Collinearity Condition Research Articles

ACHIEVING CLOSE RANGE PHOTOGRAMMETRY WITH NON-METRIC MOBILE PHONE CAMERAS

Close range photogrammetry (CRP) has gained increasing relevance over the years with its principles and theories being applied in diverse applications. Further supporting this trend, the current increase in the wide spread usage of mobile phones with high resolution cameras is expected to further popularize positioning by CRP. This paper presents the results of an experimental study wherein two (2) non-metric mobile phone cameras have been used to determine the 3-D coordinates of points on a building by using the collinearity condition equation in an iterative least square bundle adjustment process in MATLAB software environment. The two (2) mobile phones used were Tecno W3 and Infinix X509 phones with focal lengths of 5.432 mm and 8.391 mm respectively. Statistical tests on the results obtained shows that there is no significant difference between the 3-D coordinates obtained by ground survey and those obtained from both cameras at 99% confidence level. Furthermore, the study confirmed the capability of non-metric mobile phone cameras to determine 3D point positions to centimeter level accuracy (with maximum residuals of 11.8 cm, 31.0 cm, and 5.9 cm for the Tecno W3 camera and 14.6 cm, 16.1 cm and 1.8 cm for the Infinix X509 camera in the Eastings, Northings and Heights respectively).

Photogrammetric stereo image rectification

Today, off-the-shelf digital cameras are hand on tools for conducting visual inspections in surveying and mapping related activities or in structural inspection of industrial objects. Dense point clouds of the inspected object surface are of typical data required and their accuracies are also become more demanding. Deriving these data are off particular interest of this paper instead we work on overlapping images for generating those ones. An imaging geometry of stereo images is exploited further to reveal some drawbacks of the normalized stereo imaging configurations. We utilize a photogrammetric collinearity condition to model physical reality of the imaging process. A sequence of coordinate axes rotations on both images is rearranged as well as preserving normalized images resolutions to be equal to the source ones. Also based upon the prescribed resolutions, an efficient indirect resample is performed to interpolate grey values of the warped images. To ascertain reliability of the mathematically developed method, C++ programming codes were written and implemented in the solution. It reveals that a nuisance such as y-parallax is completely eliminated. Therefore, by using our method, the normalized images can be used in the dense point clouds production in a high quality and efficiency.

Simplifying of DLT Equations for a Single Digital Image and its Application in Architectural Façades Documentation

Survey of architectural façades to obtain elevation drawings is an essential, especially in case of maintenance, restoration, ...etc. On the other hand, the rapid progress of the obtained image size captured by digital cameras opens new areas for the captured images to be used in photogrammetry. One of these new areas is the use of a single digital image for surveying and recording of architectural façades. So, the main objective of the current research is to develop a computer algorithm using least squares adjustment method for studying the practical visibility, applicability, and accuracy of using a single digital image captured by a digital camera in surveying architectural façades. To achieve the above-mentioned goal, simplified formulas obtained from collinearity condition, the basis of the Direct Linear Transformation model (DLT), to suit the architectural façades conditions, which is the façade lies in one vertical plane. The obtained formulas showed that eight transformation parameters are required (needed) between the architectural façade and the captured image. Hence, the eight parameters can be computed using four common points or more. So, two field experiments were made on two architectural façades to test the practical visibility, applicability, and accuracy of the supposed technique. The obtained results proved the success of the supposed technique and its related computer algorithm in the survey and the record of the vertical architectural façades.

Robust external calibration of terrestrial laser scanner and digital camera for structural monitoring

Abstract In the last two decades, the integration of a terrestrial laser scanner (TLS) and digital photogrammetry, besides other sensors integration, has received considerable attention for deformation monitoring of natural or man-made structures. Typically, a TLS is used for an area-based deformation analysis. A high-resolution digital camera may be attached on top of the TLS to increase the accuracy and completeness of deformation analysis by optimally combining points or line features extracted both from three-dimensional (3D) point clouds and captured images at different epochs of time. For this purpose, the external calibration parameters between the TLS and digital camera needs to be determined precisely. The camera calibration and internal TLS calibration are commonly carried out in advance in the laboratory environments. The focus of this research is to highly accurately and robustly estimate the external calibration parameters between the fused sensors using signalised target points. The observables are the image measurements, the 3D point clouds, and the horizontal angle reading of a TLS. In addition, laser tracker observations are used for the purpose of validation. The functional models are determined based on the space resection in photogrammetry using the collinearity condition equations, the 3D Helmert transformation and the constraint equation, which are solved in a rigorous bundle adjustment procedure. Three different adjustment procedures are developed and implemented: (1) an expectation maximization (EM) algorithm to solve a Gauss-Helmert model (GHM) with grouped t-distributed random deviations, (2) a novel EM algorithm to solve a corresponding quasi-Gauss-Markov model (qGMM) with t-distributed pseudo-misclosures, and (3) a classical least-squares procedure to solve the GHM with variance components and outlier removal. The comparison of the results demonstrates the precise, reliable, accurate and robust estimation of the parameters in particular by the second and third procedures in comparison to the first one. In addition, the results show that the second procedure is computationally more efficient than the other two.

Tri-Camera High-Speed Videogrammetry for Three-Dimensional Measurement of Laminated Rubber Bearings Based on the Large-Scale Shaking Table

Laminated rubber bearings are widely used for the mitigation of seismic damage of large-scale structures and equipment. However, owing to the flexibility in horizontal direction, the traditional contacted transducer is difficult to acquire the displacement data accurately in the three directions, respectively. In this paper, three-dimensional displacement measurement of laminated rubber bearing based on the large-scale shaking table is achieved by the use of a tri-camera high-speed videogrammetric system consisting of three complementary-metal-oxide-semiconductor (CMOS) cameras, one synchronous controller, and one pair of 1000 watt light sources, which are used to simultaneously acquire the tri-camera image sequences of laminated rubber bearing at a speed of 300 frames per second (fps). Firstly, this paper proposes a fast image block technique for detecting and tracking targets in tri-camera image sequences by integration of techniques morphological edge detection, attribute based ellipse extraction and least-squares-based fitting adjustment. Secondly, this paper presents an integrated bundle adjustment approach, which brings continuous tracking points into one collinearity condition equation, to reconstruct the three dimensional coordinates of continuous tracking points, for the purpose of improving the accuracy of three-dimensional coordinates of tracking points based on tri-camera image sequences. At last, an empirical experiment was conducted to measure the three-dimensional displacement of laminated rubber bearings on the shaking table by the use of the proposed method. The experimental results showed that the proposed method could obtain three-dimensional displacement of laminated rubber bearings with an accuracy of more than 0.5 mm.

Progressive registration of image features and 3D vector lines for orientation modelling

AbstractThis paper proposes a progressive method to register image features and three‐dimensional (3D) vector lines for image orientation modelling. Directed at images acquired with an approximate direct‐georeferencing capability, this study employed 3D vector lines extracted from a geographical information system (GIS) database as ground control. The major work comprises three steps: (1) straight lines were parameterised and applied as control in the collinearity condition equations to determine exterior orientation parameters (EOPs); (2) coarse registration employed quadrangle features formed from straight lines to modify the initial EOPs; and (3) a two‐step fine registration, initially involving all line‐feature candidates, followed by the selection of the most probable one from neighbouring lines. Experimental results indicate that the proposed methods can achieve an accuracy of about 2 pixels (close to the accuracy of the employed GIS database) from initial EOPs with more than 800 pixel errors. Accuracy improvements in each step of the proposed coarse‐to‐fine registration are also demonstrated.

Vanishing Point Extraction and Refinement for Robust Camera Calibration.

This paper describes a flexible camera calibration method using refined vanishing points without prior information. Vanishing points are estimated from human-made features like parallel lines and repeated patterns. With the vanishing points extracted from the three mutually orthogonal directions, the interior and exterior orientation parameters can be further calculated using collinearity condition equations. A vanishing point refinement process is proposed to reduce the uncertainty caused by vanishing point localization errors. The fine-tuning algorithm is based on the divergence of grouped feature points projected onto the reference plane, minimizing the standard deviation of each of the grouped collinear points with an O(1) computational complexity. This paper also presents an automated vanishing point estimation approach based on the cascade Hough transform. The experiment results indicate that the vanishing point refinement process can significantly improve camera calibration parameters and the root mean square error (RMSE) of the constructed 3D model can be reduced by about 30%.

Noise Effect and Noise-Assisted Ensemble Regression in Power System Online Sensitivity Identification

Recently developed data acquisition equipment and data processing methods have ignited the possibility of power system online sensitivity identification (OSI). Despite the existing OSI algorithms, practical issues such as data collinearity and the noise effect on the identification algorithm must be considered to realize OSI in real-power systems. In this study, the negative and positive aspects of noise to OSI are first studied. Then, under the data collinearity condition and by making use of the positive aspects of noise, a noise-assisted ensemble regression method is proposed to simultaneously solve the data collinearity problem and manage the negative aspects of noise. Moreover, the proposed method is proven equivalent to one of the most effective measures, the norm-2 regularization method, to address the collinearity problem, and therefore provides satisfactory OSI results. The proposed method is tested in an 8-generator 36-node system with original operations data from a real-power system, and the results validate its effectiveness.

Novel Image Mosaicking of UAV’s Imagery using Collinearity Condition

This paper presents a preliminary result of ongoing research on unmanned aerial vehicle (UAV) for cooperative mapping to support a large-scale urban city mapping, in Malang, Indonesia. A small UAV can carry an embedded camera which can continuously take pictures of landscapes. A convenient way of monitoring landscape changes might be through accessing a sequence of images. However, since the camera’s field of view is always smaller than human eye’s field of view, the need to combine aerial pictures into a single mosaic is eminent. Through mosaics, a more complete view of the scene can be accessed and analyzed. A semi-automated generation of mosaics is investigated using a photogrammetric approach, namely a perspective projection which is based on collinearity condition. This paper reviews the general projection model based on collinearity condition and uses that to determine a common projective plane from images. The overlapped points for each RGB channel are interpolated onto that of orthographic plane to generate mosaics. An initial attempt shows a promising result.

GEOMETRIC CALIBRATION OF FULL SPHERICAL PANORAMIC RICOH-THETA CAMERA

Abstract. A novel calibration process of RICOH-THETA, full-view fisheye camera, is proposed which has numerous applications as a low cost sensor in different disciplines such as photogrammetry, robotic and machine vision and so on. Ricoh Company developed this camera in 2014 that consists of two lenses and is able to capture the whole surrounding environment in one shot. In this research, each lens is calibrated separately and interior/relative orientation parameters (IOPs and ROPs) of the camera are determined on the basis of designed calibration network on the central and side images captured by the aforementioned lenses. Accordingly, designed calibration network is considered as a free distortion grid and applied to the measured control points in the image space as correction terms by means of bilinear interpolation. By performing corresponding corrections, image coordinates are transformed to the unit sphere as an intermediate space between object space and image space in the form of spherical coordinates. Afterwards, IOPs and EOPs of each lens are determined separately through statistical bundle adjustment procedure based on collinearity condition equations. Subsequently, ROPs of two lenses is computed from both EOPs. Our experiments show that by applying 3*3 free distortion grid, image measurements residuals diminish from 1.5 to 0.25 degrees on aforementioned unit sphere.

Notice of Retraction: Nonlinear Observation Models With Additive Gaussian Noises and Efficient MLEs

We investigate the statistical efficiency of maximum likelihood estimates (MLE) with nonlinear observation models and additive Gaussian noises. Although we and many others have found empirically that in many cases the sample variances of the parameter estimates via Monte Carlo runs match statistically their corresponding Cramer–Rao lower bounds, there has been, up to now, no formal proof to support this. In this letter, we show that the MLE obtained using the iterated least squares method is unbiased and satisfies the generalized collinearity condition between the estimation error and the gradient of the log-likelihood function w.r.t. the parameter vector that is sufficient for the MLE to be efficient. In other words, the MLE is an efficient estimator regardless of the sample size, although, admittedly, this may not be observed for very ill-conditioned problems. A numerical example is provided to demonstrate the statistical efficiency for such problems.

A Flexible, Generic Photogrammetric Approach to Zoom Lens Calibration

Compared with prime lenses, zoom lenses have inherent advantages in terms of operational flexibility. Zoom lens camera systems have therefore been extensively adopted in computer vision where precise measurement is not the primary objective. However, the variation of intrinsic camera parameters with respect to zoom lens settings poses a series of calibration challenges that have inhibited widespread use in close-range photogrammetry. A flexible zoom lens calibration methodology is therefore proposed in this study, developed with the aim of simplifying the calibration process and promoting practical photogrammetric application. A zoom-dependent camera model that incorporates empirical zoom-related intrinsic parameters into the collinearity condition equations is developed. Coefficients of intrinsic parameters are solved in a single adjustment based on this zoom lens camera model. To validate the approach, experiments on both optical- and digital-zoom lens cameras were conducted using a planar board with evenly distributed circular targets. Zoom lens calibration was performed with images taken at four different zoom settings spread throughout the zoom range of a lens. Photogrammetric accuracies achieved through both mono-focal and multi-focal triangulations were evaluated after calibration. The relative accuracies for mono-focal triangulations ranged from 1: 6300 to 1: 18,400 for the two cameras studied, whereas the multi-focal triangulation accuracies ranged from 1: 11,300 to 1: 16,200. In order to demonstrate the applicability of the approach, calibrated zoom lens imagery was used to render a laser-scanned point cloud of a building façade. Considered alongside experimental results, the successful application demonstrates the feasibility of the proposed calibration method, thereby facilitating the adoption of zoom lens cameras in close range photogrammetry for a wide range of scientific and practical applications.

Accuracy Assessment of Analytical Orientation Process in Close range Photogrammetry

This paper studies data manipulation using analytical relative orientation process to determine the exterior orientation parameters. This process can be accomplished by two scenarios; the first one is implemented using collinearity condition while the second one is implemented using coplanarity condition. The final results of both scenarios will be a three dimensional (3D) model and in order to specify the more precise scenario in reconstruction of (3D) models Root Mean Square Error (RMSE) for each scenario was computed. Absolute orientation process was used to transform coordinates of the model system into coordinates of the ground system then (RMSE) for each was computed. The difficulty of obtaining Ground Control Points (GCPs) that covers the photogrammetric project had been overcome by establishing a Portable Control System (PCS) .This (PCS) is a block made of an Aluminum alloy that had been shaped by a (TNC) milling machine to produce plane surfaces on it. The points of intersection of the produced surfaces on the block was labeled or coded and the distances between these points were measured manually by the digital vernier caliper 150mm and micrometer. These points represented control points in the captured images also a specific point within the block was chosen to be the center of this control system and all the remaining points was calculated with reference to it. The calculated (RMSE) for collinearity condition was 1.0212 mm. while for coplanarity condition was 1.0230 mm. So the precision of both models is nearly identical moreover coplanarity condition was more feasible for the requirements of close range photogrammetry.

Development of a Large-Format UAS Imaging System With the Construction of a One Sensor Geometry From a Multicamera Array

For the purpose of large-area topographic mapping, this study proposes an imaging system based on a multicamera array unmanned aerial system (UAS) comprised of five small-format digital cameras with a total field of view of 127°. The five digital cameras are aligned in a row along the across-track direction with overlap between two neighboring cameras. The suggested system has higher data acquisition efficiency than the single-camera UAS imaging system. For topographic mapping purposes, we develop a modified projective transformation method to stitch all five raw images into one sensor geometry. In this method, the transformation coefficients are obtained by on-the-job multicamera self-calibration, including interior and relative orientations. During the stitching process, two systematic errors are detected and corrected. In the end, a large-format digital image can be produced for each trigger event independently. The photogrammetric collinearity condition is evaluated using several external accuracy assessments, such as conventional aerial triangulation, stereoplotting, and digital surface model generation procedures. From the accuracy assessment results, we conclude that the presented raw image stitching method can be used to construct a one sensor geometry from a multicamera array and is feasible for 3-D mapping applications.

ORIENTATION MODELING FOR AMATEUR CAMERAS BY MATCHING IMAGE LINE FEATURES AND BUILDING VECTOR DATA

Abstract. With the popularity of geospatial applications, database updating is getting important due to the environmental changes over time. Imagery provides a lower cost and efficient way to update the database. Three dimensional objects can be measured by space intersection using conjugate image points and orientation parameters of cameras. However, precise orientation parameters of light amateur cameras are not always available due to their costliness and heaviness of precision GPS and IMU. To automatize data updating, the correspondence of object vector data and image may be built to improve the accuracy of direct georeferencing. This study contains four major parts, (1) back-projection of object vector data, (2) extraction of image feature lines, (3) object-image feature line matching, and (4) line-based orientation modeling. In order to construct the correspondence of features between an image and a building model, the building vector features were back-projected onto the image using the initial camera orientation from GPS and IMU. Image line features were extracted from the imagery. Afterwards, the matching procedure was done by assessing the similarity between the extracted image features and the back-projected ones. Then, the fourth part utilized line features in orientation modeling. The line-based orientation modeling was performed by the integration of line parametric equations into collinearity condition equations. The experiment data included images with 0.06 m resolution acquired by Canon EOS Mark 5D II camera on a Microdrones MD4-1000 UAV. Experimental results indicate that 2.1 pixel accuracy may be reached, which is equivalent to 0.12 m in the object space.

ORIENTATION MODELING FOR AMATEUR CAMERAS BY MATCHING IMAGE LINE FEATURES AND BUILDING VECTOR DATA

With the popularity of geospatial applications, database updating is getting important due to the environmental changes over time. Imagery provides a lower cost and efficient way to update the database. Three dimensional objects can be measured by space intersection using conjugate image points and orientation parameters of cameras. However, precise orientation parameters of light amateur cameras are not always available due to their costliness and heaviness of precision GPS and IMU. To automatize data updating, the correspondence of object vector data and image may be built to improve the accuracy of direct georeferencing. This study contains four major parts, (1) back-projection of object vector data, (2) extraction of image feature lines, (3) object-image feature line matching, and (4) line-based orientation modeling. In order to construct the correspondence of features between an image and a building model, the building vector features were back-projected onto the image using the initial camera orientation from GPS and IMU. Image line features were extracted from the imagery. Afterwards, the matching procedure was done by assessing the similarity between the extracted image features and the back-projected ones. Then, the fourth part utilized line features in orientation modeling. The line-based orientation modeling was performed by the integration of line parametric equations into collinearity condition equations. The experiment data included images with 0.06 m resolution acquired by Canon EOS Mark 5D II camera on a Microdrones MD4-1000 UAV. Experimental results indicate that 2.1 pixel accuracy may be reached, which is equivalent to 0.12 m in the object space.

A COMPARISON STUDY BETWEEN COLLINEARITY CONDITION, COPLANARITY CONDITION, AND DIRECT LINEAR TRANSFORMATION (DLT) METHOD FOR CAMERA EXTERIOR ORIENTATION PARAMETERS DETERMINATION

The present work emphasizes on using collinearity condition, coplanarity condition and DLT method for determining the camera exterior orientation parameters. The derivation of the mathematical formulation based on each suggested methods is explained. The comparison of the results of the methods was performed based on accuracy aspects using mathematical and actual photogrammetric data. The used data shows that the suggested methods are suitable for camera exterior orientation parameters determination for a block of photographs of any size. The results of this investigation prove that the accuracy of using coplanarity equations is slightly better than using collinearity equations or DLT method. Although the results of the DLT method are less accurate than those of using collinearity or coplanarity equation, DLT method is essential when the necessary information for the collinearity or coplanarity model is not available. This paper shows the necessity for the mathematical photogrammetric data for testing the photogrammetric studies.

VLBI 안테나와 모바일폰 카메라를 활용한 근접수치사진측량의 캘리브레이션 초기값 결정에 따른 3차원 정확도 분석

본 논문에서는 세종시 우주측지관측센터에 위치한 VLBI 안테나를 대상으로 모바일폰 카메라의 캘리브레이션을 수행하고, 촬영된 스테레오 영상으로부터 3차원 위치좌표를 산출하였다. 모바일폰에 탑재된 카메라의 캘리브레이션을 위한 초기값으로 DLT방법과 상용 수치사진측량시스템인 PhotoModeler <TEX>$Scanner^{(R)}$</TEX> ver. 6.0을 활용하였다. DLT와 PhotoModeler방법으로 산출한 표정결과를 초기값으로 사용하여 광속조정을 통해 카메라 내 외부 표정요소를 계산하고, 두 결과의 정확도를 비교하였다. 두 가지 방법으로 산출한 표정결과는 상당한 편차가 발생하지만, 비선형의 공선조건식을 이용한 광속조정계산으로 두 가지 방법의 최종 표정결과가 거의 일치함을 알 수 있었다. 또한, 두 가지 방법으로 결정된 카메라 내 외부 표정요소들을 이용하여 VLBI 안테나 특징점에 대한 3차원 좌표를 계산하고 토탈스테이션을 통해 측정된 기준좌표들과 비교하였다. 그 결과, 두 가지 방법 모두 표준편차가 <TEX>$X=0.004{\pm}0.010m$</TEX>, <TEX>$Y=0.001{\pm}0.015m$</TEX>, <TEX>$Z=0.009{\pm}0.017m$</TEX>로서, cm급의 높은 정확도를 나타내었다. 이러한 결과를 통해 정밀 사진측량의 목적이 아닌 허용오차의 범위가 상대적으로 큰 다양한 사진측량 분야에 모바일폰 카메라를 활용할 수 있을 것이라 판단된다. This study had been aimed to conduct the camera calibration on VLBI antenna in the Space Geodetic Observation Center of Sejong City with a low-cost digital camera, which embedded in a mobile phone to determine the three-dimension position coordinates of the VLBI antenna, based on stereo images. The initial values for the camera calibration have been obtained by utilizing the Direct Linear Transformation algorithm and the commercial digital photogrammetry system, PhotoModeler <TEX>$Scanner^{(R)}$</TEX> ver. 6.0, respectively. The accuracy of camera calibration results was compared with that the camera calibration results, acquired by a bundle adjustment with nonlinear collinearity condition equation. Although two methods showed significant differences in the initial value, the final calibration demonstrated the consistent results whichever methods had been performed for obtaining the initial value. Furthermore, those three-dimensional coordinates of feature points of the VLBI antenna were respectively calculated using the camera calibration by the two methods to be compared with the reference coordinates obtained from a total station. In fact, both methods have resulted out a same standard deviation of <TEX>$X=0.004{\pm}0.010m$</TEX>, <TEX>$Y=0.001{\pm}0.015m$</TEX>, <TEX>$Z=0.009{\pm}0.017m$</TEX>, that of showing a high degree of accuracy in centimeters. From the result, we can conclude that a mobile phone camera opens up the way for a variety of image processing studies, such as 3D reconstruction from images captured.

Bond theory for pentapods and hexapods

This paper deals with the old and classical problem of determining necessary conditions for the overconstrained mobility of some mechanical device. In particular, we show that the mobility of pentapods/hexapods implies either a collinearity condition on the anchor points, or a geometric condition on the normal projections of base and platform points. The method is based on a specific compactification of the group of direct isometries of $\mathbb{R}^3$.

Collinear Latent Variables in Multilevel Confirmatory Factor Analysis: A Comparison of Maximum Likelihood and Bayesian Estimations.

Because variables may be correlated in the social and behavioral sciences, multicollinearity might be problematic. This study investigates the effect of collinearity manipulated in within and between levels of a two-level confirmatory factor analysis by Monte Carlo simulation. Furthermore, the influence of the size of the intraclass correlation coefficient (ICC) and estimation method; maximum likelihood estimation with robust chi-squares and standard errors and Bayesian estimation, on the convergence rate are investigated. The other variables of interest were rate of inadmissible solutions and the relative parameter and standard error bias on the between level. The results showed that inadmissible solutions were obtained when there was between level collinearity and the estimation method was maximum likelihood. In the within level multicollinearity condition, all of the solutions were admissible but the bias values were higher compared with the between level collinearity condition. Bayesian estimation appeared to be robust in obtaining admissible parameters but the relative bias was higher than for maximum likelihood estimation. Finally, as expected, high ICC produced less biased results compared to medium ICC conditions.