3D Similarity Transformation Research Articles

Iteratively weighted least squares solution for universal 3D similarity transformation

The 3D similarity coordinate transformation is widely used to estimate the transformation parameters for measurement datum transformation. Accurate and reliable transformation parameters are crucial for accurate and reliable data integration. However, the accuracy of the transformation parameters can be significantly affected or even severely distorted when the observed coordinates are contaminated by gross errors. To address this problem, an advanced iteratively weighted least squares solution based on the weighted least squares is proposed. This solution utilizes the singular value decomposition method to obtain the rotation matrix and introduces a novel weight estimation approach based on Gaussian function. This approach enables the weight to be normalized and optimized iteratively. To verify the accuracy and reliability of the proposed algorithm, the root mean square errors from both true and pseudo-observed values are analyzed by simulation experiments. Furthermore, the results of simulated and empirical experiments show that the proposed algorithm can effectively reduce the influence of gross errors to obtain reliable measurement datum transformation parameters. It should be noted that the new algorithm can easily be extended to the 2D/3D affine and rigid transformation cases, such as image matching, point cloud registration, and absolute orientation of photogrammetry.

A generalized weighted total least squares-based, iterative solution to the estimation of 3D similarity transformation parameters

The essence of point cloud registration is to estimate the seven similarity transformation parameters of the Bursa-Wolf model which describes the relative position of the two neighboring LiDAR stations. So far, Gauss–Markov (GM) model has been widely used in point cloud registration, however, only the random errors contained in one LiDAR station are considered, which is seriously inconsistent with the reality. To simultaneously take the random errors contained in both of the two neighboring LiDAR stations into account in registration, the EIV model is constructed based on the introduction of unit quaternions to describe the spatial rotation in 3D similarity transformation, and an iterative solution to the estimation of the transformation parameters is systematically discussed. Detailed derivation of the formulas for the estimation of the seven transformation parameters are displayed step by step. Finally, three experiments are designed to verify the correctness and effectiveness of the solution.

A new algorithm for 3D similarity transformation with dual quaternion

A new algorithm for 3D similarity transformation with dual quaternion

Extended WTLS iterative algorithm of 3D similarity transformation based on Gibbs vector

Extended WTLS iterative algorithm of 3D similarity transformation based on Gibbs vector

A complete solution of an improved universal 3D coordinate similarity transformation model

When linearizing three-dimensional (3D) coordinate similarity transformation model with large rotations, we usually encounter the ill-posed normal matrix which may aggravate the instability of solutions. To alleviate the problem, a series of conversions are contributed to the 3D coordinate similarity transformation model in this paper. We deduced a complete solution for the 3D coordinate similarity transformation at any rotation with the nonlinear adjustment methodology, which involves the errors of the common and the non-common points. Furthermore, as the large condition number of the normal matrix resulted in an intractable form, we introduced the bary-centralization technique and a surrogate process for deterministic element of the normal matrix, and proved its benefit for alleviating the condition number. The experimental results show that our approach can obtain the smaller condition number to stabilize the convergence of the interested parameters. Especially, our approach can be implemented for considering the errors of the common and the non-common points, thus the accuracy of the transformed coordinates improves.

Retrieval of Euler rotation angles from 3D similarity transformation based on quaternions

ABSTRACT Recently, it has been shown how quaternion-based representation of a rotation matrix has advantages over conventional Eulerian representation in 3D similarity transformations. The iterative estimation procedure in similarity transformations based on quaternions results in translations and (scaled) quaternion elements. One needs, therefore, an additional procedure for evaluating the rest of the transformation parameters (translation, scale factor and rotation angles) after this solution.This contribution shows how to evaluate the rotation angles and the full covariance matrix of the transformation parameters from the estimation results in asymmetric and symmetric 3D similarity transformations based on quaternions.

A universally efficient algorithm and precision assessment for seamless 3D similarity transformation

Three-dimensional (3D) similarity transformation is popularly applied for measurement datum transformation. In this study, a seamless partial errors-in-variables (EIV) model with equality constraints is established to describe the universal 3D similarity transformation problem (with arbitrary rotation angles and scale factor). Unlike the traditional transformation model, all of the random errors in the measured coordinates for common and non-common points, and their variance–covariance information can be considered. To obtain the least squares solution of this model, the constrained total least squares prediction (CTLSP) algorithm is derived using Gauss–Newton iteration and the Euler–Lagrange approach. Unnecessary matrix calculations in the application of the CTLSP algorithm for 3D datum transformation are avoided and the efficient iterative formulae are derived. Compared with the existing generalized total least squares prediction (GTLSP) algorithm, in which the transformation model is nonlinear with respect to the parameters, the CTLSP algorithm avoids the complex calculations related to the rotation matrix expressed by the trigonometric functions, and allows us to use the simple linear least squares to obtain a satisfactory initial value of the parameter vector. In addition, the linearly approximate cofactor propagation law is employed to assess the precision of the transformed coordinates of non-common points based on the CTLSP algorithm. Finally, the superiorities of CTLSP in transformation accuracy and computational efficiency are verified using an experiment. It should be noted that the new algorithm along with the precision evaluation formulae can easily be extended to the 2D/3D affine and rigid transformation cases as well, such as the map rectification, the point clouds registration, and the image matching.

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.

Indoor Layout Estimation by 2D LiDAR and Camera Fusion

This paper presents an algorithm for indoor layout estimation and reconstruction through the fusion of a sequence of captured images and LiDAR data sets. In the proposed system, a movable platform collects both intensity images and 2D LiDAR information. Pose estimation and semantic segmentation is computed jointly by aligning the LiDAR points to line segments from the images. For indoor scenes with walls orthogonal to floor, the alignment problem is decoupled into top-down view projection and a 2D similarity transformation estimation and solved by the recursive random sample consensus (R-RANSAC) algorithm. Hypotheses can be generated, evaluated and optimized by integrating new scans as the platform moves throughout the environment. The proposed method avoids the need of extensive prior training or a cuboid layout assumption, which is more effective and practical compared to most previous indoor layout estimation methods. Multi-sensor fusion allows the capability of providing accurate depth estimation and high resolution visual information.

PERFORMING 3D SIMILARITY TRANSFORMATION WITH LARGE ROTATION ANGLES USING CONSTRAINED MULTIVARIATE TOTAL LEAST SQUARES

Abstract: 3D similarity transformation is frequently encountered operation in the field of geodetic data processing, and there are many applications that involve large rotation angles. In previous studies, the errors of the coefficient matrix were usually neglected and a least squares algorithm was applied to calculate the transformation parameters. However, the coefficient matrix is composed of the point coordinates in source coordinate system, i.e., the coefficient matrix is also contaminated by errors. Therefore, a total least squares algorithm should be applied. In this paper, a new method is proposed to address the 3D similarity transformation problem with large rotation angles. Firstly, the scale factor and rotation matrix are put together as the parameter matrix to avoid the rank-defect problem. Then, the translation vector is removed and the multivariate model is constructed. Finally, the constraints are introduced according to the properties of the parameter matrix and the constrained multivariate total least squares algorithm is derived to obtain the transformation parameters. The experimental results show that the proposed method has a high computational efficiency.

Monte-Carlo-based uncertainty propagation in the context of Gauss–Markov model: a case study in coordinate transformation

One of the main tasks of professionals in the Earth sciences is to convert coordinates from one reference frame into another. The coordinates transformation is widely needed and applied in all branches of modern geospatial activities. To convert from one reference frame to another, it is necessary initially to determine the parameters of a coordinate transformation model. A common way to estimate the transformation parameters is using the least-squares theory within a linearized Gauss-Markov Model (GMM). Another approach arises from a numerical method. Here, a Monte Carlo Method (MCM) is used to infer the uncertainty of estimators. This method is based on: (i) the assignment of probability distributions to the coordinates in the two reference frames, (ii) the determination of a discrete representation of the probability distribution for the transformation parameters, and (iii) the determination of the associated uncertainties from this discrete representation of the estimates of the transformation parameters. In this contribution, we compare the weighted least-squares within the GMM (WLSE-GM) and the proposed method regarding the transformation problem of computing 2D similarity transformation parameters. The results show that, the transformation parameters uncertainties are higher for the LS-MC than the WLSE-GM. This is due to the fact that the WLSE-GM solution does not take into account the uncertainties associated with the system matrix. In future studies the Monte Carlo method should be applied to the nonlinear least-squares solution.

Low‐rank path‐following algorithm for 3D similarity registration

To address the 3D point matching problem where the pose difference between two point sets is unknown, the authors propose a path following (PF)–based algorithm. This method works by reducing the objective function of robust point matching (RPM) algorithm to a function of point correspondence variable and then using PF for optimisation. By using the 3D similarity transformation which has few parameters, authors’ method needs no regularisation on transformation and, therefore, can handle the case when the pose difference between two point sets is unknown. The authors also propose a novel convex term for use in the PF algorithm which is based on the low‐rank nature of authors’ objective function and leads to a PF algorithm which converges quickly. Experimental results demonstrated better robustness of the proposed method over state‐of‐the‐art methods and authors’ method is also efficient.

Shape based local affine invariant texture characteristics for fabric image retrieval

The rapid growth of fabric images needs fast retrieval for related applications, such as fashion design. The goal of fabric image retrieval is to retrieve and rank relevant fabrics from a large scale fabric database to visually assist users’ online shopping process in e-commerce. Most of existing solutions to this issue are not invariant with respect to 2D similarity or affine transformations, much less to 3D transformations of textured surface. In this paper, we propose a new search method with a shape based local affine invariant texture characteristics. By employing topographic map to represent fabric images, which is a complete, multi-scale and contrast invariant representation, the proposed method first obtains a tree of shapes from the topographic map. Then, a group of statistics is applied on these shapes to acquire a set of features that are invariant to 3D transformations. We finally represent these features combing relations between shapes, and based on the representation the similarity of pairs of fabric images can be estimated. To evaluate the performance of our algorithm, we conducted a series of experiments on a real-world fabric image dataset, and compared the proposed method with other previous ones. Experimental results demonstrate that the time of the proposed method spending in searching is less than 1 second, and meanwhile a higher PR score than others is obtained.

The extraction of urban road inventory from mobile lidar system

Road inventory is an important infrastructure for the intelligent transportation system. The mobile mapping system is an advanced technology to collect high-quality road data efficiently. The objective of this research is to present a procedure to extract the road inventory using the mobile mapping system. The major works of this study include data preprocessing and feature extraction. Data preprocessing is used to improve the accuracy of trajectory using co-registration and 3D similarity transformation. Feature extraction comprises road point selection, road mark extraction, and object measurement. The test data was acquired by Optech Lynx system. The experimental results show the potential for using a lidar-based mobile mapping system in road inventory extraction. The experimental results indicate that the registration error can be reduced to 5cm after co-registration. The automatic filtering is able to select ground point efficiently. Moreover, the mobile lidar system is capable of creating a 3D road inventor wherein the root-mean-square-error of road mark is greater than 20cm. The digitized road marks and road objects can be integrated with geospatial data for analysis and 3D visualization.

Increasing numerical efficiency of iterative solution for total least-squares in datum transformations

Cartesian coordinate transformation between two erroneous coordinate systems is considered within the Errors-In-Variables (EIV) model. The adjustment of this model is usually called the total Least-Squares (LS). There are many iterative algorithms given in geodetic literature for this adjustment. They give equivalent results for the same example and for the same user-defined convergence error tolerance. However, their convergence speed and stability are affected adversely if the coefficient matrix of the normal equations in the iterative solution is ill-conditioned. The well-known numerical techniques, such as regularization, shifting-scaling of the variables in the model, etc., for fixing this problem are not applied easily to the complicated equations of these algorithms. The EIV model for coordinate transformations can be considered as the nonlinear Gauss-Helmert (GH) model. The (weighted) standard LS adjustment of the iteratively linearized GH model yields the (weighted) total LS solution. It is uncomplicated to use the above-mentioned numerical techniques in this LS adjustment procedure. In this contribution, it is shown how properly diminished coordinate systems can be used in the iterative solution of this adjustment. Although its equations are mainly studied herein for 3D similarity transformation with differential rotations, they can be derived for other kinds of coordinate transformations as shown in the study. The convergence properties of the algorithms established based on the LS adjustment of the GH model are studied considering numerical examples. These examples show that using the diminished coordinates for both systems increases the numerical efficiency of the iterative solution for total LS in geodetic datum transformation: the corresponding algorithm working with the diminished coordinates converges much faster with an error of at least 10-5 times smaller than the one working with the original coordinates.

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.

Solution of the weighted symmetric similarity transformations based on quaternions

A new method through Gauss–Helmert model of adjustment is presented for the solution of the similarity transformations, either 3D or 2D, in the frame of errors-in-variables (EIV) model. EIV model assumes that all the variables in the mathematical model are contaminated by random errors. Total least squares estimation technique may be used to solve the EIV model. Accounting for the heteroscedastic uncertainty both in the target and the source coordinates, that is the more common and general case in practice, leads to a more realistic estimation of the transformation parameters. The presented algorithm can handle the heteroscedastic transformation problems, i.e., positions of the both target and the source points may have full covariance matrices. Therefore, there is no limitation such as the isotropic or the homogenous accuracy for the reference point coordinates. The developed algorithm takes the advantage of the quaternion definition which uniquely represents a 3D rotation matrix. The transformation parameters: scale, translations, and the quaternion (so that the rotation matrix) along with their covariances, are iteratively estimated with rapid convergence. Moreover, prior least squares (LS) estimation of the unknown transformation parameters is not required to start the iterations. We also show that the developed method can also be used to estimate the 2D similarity transformation parameters by simply treating the problem as a 3D transformation problem with zero (0) values assigned for the z-components of both target and source points. The efficiency of the new algorithm is presented with the numerical examples and comparisons with the results of the previous studies which use the same data set. Simulation experiments for the evaluation and comparison of the proposed and the conventional weighted LS (WLS) method is also presented.

A weighted adjustment of a similarity transformation between two point sets containing errors

Abstract For an adjustment of a similarity transformation, it is often appropriate to consider that both the source and the target coordinates of the transformation are affected by errors. For the least squares adjustment of this problem, a direct solution is possible in the cases of specific-weighing schemas of the coordinates. Such a problem is considered in the present contribution and a direct solution is generally derived for the m-dimensional space. The applied weighing schema allows (fully populated) point-wise weight matrices for the source and target coordinates, both weight matrices have to be proportional to each other. Additionally, the solutions of two borderline cases of this weighting schema are derived, which only consider errors in the source or target coordinates. The investigated solution of the rotation matrix of the adjustment is independent of the scaling between the weight matrices of the source and the target coordinates. The mentioned borderline cases, therefore, have the same solution of the rotation matrix. The direct solution method is successfully tested on an example of a 3D similarity transformation using a comparison with an iterative solution based on the Gauß-Helmert model.

COMPARISON OF 2D AND 3D APPROACHES FOR THE ALIGNMENT OF UAV AND LIDAR POINT CLOUDS

Abstract. The automatic alignment of 3D point clouds acquired or generated from different sensors is a challenging problem. The objective of the alignment is to estimate the 3D similarity transformation parameters, including a global scale factor, 3 rotations and 3 translations. To do so, corresponding anchor features are required in both data sets. There are two main types of alignment: i) Coarse alignment and ii) Refined Alignment. Coarse alignment issues include lack of any prior knowledge of the respective coordinate systems for a source and target point cloud pair and the difficulty to extract and match corresponding control features (e.g., points, lines or planes) co-located on both point cloud pairs to be aligned. With the increasing use of UAVs, there is a need to automatically co-register their generated point cloud-based digital surface models with those from other data acquisition systems such as terrestrial or airborne lidar point clouds. This works presents a comparative study of two independent feature matching techniques for addressing 3D conformal point cloud alignment of UAV and lidar data in different 3D coordinate systems without any prior knowledge of the seven transformation parameters.

An integrated solution for reducing ill-conditioning and testing the results in non-linear 3D similarity transformations

To find 3D similarity transformation parameters (a scale, three rotational angles, and three translation elements) between two orthogonal coordinate systems in 3D is an ill-posed non-linear inverse problem by means of common points (their Cartesian components are known in the both systems). The problem can be solved via Linearized Least Squares (LLS) or Direct (non-iterative) Least Squares (DLS). Since the parameters in LLS take different quantities (and units) from each other, the condition problems can arise during the solution of normal equations. In this paper, we propose a combined solution to reducing ill-conditioning and to perform precision analysis and global outlier test in LLS accordingly. The way is based on column norming and uses the normalized unknowns instead of the original ones at the solution stage of the normal equations. While the global outlier test is fulfilled on the normalized unknowns, the original unknowns and their precisions obtained using the normalized matrix with linear transformation to the normalized unknowns and by the error propagation law to their variance-covariance matrix. A direct method (Direct Least Squares, DLS) is used to compute the initial values of LLS for reducing iteration number (to 1–3 cycles) with a 1.E-7 threshold in the paper. And, when the solution of 3D-ST having large rotational angles and uncertainties, it is also shown that DLS goes to a worse solution than LLS by means of a simulated transformation problem.