Radial Distortion Coefficient Research Articles

A Straightforward Camera Calibration Method Based on a Single Low-cost Cubical Target

Abstract. Calibrating optical sensors with common targets facilitates the efficient and convenient acquisition of the sensor's internal parameters. In this paper, we present a new method of camera calibration utilizing a low-cost foamy cube, in a form of dice, which is based on the fact that arrangement of pip and cubical die surfaces is mutually orthogonal. Initially, each face and pips are identified through the color information on the die’s surfaces. Subsequently, the centers of pips are corrected using a circular projection model, and radial distortion coefficients are estimated based on centers’ one-to-one correspondences. After that, the tangent information between pairs of pips on orthogonal dice faces are utilized to compute vanishing points, leading to estimation of intrinsic parameters. Experimental results demonstrate that our method has similar effects compared to well-known checkerboard calibration method, reaching an average relative error of 2.43%, simplifying the calibration process in practical applications and showcasing good practicality and robustness.

A New Camera Calibration Technique for Serious Distortion

A new camera calibration technique based on serious distortion is proposed, which only requires the camera to observe the plane pattern in an arbitrary azimuth. It uses the geometrical imaging principle and radial distortion model to acquire radial lens distortion coefficient and the image coordinate (u0, v0), and then solves the linear equation aiming at the other parameters of the camera. This method has the following characteristics: Firstly, the position of the camera and the plane is arbitrary, and the technique needs only a single observation for plane pattern. Secondly, it is suitable for camera calibration with serious distortion. Thirdly, it does not need expensive ancillary equipment, accurate movement, or lots of photos observed from different orientations. Having been authenticated by computer emulation and actual experiment, the results of the proposed technique have proved to be satisfactory. The research has also paved a new way in camera calibration for further studies.

A method of fine size measurement for telecentricity-based error compensation

Focusing on the stable magnification, near zero distortion, and wide field of depth (DoF), the telecentric optical system is well suited for size measurements with a large range of working distances. However, suffering from the telecentricity-based error (TBE), the size measurement is connected with the working distances and imaging position of the image sensor, even within the DoF. Therefore, a model of TBE compensation is established, inspired by the radial distortion model in the camera calibration. With the correction of radial distortion coefficients for different working distances, the TBE is evidently improved. The feasibility of the proposed method is verified both in the theory and experiment. Our results show that the uncertainty of size measurement at different working distances of the DoF is improved obviously after the TBE compensation, which is attractive for fine size measurements in industry.

Integrated correction of optical distortions for global HR-EBSD techniques

Optical distortions caused by camera lenses affect the accuracy of the elastic strains and lattice rotations measured by high-angular resolution techniques. This article introduces an integrated correction of optical distortions for global HR-EBSD/HR-TKD approaches. The digital image correlation analysis is directly applied to optically distorted patterns, avoiding the pattern pre-processing step conducted so far while preserving the numerical efficiency of the Gauss-Newton algorithm. The correction implementation is first described and its numerical cost is assessed considering a homography-based HR-EBSD approach. The correction principle is validated numerically for various levels of first-order radial distortion over a wide range of disorientation angles (0 to 14°) and elastic strain (0 to 5×10−2). The errors induced when neglecting such distortions as well as the influence of both the radial distortion coefficient and the pattern centre and optical centre locations are quantified. Even when both reference and target patterns are distorted, the correction appears necessary whatever the disorientation between those patterns. The required accuracy on the true distortion parameters for an effective correction is consequently determined.

Digital camera calibration for cultural heritage documentation: the case study of a mass digitization project of religious monuments in Cyprus

ABSTRACT The paper summarizes the methodology followed, to evaluate the accuracy of different digitization methods of ecclesiastical monuments in 3D computer vision form and stresses the importance of photographic equipment calibration. In this study, a set of images were taken using the CANON EOS M5 digital camera, while the internal calibration parameters – horizontal and vertical focal length (fx, fy), principal point coordinates (x0, y0), radial distortion coefficients (K1, K2, K3), tangential distortion coefficients (P1, P2) and the affinity and the shear terms (b1, b2) were estimated. These parameters were calculated using different software applications and then analyzed. For the calibration procedure, 3D texture models were built with the Agisoft commercial software based on: (a) the aforementioned calibration parameters and (b) the self-calibration process. The overall accuracy (Root Mean Square – RMS) between these models, by comparing known geo-referenced ground-control-points (GCP) is presented through the Cloud Compare software. The results indicate that the internal calibration parameters of the digital camera used for documentation purposes are essential and should be systematically implemented for documentation purposes.

Distortion correction method of a zoom lens based on the vanishing point geometric constraint

In order to solve the problem that the distortion parameter of the zoom lens varies with focal length changes, a correction method for a zoom lens based on the minimum fitting error of a vanishing point geometric constraint is proposed. In light of a calibrated image with parallel line clusters without knowing the metric, the method can quickly correct the distortion of zoom lens in full focal length by the regression fitting of parameters. Firstly, based on the radial distortion division model, the functional relationship between the vanishing point and the radial distortion coefficient is established according to the geometric constraint of the vanishing point, and the first-order radial distortion coefficient of the camera is estimated by the Levenberg–Marquardt algorithm. And then, in terms of the principle of the deviation error minimization, the least squares can be used to fit the equation of straight line of the corrected points to optimize the center-of-distortion. Finally, the variation of distortion parameters with focal length is analyzed, the distortion parameter corresponding table between distortion parameters and focal length and the empirical formula of fitting are established. The results show that the root mean square error corrected by the fitting empirical formulas and the least squares support vector regression parameter corresponding table method for three different focal lengths are 1.16 pixels and 0.83 pixels, respectively. Moreover, those two ways are about 13% and 19% faster than Zhang’s method, respectively. By mapping this method to FPGA, the application effect of photographic UAV, security monitoring and other equipment requiring rapid response can be greatly improved.

CAMERA CALIBRATION WITH IRRATIONAL RADIAL DISTORTION MODEL WITH ANALYTICAL SOLUTIONS

Abstract. Consumer grade digital cameras are widely used in many applications including Photogrammetric mapping and 3D modelling. One common limitation found in such cameras is radial lens distortions. To produce wide angle lenses camera manufacturers reduce the amount of barrel distortion by minimizing both the central and edge distortion profiles, resulting in a mixture of pincushion and barrel distortions for a single lens. These lenses also lack symmetry, making some of the existing distortion models almost ineffective. The mostly used model for radial distortion corrections is the polynomial model which is difficult to solve analytically especially when the model possesses many quadratic terms. Suggestions were made for division models but such models are not suitable when the lens field of view exceeds 180 degrees and exhibit some instabilities when dealing with large magnitude of distortion coefficients. Moreover mathematical formulations of some models cannot handle negative distortion coefficients. Attempts to improve the division models were made with proposals for rational models which present the advantage of handling larger distortion magnitude with fewer terms. However some of these models do not account for all the distortion coefficients in their solutions, limiting the potential of the techniques. This study presents an irrational distortion model with analytical solutions. The proposed model was tested with imagery captured by wide angle lenses and the experimental results reveal that the technique produced the best estimates of radial distortion coefficients. The proposed model was also able to capture image distortions originating from projection errors by the wide angles lenses used in this study.

Calibration Method Based on the Image of the Absolute Quadratic Curve

In this paper, a new camera calibration method based on the image of the absolute quadratic curve (IAC) is proposed, and a new target is designed for this method, which is both convenient and flexible. It first extracts the characteristic points and the characteristic lines of the target and finds out the vanishing point and the vanishing line. The radial and tangential distortion coefficients are obtained by using the cross ratio invariance to correct the target image distortion. Then, the four internal parameters of the camera are obtained by IAC. The influence of the skew parameters is ignored. The rotation matrix is then calculated by the orthogonal characteristic of the coordinate system, and the translation vector is calculated by the center coordinates of the camera. In this way, the internal and external parameters of the camera can be obtained. The internal and external parameters are taken as initial values, and the optimal results are obtained by nonlinear optimization using the reprojection method. Finally, the relative position between different target images can be obtained by using the fundamental matrix, namely, the rotation angle. In the process of solving, the normalization method is used to improve the accuracy of data processing. Not requiring any prior information of the camera, the method has a wide range of applications.

A hand-eye calibration method for augmented reality applied to computer-assisted orthopedic surgery

Purpose Recent technology advances in Augmented Reality (AR) is creating an opportunity for a paradigm shift in the field of Computer Assisted Surgery (CAS). Despite having the potential of being a promising technology in clinical applications, its accuracy, robustness, and performances under specific environmental conditions have not been fully investigated and evaluated. In an attempt to address the limitations and uncertainties that arise from using computer vision, we propose a Hand-Eye (HE) based approach to determine an invariant mapping between the HoloLens headset (i.e., rigid body defined by a unique set of passive markers) and the OHMDs virtual camera. Methods In order to achieve the required precision for using the Microsoft HoloLens in a surgical application, an invariant mapping was estimated. First, a set of non-collinear retroreflective markers were rigidly attached to the Microsoft HoloLens headset. Consequently, the OHMDs's intrinsic parameters, as well as their radial and tangential lenses distortion coefficients, were determined by solving simultaneously a set of homogeneous system of equations by using nonlinear optimization techniques. The accuracy of the proposed HE based method has been evaluated by using a retro-projection error estimation method. Results Initially, a surgical frame has been anchored to the organ of interest. After loading the patient-specific dataset, a hologram of the organ of interest appears in front of the user's initial position. Correspondences between the hologram and the fiducial markers can then be established by using the tracked surgical pointer and then be mapped with respect to the surgical frame. After finishing this process, an orthonormal basis is created in the physical space, which has its counterpart in the image space and, therefore, a transformation can be determined to map the hologram with respect to the surgical frame. Conclusions OHMDs may be able to create an opportunity for a paradigm shift in the field of CAS, since all this detailed information could easily be displayed on virtual and interactive screens in operating rooms on the same time and in real time. A very important aspect of using such OHMDs is that it can be very intuitive, especially among inexperienced surgeon, since surgical instruments and maneuvers motions are represented with respect to the surgeon's visual system and, therefore, attention disruptions are more likely to be reduced in terms of the surgeon's gaze, considering that the surgeon would not need to be searching for a conventional display located at static location in the operating room.

Binocular stereo vision calibration based on alternate adjustment algorithm

This paper presented a high precision camera calibration algorithm of binocular stereo vision system based on alternative adjustment. This method could be achieved by taking multiple pictures on a professionally designed calibration board in different directions. The watershed algorithm is used to extract feature points, and time domain coding is used to match image pairs. The camera initialization parameters including radial and tangential distortion coefficient are obtained by the classical monocular camera calibration algorithm. We built a binocular vision calibration system which is based on the left and right camera coordinates as the reference coordinate system respectively, and the internal parameters of the two cameras are optimized by alternately adjusting experiments to obtain the optimal value. Then, all the internal parameters are used to optimize the external parameters, so as to obtain the optimum distortion parameters and internal and external parameters. Compared with Zhang’s calibration method, the results show that the mean absolute relative error of the two camera parameters is less than 3.45e−6, and the mean re-projection error of the 10 sets of data is about 0.0725 pixels. The mean re-projection error obtained from the distorted model of Zhang is about 0.0852 pixels. Compared with other calibration methods, the proposed algorithm has higher accuracy and convenient, cost-effective and easy to implement.

A stereo photographic physical model for ocean wave measurement

Unlike the traditional models, a new stereo photographic model requiring no sea control points that are difficult to lay on a sea surface is developed in this paper. It is realized according to the order of camera intrinsic parameters calibration, relative orientation and absolute orientation based on average sea surface, and we give the governing equations and its algorithms. First of all, in the paper we present an imaging model that adopts non-measurement camera with considering only the radial distortion coefficients, and then give the method that calculates the camera intrinsic parameters by combining the co-planar equations and the distance equations to form a closed system of equations. For verifying the convergence of the governing equations, we lay four parallel sand ridges on a level ground in laboratory, measure seven lines of different heights and take a pair of the images by two cameras. The intrinsic parameters are successfully calculated by using the method, and further analysis shows that the results are highly precise. When observing the ocean waves in offshore sea, we need to recalculate the relative orientation parameters. At this time the governing equations are the co-planar equations, but the accuracy of the image matching is seriously related to the accuracy of the relative orientation parameters. However, as the mirror reflection of the sea surface, the matching accuracy of the sea image is often low, so we select 407 optimal matching points from among 1600 conjugate points and calculate the parameters. It needs a sequence of sea image pairs to obtain the absolute orientation parameters, 256 pairs in the paper. For each pair, we select firstly 300300 feature points on the left image approximately evenly, then find the conjugate points on the right image by image matching, calculate the coordinates of the corresponding sea surface points in the left camera coordinate system, and then fit the points into a plane; thus a plane sequence can be obtained. We establish a special object coordinate system on the average sea surface that is obtained by summing the plane sequence, and calculate the absolute orientation parameters by the geometric relationship between the two coordinate systems. And it is proved that the methods of relative orientation and absolute orientation proposed in this paper are feasible by reconstructing sea surface height field. The model greatly reduces the difficulty in calibrating the stereoscopic photography for ocean wave measurement, which is beneficial to its popularization and application.

Radial distortion correction in a vision system

Accurate correction of lens distortion is the core technology in visual based measurements. Radial distortion is the most significant lens distortion in a vision system. In this paper, a radial distortion correction method named the iteration geometrical similar method is proposed. The initial method is used to obtain a geometrical similar degree of image points. The distortion coefficients are then estimated by using the iteration approximation model. The radial distortion coefficients and other camera parameters can be successfully decoupled. Both simulative and practical experiments have been carried out to evaluate the performance of the proposed method. First, the influence of image noise and distortion degree on correction results is discussed in computer simulations. The results show that correction performance of the proposed method is manifested well. The proposed method was also compared with linear regression analysis (LRA) and the diversion model (DM) in the practical experiment. The results indicate that the average straightness calculated by the proposed method is only 0.195 pixels, compared with 0.394 pixels and 0.438 pixels by LRA and DM, respectively. In addition, the rectified images are used to calibrate the camera. The mean reprojection error using rectified images corrected by proposed method is only 0.137 pixels, which is significant less than 0.421 pixels and 0.462 pixels by the LRA and DM, respectively.

Camera Self-Calibration with Lens Distortion from a Single Image

This paper presents an effective approach for self-calibration with lens distortion using a single image combined with geometric constraints including vanishing points and ellipses. To improve the accuracy of self-calibration, radial distortion and distortion center are included in the calibration procedure. First, assuming image center as the symmetric center, the first radial distortion coefficient and vanishing points are simultaneously optimized from line segments in the image. Second, by utilizing the optimized vanishing points and extracted ellipse, principal distance and principal point are estimated. Last, distortion center is set as the current calculated principal point, and the above steps are then repeated until the principal point reaches a stable solution. Extensive quantitative and qualitative studies of the approach are performed. The experiments pertaining to simulated and real images demonstrate that the approach is effective and suitable and that the approach obtains satisfactory results.

Camera Self-Calibration with Lens Distortion from a Single Image

Abstract This paper presents an effective approach for self-calibration with lens distortion using a single image combined with geometric constraints including vanishing points and ellipses. To improve the accuracy of self-calibration, radial distortion and distortion center are included in the calibration procedure. First, assuming image center as the symmetric center, the first radial distortion coefficient and vanishing points are simultaneously optimized from line segments in the image. Second, by utilizing the optimized vanishing points and extracted ellipse, principal distance and principal point are estimated. Last, distortion center is set as the current calculated principal point, and the above steps are then repeated until the principal point reaches a stable solution. Extensive quantitative and qualitative studies of the approach are performed. The experiments pertaining to simulated and real images demonstrate that the approach is effective and suitable and that the approach obtains satisfactory results.

A Robust Camera-Based Method for Optical Distortion Calibration of Head-Mounted Displays

One of the problems in using HMDs is that the virtual images shown through the HMDs are usually warped due to their optical distortions. In order to correctly compensate the optical distortions through a predistortion technique, accurate values of the distortion parameters are required. Although several distortion calibration methods have been developed in prior work, these methods have some limitations. In this paper, we propose a method for accurately estimating the optical distortion parameters of both immersive and (optical) see-through HMDs, without the limitations. The proposed method is based on photogrammetry and considers not only the radial distortion but also the tangential distortion. Its effectiveness, including the effects of different coefficient orders of radial and tangential distortions, are evaluated through an experiment conducted with two different HMDs, an optical see-through head-mounted projection display (HMPD) and a commercially available immersive HMD. According to the experimental results, the proposed method showed significantly lower reprojection and line-fitting errors than a previous method proposed by Owen , and the radial distortion coefficients estimated by the proposed method were significantly more accurate than the nominal values obtained from the HMD optics prescription. In addition, when both the radial and tangential distortions were considered, the proposed method was significantly more effective than when only the radial distortion was considered. We also discuss an OpenCV-based simple implementation of a distortion correction method.

Hard-Disk Stereo Vision Measurement System Calibration Using Light Plane Constraint

This paper proposed a calibration method of sheet-of-light vision measurement sensor based on light plane constraint. Through capturing 12 images of different direction from homemade circular calibration target, the center of the circle and the light stripe is extracted based on Halcon platform of Germany. The experimental results obtained the intrinsic parameters, extrinsic parameters and radial distortion coefficient of the nonlinear model. At the same time the light plane constraint equation is got based on PCA plane fitting method. The results show that the calibration method is simple and reliable, and the method does not need any auxiliary adjustment. The work laid the better foundation for hard disk planeness vision measurement.

Least Square Estimation of Spacecraft Attitude along with Star Camera Parameters

A methodology for determining spacecraft attitude and autonomous calibration of star camera, both independent of each other, is presented. In this paper, both attitude estimation and star camera calibration is done together, independent of each other, by directly utilizing the star coordinate in image plane and corresponding star vector in inertial coordinate frame. Both radial and decentering distortion of lens accounted in the analysis. Satellite attitude, camera principal point, focal length (in pixel), lens distortion coefficients are found by a simple three step method. In the first step, camera intrinsic parameters are estimated using a closed-form solution assuming lens is distortion free. In the second step lens radial distortion coefficient is estimated by linear least squares method using the solution of the first step to be used in the camera model that incorporates only radial distortion. These steps are applied in an iterative manner until the radial distortion coefficient converges. In third step, lens decentering distortion coefficients are calculated using the estimated camera parameters and lens radial coefficient estimated in the previous steps. The whole procedure is fast enough for onboard implementation.

Computer Hard Disk Planeness Vision Measurement System Calibration Based on Light Plane Constraint

This paper proposed a calibration method of sheet-of-light vision measurement sensor based on light plane constraint. Through capturing 12 images of different direction from homemade circular calibration target, the center of the circle and the light stripe is extracted based on Halcon platform of Germany. The experimental results obtained the intrinsic parameters, extrinsic parameters and radial distortion coefficient of the nonlinear model. At the same time the light plane constraint equation is got based on PCA plane fitting method. The results show that the calibration method is simple and reliable, and the method does not need any auxiliary adjustment. The work laid the better foundation for hard disk planeness vision measurement.

Effect of Calibration Error on Reconstruction Accuracy of Stereovision System

This paper investigates the relation between the calibration error and reconstruction accuracy of stereovision system through simulation experiment. According to the model of a typical non-parallel stereovision system, an array of points in the common view field of the two cameras is projected onto the image plane of the left and right camera respec- tively, forming the left image and right image. After changing the calibration parameter slightly around its ground truth, the array of points is reconstructed. By comparing the reconstruction result of the points in the array with their original position, the major factors affecting reconstruction accuracy are summarized, which include the abscissa of the principal point, the component of the rotation vector along the ordinate axis, the first order radial distortion coefficient, the compo- nent of the translation vector along the abscissa axis, and the abscissa of the principal length. Especially, the abscissa of the principal point and the component of the rotation vector along the ordinate axis have prominent effect on reconstruc- tion accuracy. In the process of calibration, the error of the major factors must be controlled strictly, so that the accuracy requirement of the stereovision system can be satisfied.

A camera calibration method for large field optical measurement

A camera calibration method is presented for large field optical measurement, where the camera is close to the ground and the control points can only be located close to the ground, too. In such conditions, the camera's optical center and the control points are approximately coplanar. Only a single image of these control points captured by the camera in measurement state is used in the method. Neither to distribute the control points in space rationally nor to calibrate the camera's intrinsic parameters in laboratory in advance is needed. By the presented method, the camera's principal point position, focal length, radial and transverse tangency lens distortion coefficients, and the camera's position and attitude parameters can be estimated precisely. Then the calibration results can be used for precise large field optical measurement in the conditions that the camera's longitudinal tangency lens distortion can be neglected or the objects’ movement field is close to the ground, which is usually factual in practical applications. The presented camera calibration method has been successfully used in applications, such as automatic landing of UAV (Unmanned Aerial Vehicle) based on optical measurement guidance, to calibrate the cameras precisely.