Hrabacek, Jan 
3.3.5 Constraint on the angle between two planes inserts an assumption on the angle between two planes into the 
system (20). It benefits from the parametrisation by planes. Thanks to the unit lenght of the normal vectors n;, n; of the 
planes, the constraint assuming o as the angle between them, has the linear form 
(nj,nj) — cosa = 0 ; (12) 
For a given standard deviation m, of the assumed angle, it is necessary to compute a propagation on the variance of cos 
of the angle a 
2 xus D 9 
Meos œ = Sim” à ma ? (13) 
3.3.6 Constraint on the distance between two parallel planes is restricted in its application on such planes, on which 
the plane-plane parallelism constraint has been applied simultaneously. Again we benefit from over-parametrisation; the 
constraint has the simple form 
Hn, i In; | "m d; =0 , (14) 
di; — assumed distance between the planes. 
In;> In; — distances between the planes and the origin of chosen coordinate system. 
The signs of la; > In; have to respect corresponding or the oposite direction of the vectors. The constraint allows the use 
of a direct measurement of the object size. Standard deviation m, of d;; reflects the precision of the distance, and thus it 
is to be assessed by the user. 
3.4 Constraint on parallelism of two planes 
The equation (12) cannot be used for a value o. — 0, when the planes turn 
to be parallel, because the adjustment will not converge. Despite the problem, 
the parallelism constraint is very usefull for validating a regular object shape, 
as experiments show. Therefore, we have developed a new original solution 
to the plane-plane parallelism constraint, using a singularity free formulation. 
The angle constraint fails due to the need to fix two degrees of freedom in this 
case. We formulate two constraints using two vectors derived from n;, and here 
denoted n7, nz. For n; and the new pair of vectors following relationship is 
valid: 
dose. 
ioni 
ni xn] 
n 
  
n; 
; (15) 
  
  
The pair of constraints for plane parallelism is then Figure 3: Errors in the relation of par- 
allel planes 
(ni j/n;) 70 
(ni, n;) = 1) 
(16) 
Forn;; = (nz; Ny,Nz), à proper pair is chosen from the tripplet 
(n,,0,—nz), (0,n;,—n,), (ny, —n5,0) , (17) 
with the aim to have the angle i» between the two vectors as closer as possible to 90?. 
How to set weights of these newly introduced constraints? For a given standard deviation of 0? angle between D;, nj, We 
have to compute standard deviations for 90° angles between decomposed vectors ni ; n; and n;. The situation is shown 
in figure 3, the parameters are explained bellow. Applying rules of sphere trigonometry, we obtained an exact formula for 
the relation between €, €, €», p. We have made the assumption m., = m., = m_. and their correlation as the most 
unfavourable. Then we estimate the variances m2 m2, of the 90° angles with 
£i? 
2 
2 sin^ v > 
ma = me : (18) 
2 (1— cos qp) 
i 75 
q — angle between n; andn;. 
€ — real error in the 0? angle between n;, nj. 
£1, €2 — real errors of 90? angles between ni n; and n; , nj respectively. 
2 2 . À 
Tmz,, mz, — variances — mean value F(e1€1), E(esez) respectively. 
  
384 International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B5. Amsterdam 2000. 
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