Coordinate Systems

Camera Coordinate System

Assume an object point X (4x1 homogenous coordinates) is mapped to the image coordinates x (3x1 homogenous coordinates) by multiplication of the projection matrix P (3x4):

x = PX

The matrix P is composed of the rotation matrix (3x3), the translation t vector (3x1) and the camera matrix (3x3). The matrix P (3x4) is shaped as follows

P = K[R|t].

Thereby the translation is t = -RC. C (3x1) represents the coordinates of projective center of the camera with respect to the object COS. The z-axis of the camera is oriented in viewing direction.

$ExtOri_RotationMatrix

R
$ExtOri_TranslationVectorC
Exterior orientation parameters for the .ori format

Image Coordinate System

Let X_cam = [R|t]X be the coordinates of an object point in the cam system. X_cam(3x1) can be mapped to homogenous image coordinates using the camera matrix K

x = KX_cam.

K is shaped as follows:

Thereby f_x, f_y represent focal lengths in pixels. Parameters c_x, c_y represent the offsets to the principle point in terms of pixel dimensions. The parameter s_x specifies the skew parameter. In the image COS (0,0) corresponds to the center of the upper left pixel.

Image coordinate system
$IntOri_CameraMatrixK
$IntOri_SensorSizeImage width, Image height in pixels
$IntOri_PixelSizePixel size in mm, if not known specify with 1.0 (not used by the algorithm)
$IntOri_FocalLengthFocal length in mm, if not known or f_x, f_y vary specify wit 1.0 (not used by the algorithm)
Interior orientation parameters for the .ori format