In standard perspective projection, the mapping from a 3D coordinate onto the image plane is accomplished via the projection Equation 11.

However, we instead use the central projection representation as depicted in Figure 7. Here, the coordinate system's origin is fixed at the image plane instead of at the center of projection (COP). In addition, the focal length is parameterized by its inverse, . This camera model has long been used in the photogrammetry community and has also been adopted by Szeliski and Kang [52] in their nonlinear least squares formulation. The projection equation thus becomes Equation 12.

Note how this projection decouples the camera focal length (** f**) from
the depth of the point (

In our representation, however, the inverse focal length
alters the imaging geometry independently of the depth value ** Z_{C}**.
State variable decoupling is known to be critical in Kalman filtering
frameworks and is applicable here since we plan on putting

Another critical property of
as opposed to ** f** is that it does
not exhibit numerical ill-conditioning. It can span the wide range of
perspective projection but also the special case of orthographic
projection which occurs when we set the focal length
and
all rays project orthogonally onto the image plane. However, under
orthographic projection,
which does not 'blow up' and
maintains numerical stability in KF frameworks. We can thus combine
both perspective and orthographic projection into the same so-called
central projection framework without any numerical instabilities (this
is demonstrated experimentally in the next section). This flexibility
is not typical in many traditional computer vision approaches where
perspective and orthographic projection must be treated quite
differently. We now begin building our internal state vector with this
well-behaved parameter,
as in Equation 13.