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# Image rectification example.

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Technical Report
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This report details the algorithmic steps involved in the well-known camera calibration method by Zhang and describes an associated open-source Java implementation that depends only upon the Apache Commons Math library. Key terms: Computer vision, camera calibration, Zhang's method, camera projection, imaging geometry, image rectification, Java imp...

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## Citations

... Thus, the geometric camera calibration is the process of estimating the camera parameters necessary to form an image of the scene on the imaging sensor. Those parameters could be categorized as: (1) intrinsic parameters that describe the internal geometry of the camera such as the focal length, principal point and lens distortions (2) the extrinsic parameters that define the position and the orientation of the camera with respect to the real world coordinates system (Burger, 2016). ...
... -f u f v are the focal length in horizontal and vertical pixel unit, respectively -c u ; c v are the coordinates of the principal point -s is the skewness coefficient -u; v are the coordinates in pixel of the point projected on the image plane -X; Y; Z are the real world coordinates of the point -£; P are respectively the intrinsic matrix and the rigid transformation matrix from the world frame to camera frame One flexible and accurate implemented technique is the Zhang calibration method ''Z-method" (Burger, 2016;Feng et al., 2008;Ricolfe-Viala and Sanchez-Salmeron, 2007), commonly used to calibrate machine vision systems. The Z-method is based on the exploitation of a flat surface with known patterns called calibration grid (checkerboard, circle-board, etc.). ...
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The inspection of large mechanical parts manufacturing in real time camera-based scanning systems are increasingly adopted in industry 4.0. It leads to take preventive actions during the manufacturing process and then to fabricate mechanical parts right-first-time with respect to specified tolerances. Therefore, the use of camera-based scanners requests a preliminary calibration process. It consists on estimating the intrinsic and extrinsic parameters required to relate the 3D world point to its projection on the image plane. Since selection of the calibration grid poses affect the calibration quality, one approach-based machine learning (ML-approach) is proposed including the polynomial approximation of the reprojection errors function of 6 degree of freedom (DoF) combined with particle swarm optimization (PSO). Synthetic and experimental evaluations have been performed while assessing the performance of the proposed ML-approach. The synthetic evaluation reveals a better convergence of the intrinsic and extrinsic parameters in comparison to recent published calibration methods by Wizard (CW-method) and Rojtberg (R-method). The experimental evaluation of the ML-approach shows an average error RE < 12µm and a sub-micrometre repeatability, which confirm the benefit of using machine vision-based scanning systems for the inspection of large volume parts in real time.
... The proposed methodology requires undistorting the RGB image layer using the Zhang calibration strategy [43]. For this purpose, a set of at least 10 color images where a chessboard is visible needs to be sampled using RGB-D camera, as shown in Figure 7, we use a public toolbox available from [44] based in [45] to get a intrinsic matrix and undistortion coefficients. ...
... The first phase of the proposed method in Figure 8 uses a well-established method [43] to de-distort color images and get an intrinsic matrix, in this step, ten different images showing a chessboard are enough. A set of ten different images with a chessboard in the scene are needed to use [43] in order to find the intrinsics and undistort RGB layer, implementation details is based in [45] and a toolkit is available by [44]. ...
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RGB-D cameras produce depth and color information commonly used in the 3D reconstruction and vision computer areas. Different cameras with the same model usually produce images with different calibration errors. The color and depth layer usually requires calibration to minimize alignment errors, adjust precision, and improve data quality in general. Standard calibration protocols for RGB-D cameras require a controlled environment to allow operators to take many RGB and depth pair images as an input for calibration frameworks making the calibration protocol challenging to implement without ideal conditions and the operator experience. In this work, we proposed a novel strategy that simplifies the calibration protocol by requiring fewer images than other methods. Our strategy uses an ordinary object, a know-size basketball, as a ground truth sphere geometry during the calibration. Our experiments show comparable results requiring fewer images and non-ideal scene conditions than a reference method to align color and depth image layers.
... Geometric calibration is the process of estimating the intrinsic and extrinsic parameters of each optical components [8]. The intrinsic parameters define the internal geometry of the camera including the focal lengths (f u , f v ), the coordinates of the principal points (c u , c v ) and the lens distortion coefficients (k 1 , k 2 ). ...
... The intrinsic parameters define the internal geometry of the camera including the focal lengths (f u , f v ), the coordinates of the principal points (c u , c v ) and the lens distortion coefficients (k 1 , k 2 ). The extrinsic parameters define the pose (position and orientation) of the camera in space [8]. The estimation of those intrinsic and extrinsic parameters is required to for the 3D scanning of objects. ...
... The process of geometric calibration for cameras differs depending on the use and application. For close range photogrammetry [8], camera calibration requires the use of a calibration object with known geometry. For aerial photogrammetry, self-calibration methods are preferred [9]. ...
... In this research, we define the model plane as the model that consists of ( ) m photos and relating the camera coordinate system to ground coordinate system. For each photo in the model plane, the relationship between ground point coordinates ( , , ) X Y Z , in metric units, and camera point coordinates ( , ) u v , in pixels, is investigated using the pinhole camera model suggested by Zhang (2000), Burger (2019), and Burger and Burge (2016): ...
... If we have ( ) m photos with ( 3) m ≥ , a singular value decomposition solution (Golub & Van Loan, 2013) is applied in order to obtain the 6 elements (actually 5 elements up to a scale factor) of ( ) b vector. Once the ( ) b vector is obtained, a unique solution for the 5 intrinsic parameters is obtained as follows (Burger, 2019): ...
... Then, we have the following relationship based on distortion coefficients 1 2 ( , ) k k 1 2 ( , ) k k (see Burger, 2019;Burger & Burge, 2016;Camer, 1971;Wei & Ma, 1994): ...
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This paper aims to calibrate smartphone’s rear dual camera system which is composed of two lenses, namely; wide-angle lens and telephoto lens. The proposed approach handles large sized images. Calibration was done by capturing 13 photos for a chessboard pattern from different exposure positions. First, photos were captured in dual camera mode. Then, for both wide-angle and telephoto lenses, image coordinates for node points of the chessboard were extracted. Afterwards, intrinsic, extrinsic, and lens distortion parameters for each lens were calculated. In order to enhance the accuracy of the calibration model, a constrained least-squares solution was applied. The applied constraint was that the relative extrinsic parameters of both wide-angle and telephoto lenses were set as constant regardless of the exposure position. Moreover, photos were rectified in order to eliminate the effect of lens distortion. For results evaluation, two oriented photos were chosen to perform a stereo-pair intersection. Then, the node points of the chessboard pattern were used as check points.
... Zhang's camera calibration was first introduced by Zhang (1998;2000), and later implemented by Burger (2019). In this approach a chessboard pattern is observed using a single camera from different orientations for the calculation of intrinsic (interior), extrinsic (exterior) and lens distortion parameters. ...
... Since two groups of photos exist, we have two model planes: a wide-angle model plane and a telephoto model plane. For each photo in each model plane, the following equation (Zhang, 2000;Burger, 2019;Burger and Burge, 2016) describes the relationship between object point coordinates ( , , ), in metric units, and the image coordinates( , ), in pixels: ...
... First, all photos captured by both wide-angle and telephoto cameras were rectified using the values of radial lens distortion coefficients ( 1 , 2 , 3 ) and decentring lens distortion coefficients ( 1 , 2 ) obtained for both cameras in the camera calibration. Given the original image , the rectified image ′ is to be calculated according to Burger (2019) and Aldelgawy and Abu-Qasmieh, Preprint) as follows: ...
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In this paper, the possibility of reconstructing object lines using a smartphone’s rear dual camera (wide-angle and telephoto) was examined through designing a semi-automatic system. After calibrating both cameras, six scenes for each of three objects were captured and rectified. Object lines were categorised into six groups based on the distance and angle to the dual camera system. Image lines were extracted using the linear Hough transform technique and points of intersection detected. Stereo pairing of conjugate points then allowed the calculation of object coordinates and the lengths of object lines were compared to their lengths measured by a digital caliper. The best line reconstruction results were achieved with the smallest distance and angle to the dual camera system. © 2021 The Authors. The Photogrammetric Record © 2021 The Remote Sensing and Photogrammetry Society and John Wiley & Sons Ltd
... The pictures of this camera were extracted first, and then they were calibrated using a program was designed by the current author for this purpose, it depended on Zhang's method [35], and the algorithm was applied is the one introduced by Burger [5], with slight modification that included more radial distortion coefficients. The camera was set at fixed position in front of the channel where it could take continuous pictures, the frequency of capturing the pictures was 1 frame per 30 s. ...
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A new model for Korteweg and de-Vries equation (KdV) is derived. The system under study is an open channel consisting of two concentric cylinders, rotating about their vertical axis, which is tilted by slope $$\tau$$ τ from the inertial vertical $$z$$ z , in uniform rate $${\Omega }_{1}=\tau \Omega$$ Ω 1 = τ Ω , and the whole tank is elevated over other table rotating at rate $$\Omega$$ Ω . Under these conditions, a set of Kelvin waves is formed on the free surface depending on the angle of tilt, characterized by the slope $$\tau$$ τ , volume of water, and rotation rate. The resonant mode in the system appears in the form of a single Kelvin solitary wave, whose amplitude satisfies the Korteweg-de Vries equation with forced term. The equation was derived following classical perturbation methods, the additional term made the equation a non-integrable one, that cannot be solved without the help of numerical methods. Invoking the simple finite difference scheme method, it was found that the numerical results are in a good agreement with the experiment.
... It is reasonable to assume that the cause of this is the non-linear optimization in the photogrammetric calibration process, which is performed in order to minimize the total projection error. 16 This process implicitly improves the collinearity of the reference points and consequently leads to smaller initial functional values. ...
... • Zhang calibration algorithm: this calibration approach uses a planar pattern shown at few different positions and orientations in space [6]. It is supposed to be easily implemented and more flexible as compared to classical techniques [29]. ...
... It only requires the camera to observe a planar pattern shown at a few different orientations; the knowledge of the plane motion is not necessary. An algorithm of this method was earlier given by Burger 36 and was used by the author of this paper, but she slightly changed his algorithm as he included only two radial distortion coefficients, whereas the program here includes five different coefficients: three radial distortion parameters and two tangential ones. The program was written using the Python programming language; the pattern used in the calibration was a checkerboard that had 8 × 8 squares, taking several pictures to cover all Euler angles, yaw, pitch, and roll or twist angles. ...
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In this paper, theoretical and experimental results of the forced oscillations in an open precessed cylindrical channel are reported. The theoretical part treats the problem using a linear inviscid irrotational approximation, where different inertial modes, such as resonance ones, are presented. A real-channel flume model was constructed for the experimental part, where three different control parameters were considered: the nutation angle, the rotation rate, and the average water depth. The experiments focused on tracking the different responses toward the provided forces to the system with comparison with the assumed theory where this was possible, as other nonlinear aspects appeared. The experimental observations were tracked using a charge-coupled device sensor type camera, which enabled the extraction of both quantitative and qualitative results. The measurement of the azimuthal velocity involved the use of an acoustic Doppler velocimeter, an approach that closely aligned with the theoretical linear model; this measured velocity was also compared with the three different control parameters. The system shows instabilities in the form of resonance collapse and triadic resonance; in addition, an experimental diagram involving variation in both Reynolds and Rossby numbers is provided.
... To project a 3D point in the camera coordinate system to a 2D homogenous image coordinate we first need to calibrate the intrinsic matrix K. The formula for the projective transformation using the intrinsic matrix is given in Equation We used Singular value decomposition (SVD) to find the least-squares solution to the homogenous system [58]. Implementations of the algorithm can easily be found in common computer vision libraries such as Open-CV. ...
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Efficient quantitative analysis of plant traits is critical to keep pace with advances in molecular and genetic plant breeding tools. Machine learning has shown impressive results in automating a lot of these analytical processes however, many of the algorithms rely on a surplus of high-quality biological imagery. This data is currently collected in labs via static camera systems, which provide consistent images but are challenging to tailor to individual plants, species, or tasks. Current research in autonomous camera systems use object detection or tracking methods to control the camera. Unfortunately, this quickly falls apart for static biological imagery as large inter- and intra-species variations, even within the same specimen, make object detection less robust and stationary targets make tracking unusable. Inspired by the success of deep learning in the autonomous driving space, we apply an end-to-end learned approach to directly map saliency-augmented input frames from an RGB monocular camera to a pan-tilt-zoom (PTZ) actuation. Our results show our model correctly classifies which direction to move the camera in 87% of instances and has an average offset error of 250 and 140 pixels for a 1920x1080 image, respectively. Results on a much smaller, plant-only dataset demonstrates the applicability of the model to biological imagery and we demonstrate saliency’s effectiveness in improving accuracy by up to 4%.