Robust Estimation Model for 3D Coordinate Transformations based on Differential Total Least Squares Algorithm

고중복도 비디오카메라 영상을 이용하여 GPS/INS 및 지상기준점 자료 없이 제작되는 자유 입체 모자이크 영상은 기준 프레임 영상의 3차원 모델좌표계로 표현되는 상호표정요소를 이용하여 제작될 수 있다. 이와같이 제작된 자유 입체 모자이크 영상으로부터 결정되는 3차원 좌표는 3차원 모델좌표계로 나타내게 된다. 따라서 자유 입체 모자이크 영상을 이용하여 절대좌표를 결정하기 위해서는 모델좌표계를 절대좌표계로 변환하기 위한 방법이 필요하다. 일반적으로 서로 다른 두 개의 3차원 직각 좌표계간의 좌표변환은 3차원 상사변환(similarity transformation)이 사용된다. 하지만 자유 입체 모자이크 영상의 3차원 모델좌표는 원점으로부터 떨어질수록 오차가 누적되어 선형변환을 이용한 좌표변환을 수행하기 어렵다. 따라서 이러한 자유 입체 모자이크 영상의 모델좌표를 절대좌표로 변환하기 위한 3차원 비선형 변환 방법이 필요하다. 또한 절대좌표계로 표현된 수치지도와 입체 모자이크 영상을 중첩하여 사용하기 위해서는 자유 입체 모자이크 영상을 실좌표 입체 모자이크 영상으로 변환하기 위한 방법이 필요하다. 따라서 본 연구에서는 자유 입체 모자이크 영상의 3차원 모델좌표를 3차원 절대좌표로 변환하기 위한 3차원 비선형 변환 방법과 이 방법을 기반으로 자유 입체 모자이크 영상을 실좌표 입체 모자이크 영상으로 제작하기 위한 2차원 비선형 변환방법을 제안하였다. The free-stereo mosaics image without GPS/INS and ground control data can be generated by using relative orientation parameters on the 3D model coordinate system. Its origin is located in one reference frame image. A 3D coordinate calculated by conjugate points on the free-stereo mosaic images is represented on the 3D model coordinate system. For determining 3D coordinate on the 3D absolute coordinate system utilizing conjugate points on the free-stereo mosaic images, transformation methodology is required for transforming 3D model coordinate into 3D absolute coordinate. Generally, the 3D similarity transformation is used for transforming each other 3D coordinates. Error of 3D model coordinates used in the free-stereo mosaic images is non-linearly increased according to distance from 3D model coordinate and origin point. For this reason, 3D model coordinates used in the free-stereo mosaic images are difficult to transform into 3D absolute coordinates by using linear transformation. Therefore, methodology for transforming nonlinear 3D model coordinate into 3D absolute coordinate is needed. Also methodology for resampling the free-stereo mosaic image to the geo-stereo mosaic image is needed for overlapping digital map on absolute coordinate and stereo mosaic images. In this paper, we propose a 3D non-linear transformation for converting 3D model coordinate in the free-stereo mosaic image to 3D absolute coordinate, and a 2D non-linear transformation based on 3D non-linear transformation converting the free-stereo mosaic image to the geo-stereo mosaic image.

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

3D similarity transformations are often used for datum transformation in Geomatic Engineering. The transformations are routinely performed between the point coordinates evaluated from GNSS (Global Navigation Satellite Systems) observations and the point coordinates given in national datum. A mathematical model established among two different 3D coordinate systems contains tree type datum parameters which are translations, rotations and a scale. Coefficients of the parameters reflect large variations from each other’s. So, the unknown coefficient matrix in normal equations is to be ill-conditioned. In this study, it is indicated how the ill-condition structure is reduced to acceptable level by means of choosing at different unit of unknown parameters. Finally, the transformation from Kocaeli Metropolitan Municipality GNSS network coordinates to Turkish Geodetic Datum is used as a numerical example.

Total least squares (TLS) is a technique that solves the traditional least squares (LS) problem for an errors-in-variables (EIV) model, in which both the observation vector and the design matrix are contaminated by random errors. Four- and seven-parameter models of coordinate transformation are typical EIV model. To determine which one of TLS and LS is more effective, taking the four- and seven-parameter models of Global Navigation Satellite System (GNSS) coordinate transformation with different coincidence pointsas examples, the relative effectiveness of the two methods was compared through simulation experiments. The results showed that in the EIV model, the errors-in-variables-only (EIVO) model and the errors-in-observations-only (EIOO) model, TLS is slightly inferior to LS in the four-parameter model coordinate transformation, and TLS is equivalent to LS in the seven-parameter model coordinate transformation. Consequently, in the four- and seven-parameter model coordinate transformation, TLS has no obvious advantage over LS.

Affine transformation that allows the axis-specific rotations and scalars to capture the more transformation details has been extensively applied in a variety of geospatial fields. In tradition, the computation of affine parameters and the transformation of non-common points are individually implemented, in which the coordinate errors only of the target system are taken into account although the coordinates in both target and source systems are inevitably contaminated by random errors. In this article, we propose the seamless affine error-in-variables (EIV) transformation model that computes the affine parameters and transforms the non-common points simultaneously, importantly taking into account the errors of all coordinates in both datum systems. Since the errors in coefficient matrix are involved, the seamless affine EIV model is nonlinear. We then derive its least squares iterative solution based on the Euler–Lagrange minimization method. As a case study, we apply the proposed seamless affine EIV model to the map rectification. The transformation accuracy is improved by up to 40%, compared with the traditional affine method. Naturally, the presented seamless affine EIV model can be applied to any application where the transformation estimation of points fields in the different systems is involved, for instance, the geodetic datum transformation, the remote sensing image matching, and the LiDAR point registration.

A quaternion method for georeferencing of satellite linear array stereo images is proposed in this paper. In this method, quaternion is used to describe the exterior orientation parameters (EOPs) of the satellite linear array stereo images, and gain the EOPs of any image line through quaternion attitude kinematics equation. Then the georeferencing results can be obtained after solving the collinearity equations with ground control points (GCPs). We validate this method through experiment with SPOT images. Experimental results indicate that quaternion attitude kinematics equation can be used for georeferencing of satellite linear array stereo images, the accuracy is as much as or even better than that of the conventional method, and the quaternion EOPs can overcome much of correlation. This method can successfully used in satellite linear array stereo images, and it can also effectively avoid a great deal of computation of trigonometric functions and the potential unstable factors. It provides another technique for georeferencing of satellite linear array stereo images. Quaternion algebra was invented by W.R.Hamilton in 1843, and it can be used to represent space orientation and rotations of space vectors very expediently. Quaternion can avoid the singularity may aroused by using Euler angles to describe attitude. So from the very beginning quaternion played a special role in the study of rotations in 3-dimensional vector spaces. For the pass few years, many photogrammetry scholars pay an attention to quaternion for their ascendant capability, and many photogrammetry algorithms are presented for processing close-range and aerial images. Jiang (2) proposed an algorithm of space resection based on unit quaternion; experiment indicated that this algorithm can obtain the EOPs independent on initial values. Wang (3) used Jiang's method in bundle block adjustment of aerial images. Guan (4) also presented a quaternion space resection which is similarity to triangular pyramid, and this method is not suit to block aerotriangulation. Liu (5) used quaternion to model the eccentric angle of IMU, and introduced it into the orientation image adjustment of airborne three-line CCD image supported by GPS/IMU. Experimental shows that the adjustment accuracy of quaternion is as much as or even better than that of the conventional method, and can avoid a great deal of computation of trigonometric functions. Zhao (6) presented a solution of 3D space similarity transformation model adjustment algorithm, which is based on quaternion rotation matrix. Some tests show that quaternion rotation matrix used in photogrammetry has a very clear superiority. Ji (7) detailedly studied the application of quaternion in aerial photogrammetry, and relative orientation and bundle block adjustment tests were done using actual aerial images. Peng (8) used quaternion for the general digital camera calibration, experimental results show that inner orientation and distortion parameters can be easily obtained using quaternion even when there were larger rotation angle between ground coordinate system and virtual photo coordinate. Tan (9) discussed the construction of rotation matrix in close- rang photogrammetry, and compared with Euler angle for the advantages and disadvantages. The test proved that the quaternion method needs fewer times in iteration and has steady result in calculation. Jiang (10) also tried to use quaternion in exterior orientation of linear array stereo images. All these show that quaternion can be used in photogrammetry very well.

The laws of nature in general, and the relations and laws in geodesy in particular can be expressed in most cases by nonlinear equations which are in general solved by transforming them to linear form and applying iteration. The process of bringing the equations to linear form implies neglections and approximation. In certain cases it is possible to obtain exact, correct solutions for nonlinear problems. In the present work we introduce parameters into the rotation matrix, and using this we derive solutions for the 2D and 3D similarity transformations. This method involves no iteration, and it does not require the transformation of the equations to linear form. The scale parameter is determined in both cases by solving a polynomial equation of second degree. This solution is already known, but our derivation is worth consideration because of its simple nature.

In cephalometric radiography a reduced dimensions radiograph system introduces a parallax effect. With the aid of a five-sided plexiglass phantom, we show that when the distance between the X-ray sources is lower than 4 metres from film and subject, the effect of parallax produces coordinates errors of a point in the space greater than the coordinate errors due to the capture on the radiographs. The objective is to improve the accuracy of the calculation of 3D coordinates of dentomaxillofacial points when the distance between patient and X-ray source is less than 6 metres. Using the stenope model of radiographic system, we describe a new method and we develop formulas, which permits the calculation of 3D real coordinates of a point viewed on any two out of the 3 radiographs, frontal, lateral and axial, from only one X-ray source. We test this method in comparing the known real coordinates of metallic markers fixed on the phantom to the computed coordinates given by this method. We test also this method by using a dried skull, we compare the computed coordinate errors to those due to capture on the radiograph by the operator and compare the dispersion of computed coordinate errors to errors due re-positioning the object on the radiographic system.

Objectives To solve the problem that the coefficient matrix and the observation vector have structured characteristics for many practical applications in the field of surveying and mapping, that is, the coefficient matrix and the observation vector contain fixed quantities (or even fixed columns) and random quantities, and the random quantities at different positions are linearly related. Methods Starting from the errors-in-variables (EIV) function model, the augmented matrix composed of the coefficient matrix and the observation vector is expressed as an affine function form, and the function model is reconstructed by the variable projection method. Then, a structured total least squares (STLS) estimation algorithm is derived by the Lagrange method. Results The example results show that the proposed algorithm is consistent with the existing weighted or structured total least squares estimation algorithms that can solve structured problem in the coefficient matrix and the observation vector. Compared with the weighted total least squares (WTLS) estimation algorithms, the proposed algorithm only needs to establish a positive definite weight matrix of independent random variables and reduces the number of estimates. Compared with other STLS estimation algorithms, the proposed algorithm takes into account the overall structure of the coefficient matrix and the observation vector. Conclusions This shows the effectiveness of the proposed algorithm. The STLS estimation and the WTLS estimation ensure the statistical optimality from the perspective of the function model and the stochastic model, respectively.

In the image reconstruction of the electrical capacitance tomography (ECT) system, the application of the total least squares theory transforms the ill-posed problem into a nonlinear unconstrained minimization problem, which avoids calculating the matrix inversion. But in the iterative process of the coefficient matrix, the ill-posed problem is also produced. For the effect on the final image reconstruction accuracy of this problem, combined with the principle of the ECT system, the coefficient matrix is targeted and updated in the overall least squares iteration process. The new coefficient matrix is calculated, and then, the regularization matrix is corrected according to the adaptive targeting singular value, which can reduce the ill-posed effect. In this study, the total least squares iterative method is improved by introducing the mathematical model of EIV to deal with the errors in the measured capacitance data and coefficient matrix. The effect of noise interference on the measurement capacitance data is reduced, and finally, the high-quality reconstructed images are calculated iteratively.

The paper addresses the identification of time-invariant bilinear system (BS) models in the errors-in-variables (EIV) framework. The proposed scheme is based on the structured total least norm (STLN) technique extended here to handle BS. The performance of the presented approach, i.e. the bilinear STLN (BSTLN) with its further extension incorporating the Tikhonov regularization is compared to several other EIV identification techniques via an extensive Monte-Carlo simulation study. The results obtained demonstrate a considerable noise robustness and therefore the applicability of the BSTLN algorithm in the EIV framework.

The European Datum 1950 (ED50) of the Turkish national geodetic network (TNGN) and the World Geodetic System 1984 (WGS84) of the Turkish national fundamental GPS network (TNFGN) are in use as geodetic reference frames in Turkey. According to the use of two reference systems, it is necessary to transform the three-dimensional (3D) coordinate data from ED50 to WGS84 or vice versa. The seven-parameter similarity transformation method is frequently used for 3D coordinate transformation in geodesy. In this study, a back propagation artificial neural network (BPANN) that has been more widely applied in engineering among all other neural network models is evaluated as an alternative 3D coordinate transformation method. BPANN is compared with a popular seven-parameter similarity transformation (Molodensky-Badekas) method over a test area, in terms of root mean square error (RMSE). The results indicated that the employment of BPANN transformed 3D coordinates (X, Y, Z) more accurate than Molodensky-Badekas method and can be useful for 3D coordinate transformation between ED50 and WGS84. Key words: 3D coordinate transformation, back propagation artificial neural network, seven-parameter similarity transformation, BPANN, Molodensky-Badekas.

Finding a new drug which is both safe and efficient is an expensive and time-consuming endeavour. In particular, establishing the ‘dose-effect relationship’ – how beneficial a drug is at different dosages – can be challenging. Predicting this curve requires gathering experimental data by exposing and recording how cells respond to various levels of the drug. However, extreme values are often observed at low and high dosages, potentially introducing errors that are hard to correct in the prediction process. Yet, these extreme observations are sometimes genuine so researchers cannot just ignore them. To improve dose-effect estimation, Zhou, Liu, Fang et al. developed a new general-purpose approach. It uses advanced statistical modelling to account for extremes in lab data. This strategy outperformed other methods when dealing with these observations while also providing higher efficiency in data analysis with more uniform data in experiments. To facilitate implementation, Zhou, Liu, Fang et al. set up a user-friendly tool baptized ‘REAP’; this free online resource allows scientists without advanced statistical experience to harness the new approach and to perform dose-effect analysis more easily and accurately. This could boost research across many different disciplines that examine the effects of chemicals on cells.

A method of fitting spatial line by applying total least squares method is presented, which can achieve multipoint measurement of gun barrel direction. Firstly, observe the multipoint of gun barrel by the double theodolite. Secondly, calculate space coordinates of these marker points, and do total least squares fitting with the object of the sum of squares of the distance from all measurement points to space line is minimal. Finally, figure out the azimuth angle and height angle of the gun tube direction of the direction vector of spatial linear. The simulation results come from Monte Carlo method show that the direction accuracy improves about 1% with the new method compared with the traditional method. The results show that the new method is feasible and available.

In the absence of sensor calibration and satellite orbit information for Ikonos satellite imagery, empirical methods have to be adopted for the geometric correction of the images. This paper addresses two major terrain related issues. First, the paper focuses on the effects of terrain variation on the rectification accuracy of Ikonos images using various two-dimensional (2D) transformation models. It was found that (1) the accuracy of rectified coordinates is significantly affected by elevation differences—the greater the differences, the lower the rectified accuracy; (2) for higher accuracy results, ground control points (GCPs) should be projected to a compensation plane before 2D transformation models are applied; and (3) an accuracy of about 0.5 m RMS error can be gained from rectified Ikonos images by utilizing most 2D transformation models when accurate ground control points are available. Second, the paper addresses the effects of terrain variation and the number of GCPs on the obtained ground points accuracy when an eight-parameter affine model is used for 3D ground points determination for stereoscopic Ikonos imagery. It was found that (1) non-collinearity-based 3D orientation and triangulation model can be used successfully in most cases for 3D ground point determination without the need for a camera model or satellite ephemeris data; (2) accuracy up to the sub-pixel level in the X-Y directions and about one pixel in the Z direction can be achieved; (3) the model works significantly better for hilly and mountainous areas than very flat areas; and (4) the accuracy of the results can be improved significantly either by increasing the number of GCPs or by adding topographic constraints.