The orthographic approximation does not take into account when camera motion is along the optical axis.  As a result, the reconstructed shape from the image sequence that contains this motion is often deformed. There are two methods of doing perspective approximations which capture this information. While remaining linear, both methods can be derived from a Taylor series expansion of the perspective projection, ignoring the higher order terms.  The first method is called scaled orthographic projection, also known as "weak perspective" projection. The second method is called paraperspective projection.  In this project, we implemented both methods.

Scaled orthographic projection is a first order approximation of the perspective projection.  It assumes that the variation in the direction of the optical axis is small compared to the viewing distance. When applying this assumption to the formulation of perspective projection, the transformation is reduced to an orthographic projection onto the image plane followed by an isotropic scaling of the image coordinates by some factor s, which is a the ratio of the focal length to the depth  [2][3].  The constraint imposed on m_f and n_f in equation (3) and (4)  (from the previous section) can thus be relaxed to |m_f|² = |n_f|² = 1 / z_f².

Paraperspective projection is another approximation to the perspective projection equations.  We also assume a small object size compared with the camera depth.  However, instead of discarding all the higher order terms as we did in the scaled orthographic projection, in paraperspective projection we try to preserve the larger portion of the second order term [2].  Thus, it is like one and a half order approximation to the perspective projection.

We will summarize the different metric constraints for these methods with the following equations.  Please see the reference for rigorous derivations.

Metric constraints

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