The world-to-image projection is given by:

where:

is a 3D column vector representing the position of the scene point in the world spatial reference system,

is a 2D column vector representing the position of the image point, in pixels. The origin is the center of the upper-left image pixel, the x-axis is oriented to the right side of the image, and the y-axis is oriented to the bottom of the image.

is a 3D column vector representing the position of the camera center.

is a 3 × 3 rotation matrix representing the rotation of the camera, which maps the axes of the world spatial reference system to the camera axes defined by camera orientation. can specified either from:

• Its coefficients:
• Omega/Phi/Kappa angles:
• Heading/Roll/Pitch angles:

• Yaw/Pitch/Roll angles:

with:

(Y, P, R) = (0, 0, 0) meaning that the camera is looking horizontally in the Y direction (X axis on the right, Z axis up).

Yaw angle being the clockwise rotation around Z axis.

Pitch angle being the counter clockwise rotation around X axis.

Roll angle being the clockwise rotation around Y axis.

is a 3 × 3 rotation matrix bringing the camera axes defined by camera orientation to the canonical camera axes (x-axis oriented to the right side of the image, y-axis oriented to the bottom of the image, and z-axis oriented to the front of the camera):

is the perspective projection function defined by:

is the distortion function defined by:

• 

(Please note that, in some other conventions, the roles of and are reversed).

is the focal matrix, where is the focal length of the camera in pixels, the skew parameter and the pixel ratio.

is a 2D column vector representing the position of the principal point of the camera, in pixels. The origin is the center of the upper-left image pixel, the x-axis is oriented to the right side of the image, and the y-axis is oriented to the bottom of the image.