C) 
irect Linear 
r solid-state 
e distortion 
| easy and 
fit for the 
tion in the 
2D DLT 
3 
ion, which 
expression 
orientation 
-orthogonal 
s a constant, 
ae 2D DLT 
duction and 
(2) 
. the relation 
between coefficients of 2D / and of 3D / can be got. A 
matrix of L can be defined as follows. 
mau (8 6 ze 
sally attdeet-—-Cgnonezel 
L 2 s 6 P 5 6 7 8-13) 
5 -k | ly ho luz 
When Z=0, the expression would be: 
+ 
IL ed, 4 
L= gif PL |= 1; oL [| 
H 72563 LS 
. —ftaud = 
d 5 such an 
=—|0 coo —M b, 2 (4) 
^ (1--d3cos/B ore Keats 
a bn 
0 0 I 
where 73 = wa Y. qu. Y. m. es. 
+ 
As a result, the meaning of coefficients l in 2D 
transformation and its relationship with coefficients / in 
3D transformation is as follows. 
* 
[, ef. -a,f, tandB —a,x,)- I, 
3 
I; -— (bf, —b,f, tandB-b,x,) 1, 
/3 
SS omes 
Men o e 1, tang — yx, )- I, 
3 
> 1 a, f. 
7 ln. Ty). =L 
73 (1+ds)cosdß 
* 1 b. f. | d 
Mo c m d -hyw) Hh 
y, (1+ds)cosdß 
"d 72, 
s V3 dico 730) s 
I, => =l, 
/5 
= A 
lg z zs 
Zs 
  
J 
The function of the above deduction lies in two ways: 
€ The 2D DLT establishes a strict projection relational 
expression between the object space 2D surface and 
International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part BS. Istanbul 2004 
film surface (digital camera chip surface). It 
establishes a relational expression for two surfaces 
that need no image interior orientation elements and 
fiducial marks. When using this expression, the image 
space coordinate can use any point as the origin and 
any direction as the axe. Theoretically speaking, it is 
not required that the image surface be paralleled with 
the object space 2D surface. 
€ There will be no effect on the distortion error 
calibration whether x axe and y axe is perpendicular 
or noi ( dB ), or whether the chip pixel is square or - 
PP 
not (ds). 
Suppose ds=0, dfi, p=œo=k=0, f. El. the 
* 
geometric meaning of each / factor would be: 
  
yr] Ad 
Leif d Eifdm ^ 
Liliana] aba bal. gc] 
: 4: 050 Lind 0 Ó| dX. 
E-—wi0 f iig, ] Q 0-1. Y 
Zs ; 
0 0.1 0 Q0. 1 0 0 Z7. 
A esr s 
f 0 IX m m 
1 : 1 
m o1) 7. |= {0 ua uu. 
Zs f um m m (6) 
9 0 As 0 30. 4 
* * 
Thus, in the above mentioned conditions, |, and {5 are 
* * * * : 
the image scales; f, ; I; ; I. and f are the decimal 
values; ls is the expression value for A according to 
- * ^ . ~ 
the scale, while d is the expression value for Y 
according to the scale. Select a suitable object spatial 
* 
coordinate system D-XYZ, when (Xs. Ys) is decimal, l 
and 4s would be decimal too. 
1.2 Methodology to establish digital camera distortion 
models 
By using control points (A, B, C, D), the self-created plane 
control network E and digital camera chip surface P may 
realize the 2D DLT. After getting factor |, the ideal