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Plane based Camera Calibration algorithms

 Most calibration algorithm dealing with plane based calibration make use of the three algorithm stated in this section may or may have inherited some of their algorithm from this three algorithm an example is the algorithm  developed by Heikkila & Silven (1997) this starts first by extracting the initial estimates of the camera parameters using Direct Linear Algorithm. Plane based Calibration algorithm makes use of reference grid and uses the images of known point array to determine the calibration matrix. This section only gives a brief outline of the different algorithm and a comparison between the algorithms.

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Direct Linear Transform

Direct Linear transform is one of the simplest methods and is adopted by different algorithms too. Calibration by direct linear transform consist of two steps, the first step solves the linear transformation from the object coordinates to image coordinates. This is represented by a 3 x 4 matrix Pi for the i-th projection and N fiducial points. The matrix parameter of the direct linear transform p11…..p34 can be obtained from the homogeneous matrix equation 

Lpi  = 0,

 Where L  is a Nx12 matrix, constituted by corresponding world and image coordinates and image coordinates and pi are the parameters of the direct linear transform (p11…..p34). The direct linear transformation matrix pi becomes singular in the case of a coplanar control point structure when this occurs; a 3×3 sub matrix has to be used. In this case the decomposition of the sub matrix can only deliver a subset of estimates of the camera parameters. After solving the system for these parameters a subset of them can be used as starting values for a classical bundle calibration 1.

Tsai Algorithm

This calibration algorithm assumes that the manufacturer provides some parameters of the camera 1. The Tsai algorithm requires n feature points per image and the calibration problem can be solved with a set of linear equations based on radial alignment constraint this linearizes a huge part of the computation. Skewness and lack of orthogonality are not recognized by Tsai model 1. The two stages can handle planar calibration grid or multiple images or single image nut it is essential to know the grid point coordinates. Also the two stages in the calibration model do not require initial guessing of the calibration parameters and is quite fast. First all the extrinsic parameters are computed except for the tz by using the parallelism constraint. In the second step, the non linear optimizations are used to evaluate all the missing parameters. In other to speed up performance the optimization does not use full camera model. The residual computed for error measurement is not really necessary and is done separately by building a full camera model. The difference between the back projected fiducial 3D world points and their corresponding image points gives the final error. The solution, generally designed for mono view calibration, was also applied for multiple viewing position calibration. In this case a pattern is moved to different levels for multiple calibration images.

 

Zhang Algorithm

The popular camera calibration method by Zhang uses at least two views of a planar calibration pattern called target, whose layout and metric dimensions are precisely known. In the Zhang algorithm, images of the model are taken under different views by either moving the model or the camera (or both). From each image sensor points are extracted (observed) and assumed to be in 1:1 correspondence with the points on the model plane. From the observed points, the associated homographies (linear mappings from the model points and the observed 2D image points) are estimated for each view. From the view homographies, the five intrinsic parameters of the camera are estimated using a linear solution, if the sensor plane is assumed to be without then images is sufficient. More views generally lead to more accurate result. Once the camera intrinsics are known, the extrinsic 3D parameters are calculated for each camera. The radial distortion is estimated by linear least-squares minimization. Using the estimated parameter values as an initial guess, all parameters are refined by non-linear optimization over all views

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