Weighted Total Least Squares Research Articles

Weighted Total Least Squares (WTLS) Solutions for Straight Line Fitting to 3D Point Data

In this contribution the fitting of a straight line to 3D point data is considered, with Cartesian coordinates xi, yi, zi as observations subject to random errors. A direct solution for the case of equally weighted and uncorrelated coordinate components was already presented almost forty years ago. For more general weighting cases, iterative algorithms, e.g., by means of an iteratively linearized Gauss–Helmert (GH) model, have been proposed in the literature. In this investigation, a new direct solution for the case of pointwise weights is derived. In the terminology of total least squares (TLS), this solution is a direct weighted total least squares (WTLS) approach. For the most general weighting case, considering a full dispersion matrix of the observations that can even be singular to some extent, a new iterative solution based on the ordinary iteration method is developed. The latter is a new iterative WTLS algorithm, since no linearization of the problem by Taylor series is performed at any step. Using a numerical example it is demonstrated how the newly developed WTLS approaches can be applied for 3D straight line fitting considering different weighting cases. The solutions are compared with results from the literature and with those obtained from an iteratively linearized GH model.

Camera pose estimation based on global structure from motion

In this paper, a new global camera pose estimation algorithm WTLS-IRLS is proposed, which can effectively solve the global rotation when there are outliers. Firstly, according to the relationship between the rotation vector and the rotation matrix, we simplify the product operation of the rotation matrix into the subtraction operation of the rotation vector, which reduces the complexity of the algorithm. Secondly, the weighted total least squares (WTLS) and the iteratively reweighted least squares (IRLS) are used to average relative rotations. As the initialization of IRLS, WTLS provides a good initial guess by correcting the linearization equation and adding weight information to the relative rotations. IRLS continues to add weight information to the relative rotation matrices to optimize the global rotations. We demonstrate the performance of our approach by a number of large-scale data sets, the results show that our method has been greatly improved in efficiency, accuracy and iteration. In order to verify the correctness of our proposed method, we completed the complete reconstruction process, the experimental results show that our proposed WTLS-IRLS rotation averaging algorithm can obtain dense point clouds with more three-dimensional points.

WTLS iterative algorithm of 3D similarity coordinate transformation based on Gibbs vectors

The 3D similarity coordinate transformation is fundamental and frequently encountered in many areas of work such as geodesy, engineering surveying, LIDAR, terrestrial laser scanning, photogrammetry, machine vision, etc. The algorithms of 3D similarity transformation are divided into two categories. One is a closed-form algorithm that is straightforward and fast. However, it cannot provide the accuracy information for the transformation parameters. The other category of algorithm is iterative, and this can offer the accuracy information for the transformation parameters. However, the latter usually needs a good initial value of the unknown. Considering the accuracy information for transformation parameters is essential or indispensable from the viewpoint of uncertainty, this contribution proposes a weighted total least squares (WTLS) iterative algorithm of the 3D similarity coordinate transformation based on Gibbs vectors. It is fast in terms of fewer iterations, reliable and does not need good initial values of transformation parameters. Two cases including the registration of LIDAR points with big rotation angles and a geodetic datum transformation with small rotation angles are demonstrated to validate the new algorithm.

Jackknife method for the location of gross errors in weighted total least squares

Because the weighted total least squares (WTLS) method lacks robustness and is sensitive to gross errors, it cannot eliminate the influence of outliers effectively. A small number of gross errors may produce a devastating effect on estimates. Focusing on the limitation of the WTLS method regarding the influence of gross errors, this work combines Jackknife resampling theory with the WTLS algorithm for the identification and detection of outliers. The gross errors located in the WTLS method by the Jackknife method are identified to further improve the qualities of the estimated values if the observation data are falsified by outliers. This paper focuses on the following two aspects: only one gross error and multiple gross errors. Detailed calculation steps and the whole procedure for outlier detection using the new method are given. This algorithm is applied to the straight-line fitting model and the plane coordinate transformation model. From the experimental estimation results, we can see that the method proposed in this paper can identify gross errors that are greater than or equal to three times the standard error, and obtain more accurate estimation values when compared with the WTLS method and the classic robust weighted total least squares (RWTLS) method. The numerical case studies verify the effectiveness and practicality of the proposed procedure.

Using Weighted Total Least Squares and 3-D Conformal Coordinate Transformation to Improve the Accuracy of Mobile Laser Scanning

With the aid of global position system (GPS), mobile laser scanning (MLS) is able to provide 3-D geo-referenced point cloud that has centimeter-level accuracy. The MLS accuracy, however, degrades significantly due to the trajectory errors of the laser scanner and the residual systematic errors from the geo-referencing transformation process in the GPS-free environments. To solve this problem, this article presents a novel integration algorithm based on the weighted total least squares (WTLS) and the 3-D conformal coordinate transformation (3DCCT). In this new method, the 3-D point measurement model and the error propagation parameter vector in the MLS can be updated in real-time, and they can also adjust the geo-referenced coordinate transformation parameters and eliminate the influences of the residual systematic errors during MLS. In this article, the MLS mathematical model is first established, followed up by a detailed analysis for MLS error budget interpreting the effects of the individual error sources. Second, WTLS is used to correct the 3-D point measurement model of MLS and the error of propagation parameter vector; 3DCCT, WTLS, and ground control target feature constraints are applied to eliminate the residual systematic errors in the geo-referencing transformation process. Finally, several data sets from outdoor scenarios are used to evaluate and validate the proposed method. The experimental results demonstrate that the proposed method can significantly improve the overall accuracy of the MLS system.

An Advanced Outlier Detected Total Least-Squares Algorithm for 3-D Point Clouds Registration

The registration of 3-D point clouds is an important procedure during the terrestrial laser scanning data processing. Recently, due to their high flexibility and the powerful mathematical model, a large amount of least-squares-based (LSs-based) methods are proposed and widely applied to estimate the transformation parameters of 3-D point clouds registration. In these LSs-based methods some based on the generalized Gauss–Markov model do not correct the influence of random errors on source 3-D point clouds. Although there are other methods based on the errors-in-variables (EIV) model, they are inapplicable for transformation problems with large rotation angles and arbitrary scale ratio. In addition, the gross errors are usually ignored in previous studies on 3-D point clouds registration, which, however, exists commonly and could distort the registration severely. Aiming to avoid the influence of gross errors and extend its application, an advanced outlier detected total least-squares (OD-TLS) method is proposed in this paper. Based on the generalized EIV model OD-TLS performs a seven-parameter 3-D similarity transformation with large rotation angles and arbitrary scale ratio. The random errors of both source and target 3-D point clouds are considered. Furthermore, outliers are detected and removed automatically by combining the data snooping method with total least-squares (TLS) estimation. In order to indicate the benefits of OD-TLS, comparative experiments with the LS3D and weighted total least squares (WTLS) on synthetic and real-world scanned 3-D point clouds were performed. The experimental results show OD-TLS not only enhances the registration accuracy but also increases its robustness.

Jackknife resampling parameter estimation method for weighted total least squares

To make the result of weighted total least squares (WTLS) parameter estimation more accurate, the Jackknife method is used to resample the observed data and make full use of Jackknife samples for multiple calculations. Combining Jackknife-1 and Jackknife-d with the weighted total least squares, the calculation methods of Jackknife-1-WTLS and Jackknife-d-WTLS are proposed, and the value of d is further studied. Meanwhile, the two methods are applied to the linear regression model, the planar coordinate transformation model and big rotation angle’s 3 D coordinate transformation model. The results show that the proposed methods of Jackknife of weighted total least squares are more effective than least squares (LS), weighted total least squares and least squares resampling methods in improving the quality of parameter estimation, which verifies the validity and feasibility of the methods.

Jackknife resample method for precision estimation of weighted total least squares

Few studies have been conducted on the precision estimation of weighted total least squares (WTLS) by using the approximate function probability distribution method. And the existing Monte Carlo method needs to simulate a lot, the amount of calculation is large and the results obtained are uncertain. In order to further improve the total least squares precision estimation theory, this paper introduces the Jackknife method into Geomatics data processing. Combining the Jackknife method and WTLS method, the delete-1 Jackknife method and delete-d Jackknife method are proposed. The biases and standard deviations or covariance of parameter estimations are calculated by these proposed methods. And the specific steps of the precision estimation of these two methods are given. Applying these methods to the linear regression model and the coordinate transformation model, and comparing with the approximate function method and the Monte Carlo method, we can see that the Jackknife methods for precision estimation can obtain more stable and reasonable precision results and are very adaptive. In order to get more reasonable precision results, the Jackknife method does spend much more time over total least squares when the amount of the observed data is large. But compared with Monte Carlo method, it can reduce the amount of calculation and improve the computational efficiency. The method in this paper could provide an idea for further study on the precision estimation for total least squares.

Weighted total least-squares joint adjustment with weight correction factors

A joint adjustment involves integrating different types of geodetic datasets, or multiple datasets of the same data type, into a single adjustment. This paper applies the weighted total least-squares (WTLS) principle to joint adjustment problems and proposes an iterative algorithm for WTLS joint (WTLS-J) adjustment with weight correction factors. Weight correction factors are used to rescale the weight matrix of each dataset while using the Helmert variance component estimation (VCE) method to estimate the variance components, since the variance components in the stochastic model are unknown. An affine transformation example is illustrated to verify the practical benefit and the relative computational efficiency of the proposed algorithm. It is shown that the proposed algorithm obtains the same parameter estimates as the Amiri-Simkooei algorithm in our example; however, the proposed algorithm has its own computational advantages, especially when the number of data points is large.

An improved RUSLE/SDR model for the evaluation of soil erosion

The accurate assessment of soil erosion is key to assess environmental parameters, such as the reduction in soil fertility, the increase in flood risk, the loss of nutrients, and degradation of water quality. In this study, we developed a methodology using the Revised Universal Soil Loss Equation (RUSLE) with the sediment delivery ratio (SDR) to estimate the annual amount of soil erosion and sediment yield in the Nozhian watershed (western Iran). The weighted total least-squares (WTLS) algorithm was applied to generate the rainfall–runoff erosivity surface using rainfall data and a digital elevation model (DEM) instead of traditional interpolation methods. The results demonstrated that the obtained sediment yield by the RUSLE/SDR model was approximately 802,000 tons per year. More than half of the watershed (61.6%) belonged to the high and severe erosion classes (20–100 t/ha year), and the mean soil erosion rate in the study area was 89.32 t/ha year. Several landslides extracted using a Google Earth map by expert interpretation were exactly consistent with areas that had high erosion rates based on the RUSLE results. This compatibility implies the compatibility between the results and reality. According to the statistical analysis, topographic features, especially slope steepness, had the greatest effect on the rate of soil erosion in the region. The results of our RUSLE/SDR analysis were also compared with the reported results from the Modify Pacific Southwest Interagency Committee (MPSIAC), Erosion Potential method (EPM), and Hydrophysical model. In situ data from the measured annual sediment yield during a 40-year interval from a hydrometric station were used for the accuracy analysis. The comparison indicated improvement in the accuracy of our approach by up to 65% in comparison to other reported results. These results can surely aid in the implementation of soil management and conservation practices to reduce soil erosion in the Nozhian watershed.

A Fast and On-Machine Measuring System Using the Laser Displacement Sensor for the Contour Parameters of the Drill Pipe Thread.

The inconvenient loading and unloading of a long and heavy drill pipe gives rise to the difficulty in measuring the contour parameters of its threads at both ends. To solve this problem, in this paper we take the SCK230 drill pipe thread-repairing machine tool as a carrier to design and achieve a fast and on-machine measuring system based on a laser probe. This system drives a laser displacement sensor to acquire the contour data of a certain axial section of the thread by using the servo function of a CNC machine tool. To correct the sensor’s measurement errors caused by the measuring point inclination angle, an inclination error model is built to compensate data in real time. To better suppress random error interference and ensure real contour information, a new wavelet threshold function is proposed to process data through the wavelet threshold denoising. Discrete data after denoising is segmented according to the geometrical characteristics of the drill pipe thread, and the regression model of the contour data in each section is fitted by using the method of weighted total least squares (WTLS). Then, the thread parameters are calculated in real time to judge the processing quality. Inclination error experiments show that the proposed compensation model is accurate and effective, and it can improve the data acquisition accuracy of a sensor. Simulation results indicate that the improved threshold function is of better continuity and self-adaptability, which makes sure that denoising effects are guaranteed, and, meanwhile, the complete elimination of real data distorted in random errors is avoided. Additionally, NC50 thread-testing experiments show that the proposed on-machine measuring system can complete the measurement of a 25 mm thread in 7.8 s, with a measurement accuracy of ±8 μm and repeatability limit ≤ 4 μm (high repeatability), and hence the accuracy and efficiency of measurement are both improved.

Weighted Total Least Squares with Constraints: Alternative Derivation without Using Lagrange Multipliers

AbstractThis technical note presents alternative derivations for the weighted total least-squares (WTLS) problem subject to weighted and hard constraints. The derivations do not take into account t...

Parameter estimation in 3D affine and similarity transformation: implementation of variance component estimation

Three-dimensional (3D) coordinate transformations, generally consisting of origin shifts, axes rotations, scale changes, and skew parameters, are widely used in many geomatics applications. Although in some geodetic applications simplified transformation models are used based on the assumption of small transformation parameters, in other fields of applications such parameters are indeed large. The algorithms of two recent papers on the weighted total least-squares (WTLS) problem are used for the 3D coordinate transformation. The methodology can be applied to the case when the transformation parameters are generally large of which no approximate values of the parameters are required. Direct linearization of the rotation and scale parameters is thus not required. The WTLS formulation is employed to take into consideration errors in both the start and target systems on the estimation of the transformation parameters. Two of the well-known 3D transformation methods, namely affine (12, 9, and 8 parameters) and similarity (7 and 6 parameters) transformations, can be handled using the WTLS theory subject to hard constraints. Because the method can be formulated by the standard least-squares theory with constraints, the covariance matrix of the transformation parameters can directly be provided. The above characteristics of the 3D coordinate transformation are implemented in the presence of different variance components, which are estimated using the least squares variance component estimation. In particular, the estimability of the variance components is investigated. The efficacy of the proposed formulation is verified on two real data sets.

Data snooping algorithm for universal 3D similarity transformation based on generalized EIV model

Three-dimensional (3D) similarity datum transformation is extensively applied in geodetic field and many other areas. In recent years, the total least squares (TLS) solution for universal 3D similarity transformation problem (with arbitrary rotation angles and scale ratio) has become a hot research issue and many algorithms have been proposed. However, the estimated transformation parameters are affected or even severely distorted when the observed coordinates are contaminated by gross errors. In this study, the 3D similarity transformation problem is described as a generalized errors-in-variables (EIV) model, and then the data snooping algorithm for this model is proposed. The weighted total least squares (WTLS) solution to the generalized EIV model is firstly derived through Euler–Lagrange method and then we reformulate it as a classical least squares problem. Two types of test statistics for data snooping are constructed based on the classical least squares theory under the conditions with known and unknown variance component, respectively. The results of the real and simulated experiments indicate that the proposed algorithm can effectively reduce the influence of the gross errors and obtain reliable transformation parameters.

Türkiye’nin Batısında 3B Datum Dönüşümü için En Küçük Kareler Yaklaşımlarının Karşılaştırma Analizi

In Turkey, the national reference frame was changed for geodetic applications in parallel with the increasing use of Global Navigation Satellite System technology. Due to the reference frame change, the three-dimensional (3D) datum transformation has become compulsory between ED50 and WGS84. Several 3D datum transformation algorithms have been developed for geodetic applications. The well-known technique is the Least-Squares (LS) method. In this study, alternative 3D datum transformation approaches (including the Total Least-Squares (TLS) and the Weighted TLS (WTLS) methods) were compared with the LS method over a test area. The results showed that the WTLS transformed 3D coordinates with better accuracy than the LS and TLS methods.

A WTLS-based rational function model for orthorectification of remote-sensing imagery

ABSTRACTThe rational function model (RFM) is widely applied to orthorectification of aerial and satellite imagery. This article proposes a new method named Ortho-WTLS to solve the RFM in remote-sensing imagery orthorectification. Based on a weighted total least squares (WTLS) estimator, the proposed method allows one to handle coordinates of ground control points (GCPs) that contain errors and are of unequal accuracies. This situation occurs, e.g. if GCPs are automatically selected. In the proposed model, first, the relationship of two linearization methods for an RFM with errors contained in GCPs is investigated and results in a hybrid linearization. Next, based on WTLS, RFM coefficients are estimated with an iterative computation function. Finally, the performance of the Ortho-WTLS method thus obtained is investigated using simulated images and remotely sensed images by collecting GCPs with varying errors. Experimental results show that the Ortho-WTLS method achieves a more robust estimation of model parameters and a higher orthorectification accuracy when compared with standard LS-based RFM estimation. We conclude that the quality of GCPs has a large impact on the accuracy and that an increasing number of low-precision GCPs may lead to a decrease in orthorectification quality.

Parameter estimation using weighted total least squares in the two-compartment exchange model.

The linear least squares (LLS) estimator provides a fast approach to parameter estimation in the linearized two-compartment exchange model. However, the LLS method may introduce a bias through correlated noise in the system matrix of the model. The purpose of this work is to present a new estimator for the linearized two-compartment exchange model that takes this noise into account. To account for the noise in the system matrix, we developed an estimator based on the weighted total least squares (WTLS) method. Using simulations, the proposed WTLS estimator was compared, in terms of accuracy and precision, to an LLS estimator and a nonlinear least squares (NLLS) estimator. The WTLS method improved the accuracy compared to the LLS method to levels comparable to the NLLS method. This improvement was at the expense of increased computational time; however, the WTLS was still faster than the NLLS method. At high signal-to-noise ratio all methods provided similar precisions while inconclusive results were observed at low signal-to-noise ratio. The proposed method provides improvements in accuracy compared to the LLS method, however, at an increased computational cost. Magn Reson Med 79:561-567, 2017. © 2017 International Society for Magnetic Resonance in Medicine.

Total least norm solution for linear structured EIV model

Structured total least norm (STLN) and weighted total least squares (WTLS) have been proposed for structured EIV (errors-in-variables) models. STLN is a principle minimizing the Lp norm of the perturbation parts of an EIV model, in which p=1, 2 or ∞. STLN permits affine structure of the matrix A or [A|y] such as Toeplitz. STLN has advantages over WTLS on having ∞-norm and robust 1-norm. However, only Hankel or Toeplitz structure was discussed explicitly in STLN, and weight of errors was not discussed. While in some applications, the matrix [A|y] has arbitrary linear structure, taking linear regression and coordinate transformation as examples. This paper aims at extending STLN to L-STLN (linear structured total least norm), which can deal with EIV models having linear structures other than Toeplitz or Hankel in [A|y]. Additionally, weighted estimation is discussed. A simulated numerical example is computed by STLN and L-STLN under 1-, 2-, and ∞-norm, the results shown that L-STLN can preserve arbitrary linear structure of [A|y]. Also, the estimated correction of [A|y] by WTLS and L-STLN under 2-norm are compared. The results show that weighted L-STLN under 2-norm is consistent with WTLS. The robustness of L-STLN under 1-norm is demonstrated by simulated outlier.

Accurate coupled lines fitting in an errors-in-variables framework

For the purpose of accurate measurement of regular polygons, the boundary lines with parallel, perpendicular or given angles are defined as coupled lines. Provided that the noisy data points are measured from each line without outliers, an accurate and numerical reliable weighted total least squares (WTLS) method is proposed for two dimensional coupled lines fitting task. The underlying problem is modelled within an errors-in-variables framework by assuming all the coordinates are subject to random errors. In order to overcome the possible ill-posedness, the lines are parameterised as constrained Hessian normal form instead of the intercept and slope one. A generic WTLS algorithm is derived in case of the random columns are corrupted by fully correlated errors. Special case that the data are corrupted by homocedastic errors are considered and solved with less computational expenses. A single line fitting and a simulated regular hexagon fitting examples are performed with comparisons and discussions. The proposed methods can be used for accurate regular polygon measurement in vision metrology or reverse engineering.

Bridge pier settlement prediction in high-speed railway via autoregressive model based on robust weighted total least-squares

The autoregressive model for time series prediction is a common method in settlement prediction. In the traditional parameter estimation of autoregressive model, least-squares (LS) is the method, which only considers the errors in the observation vector. However, the errors in the coefficient matrix have not been considered. To solve this issue, weighted total least-squares (WTLS) method is developed for parameter estimation. However, it does not consider the possible gross errors in observations, which may lead to a reduction in the robustness and reliability of parameter estimation. In order to solve this problem, in this study, robust WTLS (RWTLS) method is proposed to estimate parameters of autoregressive model for bridge pier settlement prediction in high-speed railway. A comparison with LS, robust LS (RLS) and WTLS methods is conducted for bridge pier settlement prediction and two sets of observed data are used in this evaluation. The results of experiments show that the variance components and the mean absolute values of predictive residuals obtained by WTLS and RWTLS methods are smaller than those by using LS and RLS methods in the case of modelling data without gross errors, and the variance component and the mean absolute value of predictive residuals obtained by RWTLS method is the smallest in the case of modelling data with gross errors. It shows that autoregressive model settlement prediction for bridge pier by using RWTLS method is more reliable and accurate than LS, RLS and WTLS methods in high-speed railway.