In this context, we use to represent a 2D point position in pixel coordinates and is used to represent a 3D point position in world coordinates. In both cases, they are represented in homogeneous coordinates, which is the most common notation in robotics and rigid body transforms.
Projection
Referring to the pinhole camera model, a camera matrix is used to denote a projective mapping from world coordinates to pixel coordinates. where.
Intrinsic parameters
The intrinsic matrix contains 5 intrinsic parameters of the specific camera model. These parameters encompass focal length, image sensor format, and principal point. The parameters and represent focal length in terms of pixels, where and are the scale factors relating pixels to distance and is the focal length in terms of distance. represents the skew coefficient between the x and the y axis, and is often 0. and represent the principal point, which would be ideally in the center of the image. Nonlinear intrinsic parameters such as lens distortion are also important although they cannot be included in the linear camera model described by the intrinsic parameter matrix. Many modern camera calibration algorithms estimate these intrinsic parameters as well in the form of non-linear optimisation techniques. This is done in the form of optimising the camera and distortion parameters in the form of what is generally known as bundle adjustment.
Extrinsic parameters
are the extrinsic parameters which denote the coordinate system transformations from 3D world coordinates to 3D camera coordinates. Equivalently, the extrinsic parameters define the position of the camera center and the camera's heading in world coordinates. is the position of the origin of the world coordinate system expressed in coordinates of the camera-centered coordinate system. is often mistakenly considered the position of the camera. The position,, of the camera expressed in world coordinates is . Camera calibration is often used as an early stage in computer vision. When a camera is used, light from the environment is focused on an image plane and captured. This process reduces the dimensions of the data taken in by the camera from three to two. Each pixel on the image plane therefore corresponds to a shaft of light from the original scene.
Camera resectioning
Camera resectioning determines which incoming light is associated with each pixel on the resulting image. In an ideal pinhole camera, a simple projection matrix is enough to do this. With more complex camera systems, errors resulting from misaligned lenses and deformations in their structures can result in more complex distortions in the final image. The camera projection matrix is derived from the intrinsic and extrinsic parameters of the camera, and is often represented by the series of transformations; e.g., a matrix of camera intrinsic parameters, a 3 × 3 rotation matrix, and a translation vector. The camera projection matrix can be used to associate points in a camera's image space with locations in 3Dworld space. Camera resectioning is often used in the application of stereo vision where the camera projection matrices of two cameras are used to calculate the 3D world coordinates of a point viewed by both cameras. Some people call this camera calibration, but many restrict the term camera calibration for the estimation of internal or intrinsic parameters only.
Algorithms
There are many different approaches to calculate the intrinsic and extrinsic parameters for a specific camera setup. The most common ones are:
Zhang model is a camera calibration method that uses traditional calibration techniques and self-calibration techniques. To perform a full calibration by the Zhang method at least three different images of the calibration target/gauge are required, either by moving the gauge or the camera itself. If some of the intrinsic parameters are given as data the number of images required can be reduced to two. In a first step, an approximation of the estimated projection matrix between the calibration target and the image plane is determined using DLT method. Subsequently, applying self-calibration techniques to obtained the image ofthe absolute conic matrix . The main contribution of Zhang method is how to extract a constrained instrinsic and numbers of and calibration parameters from pose of the calibration target.
Derivation
Assume we have a homography that maps points on a "probe plane" to points on the image. The circular points lie on both our probe plane and on the absolute conic. Lying on of course means they are also projected onto the image of the absolute conic , thus and. The circular points project as We can actually ignore while substituting our new expression for as follows:
Tsai's Algorithm
It is a 2-stage algorithm, calculating the pose in first stage. In second stage it computes the focal length, distortion coefficients and the z-axis translation.
Selby's method (for X-ray cameras)
Selby's camera calibration method addresses the auto-calibration of X-ray camera systems. X-ray camera systems, consisting of the X-ray generating tube and a solid state detector can be modelled as pinhole camera systems, comprising 9 intrinsic and extrinsic camera parameters. Intensity based registration based on an arbitrary X-ray image and a reference model can then be used to determine the relative camera parameters without the need of a special calibration body or any ground-truth data.
