Algorithm 19.2. Stratified auto-calibration algorithm using IAC constraints. in .NET Paint Data Matrix in .NET Algorithm 19.2. Stratified auto-calibration algorithm using IAC constraints.

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Algorithm 19.2. Stratified auto-calibration algorithm using IAC constraints. using none toencode none in web,windows application upc barcode although th none for none e infinite homography constraint seemingly provides six constraints on the five degrees of freedom of a;*, only four of these constraints are linearly independent. Removing the ambiguity. The one-parameter ambiguity may be resolved in several ways.

First, if there is another view available related by a rotation around an axis with a direction different to d r , then the combination of both sets of constraints will not have this ambiguity. A linear solution is easily obtained in the manner of (19.26).

Thus with a minimum of three views (i.e. more than one rotation) a unique solution can generally be obtained.

A second method of resolving the ambiguity is to make assumptions on the internal parameters of the cameras: for instance an assumption of zero skew (see table 19.4). The equations enforcing zero skew may be added as hard constraints to the set of equations being solved.

An alternative (but equivalent) method enforces the constraints a posteriori in the following manner. An ambiguity in solving for c, from the linear equation system. 19 Auto-Calibration Ac = 0, occ none for none urs when A has a 2-dimensional (or greater) right null-space. In this case in solving for u> there would be a family of solutions of the form u{a) Ui + au>2Here U\ and u;2 are known from the null-space generators, and a must be determined. It remains simply to find the value of a that leads to a solution satisfying the chosen constraint condition in table 19.

4. This is solved linearly. One could do the same thing solving for the DIAC, but then the constraint condition would be quadratic (see table 19.

2(/ 465)), and one of the solutions would be spurious. In certain cases, these additional constraints do not resolve the ambiguity. For example, skew-zero does not resolve the ambiguity if the rotation is about the image x- or y-axes.

Such exceptions are described in more detail in [Zisserman-98], and we give a few commonly occurring examples now. Typical ambiguities. The one-parameter family of solutions given in (19.

27) for cu*(/i) corresponds to a one-parameter family of calibration matrices obtained from u;*(/i) as cj*(/i) = K(/x)K(/x)T. For simplicity we will assume that the true camera K (which is a member of this family) has skew zero, so K has four unknown parameters. If the rotation axis is parallel to the camera X-axis, then d r = (1, 0, 0) T and v, = Kdr = a T ( l , 0 , 0 ) T .

From the form (19.11-p464) of u>* with no skew, the family (19.27) is.

U*(p) = W* rue + /iV r vT a2x(l + (i) + xl XoVo x0 xQy0 + vl yo '0. yo l (19.28). Note that t he entire family has skew-zero, and in this case only the element w ^ is varying. This means that the principal point and ay are unambiguously determined - since they may be read-off from elements which are unaffected by the ambiguity. However, it is apparent that ax cannot be determined because it only appears in the varying element to^ip).

To summarize this, and two other canonical cases: If K is computed from the infinite homography relation (19.25) assuming a zeroskew camera, then for some motions, there remains one undetermined calibration parameter. For rotation about various axes this ambiguity is as follows.

(i) X-axis: ax is undetermined; (ii) Y-axis: ay is undetermined; (iii) Z-axis (principal axis): ax and ay are undetermined, but their ratio ay/ax is determined. Geometric note. These ambiguities are not limited to calibration from a pair of views, but apply to complete sequences.

For instance if the set of rotations in a camera motion are all about the X-axis of the camera, then there will be a reconstruction ambiguity, and.
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