chemiluminescence. Simple linear and partial correlation analysis, locally weighted regression smooth scatterplot (LOESS), nonlinear least squares estimation (NLS), and segmented regression (SR) were utilized to estimate the relationship of 25(OH)D and PTH, and to determine the threshold of 25(OH)D. Results: A total of 1568 serum samples of 25(OH)D concentration and PTH concentration were analyzed. A significant inverse relationship between 25(OH)D and PTH concentration was observed below 15.25 (14.22–16.28) ng/mL, and PTH decreased slowly with the increase of 25(OH)D above 16.75 (15.43–18.06) ng/mL after adjusting by age, latitude, city type, season, corrected calcium, and phosphorus. A very short plateau of PTH was found at 15.25 ng/mL and 16.75 ng/mL in terms of 25(OH)D according to LOESS, NLS, and SR. Conclusions: The serum 25(OH)D was negatively correlated with the serum PTH. The threshold of VitD sufficiency was found in the range of 14.22–18.06 ng/mL in terms of serum 25(OH)D concentration for Chinese childbearing women aged 18–44 years old.


Author(s):  
Philipp Trunschke ◽  
Martin Eigel ◽  
Reinhold Schneider

We consider best approximation problems in a nonlinear subset  [[EQUATION]] of a Banach space of functions [[EQUATION]] . The norm is assumed to be a generalization of the [[EQUATION]] -norm for which only a weighted Monte Carlo estimate [[EQUATION]] can be computed. The objective is to obtain an approximation [[EQUATION]] of an unknown function [[EQUATION]] by minimizing the empirical norm [[EQUATION]] . We consider this problem for general nonlinear subsets and establish error bounds for the empirical best approximation error. Our results are based on a restricted isometry property (RIP) which holds in probability and is independent of the nonlinear least squares setting. Several model classes are examined where analytical statements can be made about the RIP and the results are compared to existing sample complexity bounds from the literature. We find that for well-studied model classes our general bound is weaker but exhibits many of the same properties as these specialized bounds. Notably, we demonstrate the advantage of an optimal sampling density (as known for linear spaces) for sets of functions with sparse representations.


Electronics ◽  
2021 ◽  
Vol 10 (23) ◽  
pp. 2910
Author(s):  
Andreas Andreou ◽  
Constandinos X. Mavromoustakis ◽  
George Mastorakis ◽  
Jordi Mongay Batalla ◽  
Evangelos Pallis

Various research approaches to COVID-19 are currently being developed by machine learning (ML) techniques and edge computing, either in the sense of identifying virus molecules or in anticipating the risk analysis of the spread of COVID-19. Consequently, these orientations are elaborating datasets that derive either from WHO, through the respective website and research portals, or from data generated in real-time from the healthcare system. The implementation of data analysis, modelling and prediction processing is performed through multiple algorithmic techniques. The lack of these techniques to generate predictions with accuracy motivates us to proceed with this research study, which elaborates an existing machine learning technique and achieves valuable forecasts by modification. More specifically, this study modifies the Levenberg–Marquardt algorithm, which is commonly beneficial for approaching solutions to nonlinear least squares problems, endorses the acquisition of data driven from IoT devices and analyses these data via cloud computing to generate foresight about the progress of the outbreak in real-time environments. Hence, we enhance the optimization of the trend line that interprets these data. Therefore, we introduce this framework in conjunction with a novel encryption process that we are proposing for the datasets and the implementation of mortality predictions.


Sensors ◽  
2021 ◽  
Vol 21 (22) ◽  
pp. 7636
Author(s):  
Alejandra Consejo ◽  
Arwa Fathy ◽  
Bernardo T. Lopes ◽  
Renato Ambrósio ◽  
Jr. Abass

To quantify the effect of levelling the corneal surface around the optical axis on the calculated values of corneal asphericity when conic and biconic models are used to fit the anterior corneal surface. This cross-sectional study starts with a mathematical simulation proving the concept of the effect that the eye’s tilt has on the corneal asphericity calculation. Spherical, conic and biconic models are considered and compared. Further, corneal asphericity is analysed in the eyes of 177 healthy participants aged 35.4 ± 15.2. The optical axis was determined using an optimization procedure via the Levenberg–Marquardt nonlinear least-squares algorithm, before fitting the corneal surface to spherical, conic and biconic models. The influence of pupil size (aperture radii of 1.5, 3.0, 4.0 and 5.0 mm) on corneal radius and asphericity was also analysed. In computer simulations, eye tilt caused an increase in the apical radii of the surface with the increase of the tilt angle in both positive and negative directions and aperture radii in all models. Fitting the cornea to spherical models did not show a significant difference between the raw-measured corneal surfaces and the levelled surfaces for right and left eyes. When the conic models were fitted to the cornea, changes in the radii of the cornea among the raw-measured corneal surfaces’ data and levelled data were not significant; however, significant differences were recorded in the asphericity of the anterior surfaces at radii of aperture 1.5 mm (p < 0.01). With the biconic model, the posterior surfaces recorded significant asphericity differences at aperture radii of 1.5 mm, 3 mm, 4 mm and 5 mm (p = 0.01, p < 0.01, p < 0.01 & p < 0.01, respectively) in the nasal temporal direction of right eyes and left eyes (p < 0.01, p < 0.01, p < 0.01 & p < 0.01, respectively). In the superior–inferior direction, significant changes were only noticed at aperture radii of 1.5 mm for both right and left eyes (p = 0.05, p < 0.01). Estimation of human corneal asphericity from topography or tomography data using conic and biconic models of corneas are affected by eyes’ natural tilt. In contrast, the apical radii of the cornea are less affected. Using corneal asphericity in certain applications such as fitting contact lenses, corneal implant design, planning for refractive surgery and mathematical modelling when a geometrical centre of the eye is needed should be implemented with caution.


Author(s):  
Zichen Wang ◽  
Tao Huang ◽  
Yueting Kang ◽  
Zhihong Luo

Abstract Small angle measuring tools with pendulum mechanical structure has two major challenges. One is that it takes long time to wait for the pendulum structure to completely stop swinging, and the other is that the instrument has poor anti-interference ability. Inspired by the two challenges, this paper proposes a dynamical measuring method to achieve a fast and accurate measurement synchronously. Specially, a damped oscillation modeling of the pendulum with nonlinear least squares fitting is presented to estimate the model parameters which includes the angle of the oblique plane. Besides, it is proved that the minimum sample size required for data fitting is decided by one oscillation period, which guarantees the shortest measuring time within high precision. Moreover, this paper also presents the precision criterion of fitting parameters to judge whether the fitting results meet the requirements. This proposed method is applied to a developed small angle measuring tool. The error of the proposed measurement system is better than 3.25 '' and the uncertainty is 4.8 '' within the measurement angle of 2000 '', and the settling time is 0.45 s. The experimental results confirm that this method can not only greatly shorten the measuring time, but also enhance the anti-interference ability and realize high-precision measurement.


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