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Affine Structure

• Consider a set of 3D world points in affine (non-rigid) motion described by

where is the new 3D position of the point, is an arbitrary 3 x 3 matrix and is a 3-vector representing translation.
• Removing the effects of translation
by registering the points with respect to a reference point to obtain

• Affine projections
If the affine camera models for the two views are given by the parameters and respectively, then

• Basis and Affine structure
Now, consider four non-coplanar scene points with as the origin. We define three axis vectors for . form a basis for the 3D affine space and any of the vectors can be represented in this basis as

where are the 3D affine coordinates of . We call the 3D affine coordinates the affine structure of the point . It can be shown that the affine structure remains invariant to affine motion with respect to the transformed basis, that is,

where .
• Computation of Affine structure
From the above we obtain
 (1)

where, and .

Thus, to compute the affine structure, we require two images with at least four points in correspondence, i.e.,

These correspondences establish the bases and provided no two axes, in either images, are collinear. Each additional point gives four equations in 3 unknowns

and the affine structure can be computed. The redundancy in the system enables us to verify whether the affine projection model is valid.

Subsections

Next: Tomasi and Kanade factorization Up: Affine Multiple Views Geometry Previous: Computation of Affine epipolar
Subhashis Banerjee 2002-02-18