Nonlinear Gauss Helmert Model Research Articles

A Novel Robust Point Cloud Fitting Algorithm Based on Nonlinear Gauss–Helmert Model

Point cloud fitting plays an important role in various applications of laser scanning technology. Currently, the most dominant algorithms are the linear least-squares (LS) method and the random sample consensus (RANSAC) algorithm, in which the random noises of the point cloud coordinates and the influence of the corresponding stochastic model cannot be fully considered. In this study, the mathematical model of point cloud fitting is uniformly described using the nonlinear Gauss–Helmert (NGH) model, in which the nonlinear relationship between the measurements and parameters, as well as the effects of all random errors, can be considered. In addition, the covariance matrix of point cloud coordinates obtained from the precision of the original measurements is employed to construct the corresponding stochastic model. By further introducing an equivalent weight scheme of robust estimation, a novel point cloud fitting method is proposed based on the robust NGH (RNGH) model. Unlike the previous studies, the RNGH-based algorithm can effectively resist the negative effects of outliers with full consideration of all random errors in various linear and nonlinear fitting problems. To verify the performance of the proposed RNGH, comparative experiments of point cloud fitting are employed in both simulated and real-world scenarios, respectively. The results demonstrate that the proposed point cloud fitting algorithm has significant advantages over LS and RANSAC in terms of accuracy and robustness.

A Nonlinear Gauss-Helmert Model and Its Robust Solution for Seafloor Control Point Positioning

Using GNSS-Acoustic (GNSS-A) technology to establish the seafloor geodetic datum is both feasible and flexible and thus has become an important way to obtain the absolute positions of seafloor control points. However, numerous errors are inevitable in marine surveying, including systematic errors and gross errors caused by GNSS dynamic positioning, inaccurate sound velocity profile measurements, and ocean ambient noise, and their interference will be directly reflected in the positioning results. To accurately calculate the seafloor control point coordinates, this paper first notes that the general error propagation law (EPL) method is defective in dealing with various error factors in GNSS-A positioning. A more rigorous method incorporates the time-varying term of the sound velocity ranging error into the coefficient matrix of the underwater observation equation, and the transducer position error should be considered. Therefore, a Gauss-Helmert (GH) model is used for seafloor control point positioning. Then, considering the dual nonlinearity of the model, a Lagrange objective function is constructed to derive its solution algorithm. On this basis, considering the gross errors polluting of the observations, the robust estimation principle is introduced, and the robust solution steps are given. Finally, simulation experiments and a testing experiment in the sea area near Jiaozhou Bay are used to verify the performance of the new method. The results show that the functional relationship and stochastic model of the nonlinear GH model for seafloor point positioning are reasonably described. Under ideal conditions with no gross errors and either different water depths or different transducer position errors, the accuracy and stability of the new method are both higher than those of the EPL method. When the observations are polluted by gross errors, the robust algorithm of the new method can accurately identify the abnormal information. By improving the robustness of the observation and structure spaces, the positioning precision of the 3D point deviation results can be optimized, and the solution performance of the new method is superior to that of the general method.

Erratum for “Data Snooping for the Equality Constrained Nonlinear Gauss–Helmert Model Using Sensitivity Analysis” by Bin Wang, Xing Fang, Chao Liu, and Bangyan Zhu

Erratum for “Data Snooping for the Equality Constrained Nonlinear Gauss–Helmert Model Using Sensitivity Analysis” by Bin Wang, Xing Fang, Chao Liu, and Bangyan Zhu

Data Snooping for the Equality Constrained Nonlinear Gauss–Helmert Model Using Sensitivity Analysis

To develop a universal-outliers processing algorithm under the conditions with equality constraints, the equality-constrained nonlinear Gauss–Helmert (GH) model, which contains the equality-constrained Gauss–Markov (GM) and errors-in-variables (EIV) models as special cases, is selected as the research object in this paper. The least squares solution for the nonlinear GH model with equality constraints is obtained using the Euler–Lagrange approach, and then, it is equivalently formulated as the standard constrained least squares (CLS) problem. To construct the test statistics for the outliers detection, a distinctive sensitivity analysis approach is introduced into this CLS problem. The local sensitivity of the weighted sum of squared residuals to the perturbations of observations in the CLS problem is discussed, and then, the local test statistics are constructed based on these sensitivity indicators. To verify the performance of the sensitivity-based test statistics, the proposed data-snooping algorithm for the equality-constrained nonlinear GH model is applied to a three-dimensional (3D) symmetric similarity transformation. The computational results of the simulated and real examples manifest that the proposed data-snooping algorithm using the sensitivity-based test statistics can effectually decrease the negative impact of the outliers and derive reliable parameters. It should be pointed out that the new algorithm is applicable in various kinds of equality-constrained least squares and total least squares problems.

Line fitting in Euclidean 3D space

Over a century ago Pearson solved the problem of fitting lines in 2D space to points with noisy coordinates in both dimensions. Surprisingly, however, the case of fitting lines in 3D space has seen little attention, though Adcock long ago published a brief (one page) article claiming that the solution that minimized orthogonal distances is the most probable. We solve this problem using a new algorithm for the Total Least-Squares (TLS) solution within an Errors-In-Variables Model, respectively an equivalent nonlinear Gauss-Helmert Model. Following Roberts, only four parameters are estimated, thereby avoiding over-parametrization that may lead to unnecessary singularities and, hence, require the introduction of constraints to the model. The current pervasiveness of Global Navigation Satellite Systems, robotic total stations, and digital laser scanners as sources of geodetic observations means that geodetic engineers and scientists now commonly work with observational models in 3D space as opposed to classical geodetic methods that often separated horizontal and vertical observational models. And while several papers have been written describing a TLS solution for line fitting problems in 2D space, the extension to 3D space is not readily apparent from these works. This further motivates the treatment of the 3D problem in some detail in this contribution.

Surface-based matching of 3D point clouds with variable coordinates in source and target system

The automatic co-registration of point clouds, representing three-dimensional (3D) surfaces, is an important technique in 3D reconstruction and is widely applied in many different disciplines. An alternative approach is proposed here that estimates the transformation parameters of one or more 3D search surfaces with respect to a 3D template surface. The approach uses the nonlinear Gauss–Helmert model, minimizing the quadratically constrained least squares problem. This approach has the ability to match arbitrarily oriented 3D surfaces captured from a number of different sensors, on different time-scales and at different resolutions. In addition to the 3D surface-matching paths, the mathematical model allows the precision of the point clouds to be assessed after adjustment. The error behavior of surfaces can also be investigated based on the proposed approach. Some practical examples are presented and the results are compared with the iterative closest point and the linear least-squares approaches to demonstrate the performance and benefits of the proposed technique.

On the errors-in-variables model with equality and inequality constraints for selected numerical examples

It is well known that the errors-in-variables (EIV) model has been treated as a special case of the traditional geodetic model, the nonlinear Gauss–Helmert model (GHM), for more than a century. In this contribution, an adjustment of the EIV model with equality and inequality constraints is investigated based on the nonlinear GHM. In each iteration, the constrained EIV model is linearized to form a quadratic program. Furthermore, the precision description is investigated for the mixed constrained problem. The demonstrated results from the numerical examples show that this approach avoids the large computational expenses of the existing combinatorial solution that normally accompany the number of inequality constraints.

Robust estimations for the nonlinear Gauss Helmert model by the expectation maximization algorithm

For deriving the robust estimation by the EM (expectation maximization) algorithm for a model, which is more general than the linear model, the nonlinear Gauss Helmert (GH) model is chosen. It contains the errors-in-variables model as a special case. The nonlinear GH model is difficult to handle because of the linearization and the Gauss Newton iterations. Approximate values for the observations have to be introduced for the linearization. Robust estimates by the EM algorithm based on the variance-inflation model and the mean-shift model have been derived for the linear model in case of homoscedasticity. To derive these two EM algorithms for the GH model, different variances are introduced for the observations and the expectations of the measurements defined by the linear model are replaced by the ones of the GH model. The two robust methods are applied to fit by the GH model a polynomial surface of second degree to the measured three-dimensional coordinates of a laser scanner. This results in detecting more outliers than by the linear model.

Generalization of total least-squares on example of unweighted and weighted 2D similarity transformation

In this contribution it is shown that the so-called “total least-squares estimate” (TLS) within an errors-in-variables (EIV) model can be identified as a special case of the method of least-squares within the nonlinear Gauss–Helmert model. In contrast to the EIV-model, the nonlinear GH-model does not impose any restrictions on the form of functional relationship between the quantities involved in the model. Even more complex EIV-models, which require specific approaches like “generalized total least-squares” (GTLS) or “structured total least-squares” (STLS), can be treated as nonlinear GH-models without any serious problems. The example of a similarity transformation of planar coordinates shows that the “total least-squares solution” can be obtained easily from a rigorous evaluation of the Gauss–Helmert model. In contrast to weighted TLS, weights can then be introduced without further limitations. Using two numerical examples taken from the literature, these solutions are compared with those obtained from certain specialized TLS approaches.

Total Least-Squares regularization of Tykhonov type and an ancient racetrack in Corinth

In this contribution a variation of Golub/Hansen/O’Leary’s Total Least-Squares (TLS) regularization technique is introduced, based on the Hybrid APproximation Solution (HAPS) within a nonlinear Gauss–Helmert Model. By applying a traditional Lagrange approach to a series of iteratively linearized Gauss–Helmert Models, a new iterative scheme has been found that, in practice, can generate the Tykhonov regularized TLS solution, provided that some care is taken to do the updates properly. The algorithm actually parallels the standard TLS approach as recommended in some of the geodetic literature, but unfortunately all too often in combination with erroneous updates that would still show convergence, although not necessarily to the (unregularized) TLS solution. Here, a key feature is that both standard and regularized TLS solutions result from the same computational framework, unlike the existing algorithms for Tykhonov-type TLS regularization. The new algorithm is then applied to a problem from archeology. There, both the radius and the center-point coordinates of a circle have to be determined, of which only a small part of the arc had been surveyed in-situ, thereby giving rise to an ill-conditioned set of equations. According to the archaeologists involved, this circular arc served as the starting line of a racetrack in the ancient Greek stadium of Corinth, ca. 500 BC. The present study compares previous estimates of the circle parameters with the newly developed “Regularized TLS Solution of Tykhonov type.”