Er 
P = oêD-!= | (49') 
  
_ Where 
: n n 
nr -2(n-2ng) A (n-2)7 Som i 
= I= 
ny=3(ngtng) , nc=3ns 
Vfz(iVT 2VF ... a VI), 
Vi= (Vf, Vi, Tee VE 
1Cr 
P 
Cr (Ch Cr, t Ct. (50) 
Gi- (GT :Gl ....GT ), 
Gr= (Gy, Gp, +++ Gr, 
Vi. A'S NE ee VE) 
BB 
1C5 
2 Cs 
Gi- (GI &G}, + GF ) 
I= 0" QF — QF) 
U'- (Uf Uf I US 
U!= ( Xo Yo Zo ) 
v={XYZ2) 
Ds GA 205 IT) 
{= (LA Le: Lion 
Ly (Le 15 see l5] 
In eq. (49) and (50), right subs are about photos or 
ground points, left subs are equivolent to that of eqs (43) 
-(48). 
Eq. (49) is developed from co-angularity condition and 
crown finding method, and belongs to the Gauss-Markoff 
model, so we can call it Crown Co-angularity Gauss- 
Markoff Model(CCGMM) or simplely Crown Model (CM). 
The adjustment based on CCGMM is called Crown 
Adjustment. 
Refer to eqs. (49)-(50) we know that the parameter 
vectors Q and Ug are both photo-invarriants, so we can 
place them together in practical adjustment. From eq. (49) 
we can easily find the properties of the model: 
1 The normal equation can be composed one by one 
image point (vertex) and photo (base), and reduced one 
by one ground point (or photo) such as bundle 
adjustment. 
2) The sum of observation equations and the sum of the 
parameters in crown method are both equal to that of 
bundle method. 
3) The angular orientations are not any more parameters 
in this metod. This will make the computation convenient, 
and broaden the application range of photogrammetry, for 
example, in terrestral or close-range photogrammetry, we 
For eqs. (40), we let 
Ve=Q (42) 
The residual vector Vc has been transformed to parameter 
to parameter Q, we call it the first transformation. 
The second equation of eqs. (40) left-handly multiplied by 
p! , and substitute eq. (42) into it, we obtain 
Vg-CsQ4GgU-Ls (43) 
where 
C p=-F glF c 
--Fg!Bg (44) 
L =F 8 Ws 
The second equatiopn is transformed to an observation 
one, we call this the second teansformution. 
Substitute eqs. (42) and (43) into the first equation of eq. 
(40), and merge the identical terms, we obtain 
where 
C' ra=A paC Bt ca 
G' r1=ApsGstBra (46) 
W' Ta7W r4-A pal B 
At this time, we have eliminated the residual vector Vg 
from the first equation of eqs. (40). 
Eq. (45) left-handly multiplied by matrix Ars?, we obtain 
Vra=Cr2Q+GraU-Lra (47) 
where 
C 137-A rà 7C ra 
Gra=-Ara"Ges (48) 
L r1=A ra Wra 
Now that, the first equation of eq. (39) has been 
transformed into an observation equation, this is called the 
third and last transformation. Up to now, the general 
form of co-angularity condition eq. (39) has been 
thoroughly transformed into the expected observation 
form. The approach consists of three transformations and 
an elimination, so we call it Transformation with 
Elimination Technique. 
From eq. (42), (43) and (47), we know that the 
observation form of co-angularity condition consists of 
three types of equations. eq. (42) plays a part of control, 
so is called the Control Equation; eq. (46) presents the 
relation between the base points of the crown, then is 
called the Base Equation; eq. (47) is for the vertex of the 
crown, therefore is called verTex Equation. 
The above equations are about just one vertex with the 
base related to it. They can be conveniently expanded to 
fit a block. Assume that a block consists of ng photos, ng 
ground points and n image points, then the corresponding 
Gauss-Markoff can be written as 
Vr Cr Gr] | Q Lr 
npxl ncxi 
El Cs Gs - | Ls (49) 
Vc Ec 0 U 0 
ncxí 
ngx1 
< 
w