Metric constraints under affine projections

Let be an affine camera motion obtained by factoring the measurement matrix. Then is related to the ``true'' motion described with respect to the Euclidean coordinate frame by (9). Our goal here is to obtain the affine transformation (up to an arbitrary rotation) in (9) with which affine descriptions of motion and shape are corrected to Euclidean ones .

Let two 3-vectors and denote two rows of . Then, for each affine camera model introduced in §2.2, the affine transformation is determined as follows;

Paraperspective camera:

Substituting (3) into (9) and eliminating using its orthogonality, we have constraints on the affine transformation called metric constraints:

 

Let and be two elements of . Then, eliminating from (19), we obtain

which gives 2F linear homogeneous equations in terms of 6 elements of the 33 symmetric matrix . If three or more views are available, we can obtain a least-squares solution determined up to a scale as a solution of the ordinary eigenvalue problem. Once is determined, is recovered through, for instance, Cholesky decomposition.

Weak perspective camera:

From (4) and (9), the metric constraints has a form:

 

and the affine transformation is determined as a least-squares solution of the following equations:

Orthographic camera:

From (5) and (9), the metric constraints for orthographic camera become

 

and is recovered by solving, in the least-squares sense,