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2.2 Constraints between four images
Among any four images and their six possible image pairs, one singular correlation condition can be expressed by using
the five other singular correlation matrices. For example, if among four images i, J» k, and [, the pairs ij, ik, il, jl, and
kl belong to the image pairs taken in the adjustment then the image points xj, x} Observed from the missing image pair
jk can also be adjusted by using the condition equation
(Mijrj x Mipxp)" Mu(Mj x; x M4) — 0. (5)
Above, the operator x means cross product between vectors. Thanks to the equation 5, all observations in a block of four
images can be used with projective parameters. In larger blocks, this constraint can be used for an image pair without its
own singular correlation parameters only if the five neighbouring singular correlation matrices are in the adjustment.
23 Physical form of singular correlation matrix
The physical decomposition of the singular correlation matrix M, used in the physical version of the adjustment, has a
simple form, as follows
Rt; and Fij are orthogonal rotation matrices, C; and C; are the upper triangular interior orientation matrices, and B ; and
Bj; are skew symmetric relative base component matrices. In a block adjustment, seven of the orientation parameters have
to be fixed in order to fix the datum of the system. No additional constraint equations are needed with this decomposition.
3 NUMBER OF PARAMETERS AND CONSTRAINT EQUATIONS
It is known from projective geometry that there are, in general, 11 — 15 independent projective parameters in an arbitrary
block of n images. This is also the maximum number of free physical orientation parameters in a bundle block when the
number of possible non-linear parameters for the lens distortion are not taken into account.
With physically meaningful parameters, there are six exterior orientation parameters for each image in an arbitrary block,
three for the position, and three for the rotation. What is the number of free interior orientation parameters? The datum
can always be fixed with seven parameters (e.g. position and rotation of the first images, and the scale of the block) or
with seven so called inner constraints (Fraser, 1982). This gives 6n — 7 free parameters in the block. But, because there
can be as many as 11n — 15 free parameters, there has to be at most (11n — 15) — (6n — 7) — 5n — 8 free parameters
reserved for the interior orientation.
There are at most five linear interior orientation parameters per image (affinity, non-orthogonality, two principal point co-
ordinates, and focal length. This means that, if object constraints cannot be used, eight additional constraints are needed
to solve all 5n interior orientation parameters, or conversely, at most 5n — 8 free interior orientation parameters can be
solved in an arbitrary block from image information only. Usually, the block is not entirely arbitrary. It is possible that
the interior orientation is fully or partially known, or it is the same between all or some images.
The number of equations, constraints, and unknowns of the different adjustment methods are presented in Table 1. This
table correspond to an ideal case where all image data can be used in all methods. This is possible, in general, only if all
object points can be observed from all images. Therefore, the numbers are maximum values only. Note, that because no
3-D control was assumed here, the point has to be observed from two images before it can be taken into the adjustment.
| Method | e | t el i | u | r |
Projective | (2n 3» | 3n-2) | 0] 22n- 3) - (n - 2) | (2n - 3) - m | 2np- 6n-3p-m 7
| Physical | (2n — 3)p 0 7 0 6n 4- m 2np—6n—3p—m +7
Bundle 2np 0 7 0 G6n+m+3p | 2np—- 6n—3p— m-- 1
Table 1: The characteristics of three adjustment methods. e-number of observation or singular correlation equations,
c-number of datum constraints, t-number of trifocal constraints, i=number of interior orientation constraints, u=number
of unknowns, and r=e+t+c+i-u (redundancy). n=number of images, p=number of object points, and m=number of interior
orientation parameters in the block (m <= dn — 8).
It can be seen from Table 1 that the maximum redundancy is obtainable from the projective version when all possible
constraints and data are used. In practice, all object points cannot be observed from all images in an arbitrary block, so
it is not likely to have the same redundancy in the projective and bundle methods. However, in the physical adjustment,
the same redundancy with the bundle case can be achieved because the equations between pairs of observations can be
chosen independently for each point.
International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B3. Amsterdam 2000. 645
