Pinhole Camera Model - The Geometry and Mathematics of The Pinhole Camera

The Geometry and Mathematics of The Pinhole Camera

The geometry related to the mapping of a pinhole camera is illustrated in the figure. The figure contains the following basic objects

• A 3D orthogonal coordinate system with its origin at O. This is also where the camera aperture is located. The three axes of the coordinate system are referred to as X1, X2, X3. Axis X3 is pointing in the viewing direction of the camera and is referred to as the optical axis, principal axis, or principal ray. The 3D plane which intersects with axes X1 and X2 is the front side of the camera, or principal plane.

• An image plane where the 3D world is projected through the aperture of the camera. The image plane is parallel to axes X1 and X2 and is located at distance from the origin O in the negative direction of the X3 axis. A practical implementation of a pinhole camera implies that the image plane is located such that it intersects the X3 axis at coordinate -f where f > 0. f is also referred to as the focal length of the pinhole camera.

• A point R at the intersection of the optical axis and the image plane. This point is referred to as the principal point or image center.

• A point P somewhere in the world at coordinate relative to the axes X1,X2,X3.

• The projection line of point P into the camera. This is the green line which passes through point P and the point O.

• The projection of point P onto the image plane, denoted Q. This point is given by the intersection of the projection line (green) and the image plane. In any practical situation we can assume that > 0 which means that the intersection point is well defined.

• There is also a 2D coordinate system in the image plane, with origin at R and with axes Y1 and Y2 which are parallel to X1 and X2, respectively. The coordinates of point Q relative to this coordinate system is .

The pinhole aperture of the camera, through which all projection lines must pass, is assumed to be infinitely small, a point. In the literature this point in 3D space is referred to as the optical (or lens or camera) center.

Next we want to understand how the coordinates of point Q depend on the coordinates of point P. This can be done with the help of the following figure which shows the same scene as the previous figure but now from above, looking down in the negative direction of the X2 axis.

In this figure we see two similar triangles, both having parts of the projection line (green) as their hypotenuses. The catheti of the left triangle are and f and the catheti of the right triangle are and . Since the two triangles are similar it follows that

or

A similar investigation, looking in the negative direction of the X1 axis gives

or

This can be summarized as

which is an expression that describes the relation between the 3D coordinates of point P and its image coordinates given by point Q in the image plane.