# Chinazom were inexpensive and was used commonly

Chinazom Eunice NnoliAbstract:Camera Calibration is a prerequisite for extracting precise and reliable metric information during close-range photogrammetric measurement. Camera calibration algorithm using plane calibration sheets is used for obtaining the internal parameters of a camera by viewing a plane pattern with a known geometric structure and is based on the properties of the calibration sheet or plane by minimizing their algebraic distances. The algorithm can simultaneously calibrate different views from a camera with variable intrinsic parameters and it is easy to incorporate known values of intrinsic parameters.  This paper gives an overview of camera calibration using plane calibration sheet. 1 Introduction 1      A camera which is an optical instrument for recording or capturing images has been around for a very long time.

The first set of cameras was expensive and not everyone could afford them as a result people came up with pinhole cameras which were inexpensive and was used commonly in our everyday life but this also came with a price as most pinhole cameras have significant distortion. When a camera’s principal distance, lens distortion parameters, and principal point offset are known the camera is said to be calibrated. Camera Calibration Using Plane sheet requires metric information like coordinate, angle, length ratio, radius etc about the reference plane.  Calibration sheet is cheap and can be produced easily depending on the required accuracy. Also since the calibration sheets are manmade, and their metric structure is known, it only requires knowing the homography matrices induced by world planes, whose estimations are much stable and accurate than those of inter-image transformations arising from projections of points.

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Once the internal parameters are recovered, the estimation of the relative position between planes and cameras can be achieved. The paper is organized into the following sections; the projection model, principle of plane-based calibration and three calibration algorithms commonly used in plane-based calibration. 2 The projection modelThe process of transforming or mapping from world 3D point to 2D image point is done whenever an image is captured using a camera. This goes to show that every point in the object space is transformed to the image space or plane. The image plane is positioned in front of the optical center at a distance which is the origin of the camera (optical center). The object space depicts what we are trying to capture on the image plane while the image plane is what is obtained after capturing the object. The focal length is the distance between the optical center and image plane.Figure 1: shows camera modelPoint P (X, Y, Z) yields;        (1)Image plane projection point is                                                (2)The following process is used for modeling the projection process v  The transformation from object plane 3D to Camera 3D:  The location of point from the object coordinate system in the camera coordinate system can be specified by view transformation WWhere  W  =        Location of point P in 3D camera coordinate can be given as   =        .

Simply put P’ = W. P.v  Projection on to the normalized image plane: The projection from the 3d camera coordinate system to a continuous normalized 2D coordinate system on the image plane can be described as follows; Step 1 finding the normalized projection of xStep 2 transformations from normalized coordinate x (2D affine transformation) to camera coordinate u. The affine transformation helps to map the scale and skewing of the camera coordinate. =  hom-1 A .hom (x)=   A’ . hom (x)A =  , hom-1        The W matrix captures the extrinsic parameters of the projection and A matrix captures the intrinsic properties of the camera.

v  Lens distortion:  Cameras are made up of lenses and they introduce distortions which include decentering errors and radial distortion.  The normalized 2D projection coordinates are prone to non linear radial distortion with respect to the optical center and is expressed byr =   with r and D been the lens-distorted 2d coordinates in the normalized image plane.In summary the projection process can be summarized with the diagram below.

Figure 2: projection processFrom right to left the point 3D point x in the camera plane (diagram c) is projected to the image plane to the normalized coordinate. In diagram b the lens distortion is mapped.  The affine mapping specified by the intrinsic camera transformation (matrix A) finally yields the observed sensor image coordinates u = (u, v T )  in  (a). 3 Principle of Plane-based camera calibrationPlane based calibration can be simply done through the determination of the image of the Absolute Conic (IAC) using plane homographies leading to a simple linear calibration equations 1. After scaling, the image Absolute Conic (IAC) takes this form:Implementation of the calibration constraints can be expressed as follows (Homography):The unknown camera location t and the equation holding up to scale only can be extract two different equations in ? that prove to be homogeneous linear: where h1 is the ith column of H. With these equations the fundamental calibration equation is given and if there are more than one calibration plane, the new equations are included into the linear equation system. It is not of much importance if the planes are viewed differently or if the planes are seen in same views or if the same plane is seen in numerous views, provided the calibration is constant. The equation system is of the form Ax = 0, with the vector of unknowns x = (?11; ?22; ?13; ?23; ?33)T.

After having determined ?, the intrinsic parameters are extracted via:Furthermore the principle of plane-based camera calibration can be extended either by knowing beforehand intrinsic parameters or by applying using variable intrinsic parameters with calibrating cameras. When the plane-based camera calibration is extended with prior knowledge of the intrinsic parameter, this eliminates unknowns and helps to reduce the linear equation system.Calibration procedure can be summarized as follows: ·         Print a pattern and attach it to a planar surface·         Take a few images of the model plane under different orientations by moving either the plane or the camera·         Compute plane homographies from the given features.·         Construct the equation matrix A·         Ensure good numerical conditioning of A·         Detect the feature points in the images·         Estimate the intrinsic and extrinsic parameters 4 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. Direct Linear TransformDirect 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 2.Tsai AlgorithmThis calibration algorithm assumes that the manufacturer provides some parameters of the camera 2. 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 2. 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 AlgorithmThe 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 intrinsic 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 views5 ConclusionCamera Calibration Using Plane Calibration Sheet is a flexible technique for calibration of cameras.  Calibration of camera using planar calibration targets in different views or single view has different approaches and the accuracy that can be obtained can be attributed to the differences in the camera models.

When compared to the more advanced techniques that makes use of expensive equipment the planar based gains considerable flexibility.

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