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ly express 
primitives 
yrmulation 
and their 
respective attributes as well as the transformation function. In 
this paper, the similarity measure formulation has been 
incorporated in mathematical constraints ensuring the 
coincidence of conjugate linear features after establishing the 
proper co-1 egistration between involved surfaces. 
= 
Referring to Figure 4, the two points describing the line 
segment from the photogrammetric model undergo a 3D 
similarity transformation onto the line segment from the laser 
dataset. The objective here is to introduce the necessary 
constraints to describe the fact that the model segment (12) 
coincides with the object segment (AB) after applying the 
absolute orientation transformation function. 
3D Similarity Transformation 
B 
6 NS 
1 1 
T 2 
LIDAR Model 
Figure 4. Similarity measure between photogrammetric and 
laser linear features. 
For the photogrammetric point (1), this constraint can be 
mathematically described as in Equation 3. 
X, X, X, 4, À, 
; x ; ; 3 
elt S Rpm i fidel lpr | rd, 3) 
£, od 7. Ze -27 
Equation 4 shows the constraint for point (2) 
Ex. X. fux, xX, i, 
AS Ron a le Ach X e 
Z, £a Z, Z7 7. 
where À , and A, are scale factors. 
Subtracting Equation 4 from Equation 3 yields: 
za A An X, 
(4 AY. YS ms Roow| #-H e 
Le 34 2 Z, A 
5 by (A>-4;) and substituting À 
for S//A;-A,), Equation 6 is produced. Equation 6 emphasizes 
the concept that model line segments should be parallel to the 
object line segments after applying the rotation matrix. To 
recover the elements of the rotation matrix, Equation 6 is 
further manipulated and rearranged by dividing the first and 
Dividing both parts of Equation 5 
second rows by the third to eliminate A, Equations 7 
xr : Ti [ n r 
X B A 1 X; t, 
Ye Yo i= ln. N (6) 
Lu "E. | Z, "d 
(X X À Ras zx) Aa Ya +R, (4. - Z) 
(La md) Matt: XAR =) + & d. > Zi 
(#1) A, ; ni. 3)* R,, (Z5 =) 
500 RA, XR, (Fh) + Rl i) 
173 
International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B2. Istanbul 2004 
A pair of conjugate line segments yields two equations, which 
contribute towards the estimation of two rotation angles, the 
azimuth and pitch, along the line. On the other hand, the roll 
angle across the line cannot be estimated due to singularities. 
Hence a minimum of two non-parallel lines is needed to recover 
the three elements of the rotation matrix (Q, ®, K). Fi igure 5. 
Vu ret = 
Expected Singularity Optimum Configuration 
Figure 5. Singular and optimum configuration to recover 
rotation angles 
To determine the scale factor and the shift components, apply 
the rotation matrix to the coordinates of the first point defining 
the photogrammetric line, which yields Equation 8. 
X. A 45m X. 
| 
} 7 + S y : | I Y 4 + À, Y; Y, | (8) 
2, m Za Zu] 
where, 
[x, YA zl ER exl Y, ZI 
Rearranging the terms yields: 
[AR X PAR HS, A 
| ; : ; (9) 
Zl Y= el SEN 
| Z,- Z, Zz, +82 -Z, 
In Equation 9, eliminate A; by dividing the first and second 
rows by the third, Equations 10. The same applies to point 2 
and Equations 11 can be written. 
AR MY N ES) @, =I) if, +Sy, - 
rez) 12; +55" 24 12, > 14) ©, du 
  
  
^00) 
(X, - X) s x, T Sx. e Y) S (qu uS.-r "an 
(4 4d (Z6 $SeZEbS Za. Q 58:94, 
  
  
Combining Equations 10 and 11 produces the two independent 
constraints as shown in Equations 12. 
UU x XJ) AS AN) 
(7,482 —7,) (Z, +Sz,-Z,) 
0, +541) OF VA 
Wr SE ZW (Zug) 
  
  
Equations 12 can be written for each line in one dataset and its 
conjugate in the other. Consequently, two pairs of line segments 
yielding four equations are required to solve for the four 
unknowns. If the lines were intersecting, the shift components 
can be estimated (using the intersection points) but the scale 
factor cannot be recovered. As a result, at least two non- 
coplanar line segments are needed to recover these parameters, 
Figure 6. 
In summary, a minimum of two non-coplanar line segments is 
needed to recover the seven elements of the 3D similarity 
transformation.