## 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.

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## 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 |

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## Interior orientation parameters for the .ori format |