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Image rectification example.  

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 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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... 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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... 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: ...
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... 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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... 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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