IC MODEL 
of the light ray 
ted at different 
ctive imaging 
1 Figure 3, the 
e plane caused 
t negligible. In 
]ium refraction 
rection formula 
tral perspective 
ections applied 
inciples using 
trated in Figure 
| pseudo central 
. Based on the 
dium refraction 
e have 
(1) 
  
Cover lens 
Image plane 
ic model 
2 
  
Furthermore, we have 
P"N - (D*d*H) tan, (2) 
The correction equation is obtained as 
Ar/r ^ d (1-tan8y tan;) / (D+d+H) 
+ H (1-tan60y tan9yj) / (D* dH) (3) 
where D and d are system constants. H is related to the object 
point position. 
H - OP xf/ Op" -D -d 
OP — sqrt ( (X-Xo)^* (Y-Yo) ^* (Z-Zo)?) 
Op" =~ sqrt (f+ rxr) 
4) 
The incident angle can be determined by 
tan6, = rif (5) 
and the angles 0; and 0j can be computed using Snell's 
refraction law: 
njsinO,-njsin 0, (6) 
Incorporating the above correction into the standard collinear 
equations, the reduced central perspective model becomes 
x-óx-Ax-- U/W 
y-ày-Ay--(I/ky) V/W (7) 
where, 
U=LiX+L2Y+L3Z+La 
V = LsX+L6Y+L7Z+L8 
W= LoX+L10Y+L112+1, 
and ky is a scale factor of the CCD camera, dx, dy are lens 
distortion corrections, and Ax, Ay are image corrections for 
media refraction. 
Ax 7 x (Ar/r) 
Ay — y (Ar/r) (8) 
It should be noted that the unknown H is involved in the above 
corrections. Thus, an iterative solution of Equations (7) is 
needed. 
3.2 3D Ray Tracing Model 
Within an imaging system, a light ray originating from an 
arbitrary point P in the object space with a given starting 
propagation direction can be traced through the optical system 
by successive uses of the law of refraction. Based on 
characteristics of the light ray propagation, algebraic and 
trigonometric expressions governing the precise path of a 
chosen initial ray through the optical system can be used to 
derive ray tracing equations. By applying these equations one 
can determine the exact intersecting points on the image plane 
or indeed on any chosen image surface. 
In Figure 3, a light ray originating at a point on the ith surface 
Pi (X, Yi, Zi) propagates to P;.; (X;-1, Yi-1, Zi-1) On the (i-1)th 
surface. Assume that the medium between the two surfaces is 
321 
homogenous with a refractive index nj. The length of p.- p. is 
represented as an auxiliary quantity p; as 
  
2 2 2 
pe Ka *Q-YXa4) *-224).— (9) 
Generally, a light ray between the object point P and image 
point p will be refracted at every refractive surface. In this 
case, it is assumed that the coordinates of P and p are given, as 
well as the refractive surfaces by their implicit functions F, — 
F(X, Y, Z). Each intersection point P, is situated on the 
corresponding refractive surface F: 
F(X,,Y,,Z,) 20 (10) 
At each refractive point, the law of refraction Equation (6) is 
applied. In order to trace the ray, it is necessary to find 6; and 
6; in terms of the incident ray, the normal vector to the surface 
and the refracted ray. For 6, it can be obtained from (Li, 
1995): 
cos0, - a4, * B,u, * Y,Vj (11) 
where (a; , f; yr) are the directional cosines of the ray from 
P, , to P, and (A, 4, v) are the elements of the normal vector 
of refractive surface F, at point P,. (@:, B: y) can be derived 
from: 
  
  
a, X,— X, 
5-4 x-x. 
y; Z,— Z, 
I 1 (12) 
and (A, 41 vi) can be expressed as: 
(s 
e 7 
À, -i a 
GE, arr a d | | (Cj 
=| (= pr {ns T(— — 
n zo; 5, d, | C 
Vi 1 : 1 i i 
e 
Z, if 
(13) 
Similarly, as in Equations (11) and (12), &' can be obtained 
from 
c080,'= a; Aia t Pia tYiaVia (14) 
where (a ;+1 , Bi+1, %i+1) are the directional cosines of the 
refracted ray or the incident ray referring to the next refractive 
surface. Using Equations (12-14), and given that the incident 
ray, the normal vector and the refracted ray on the same plane, 
it can be derived that 
C ju a; A; 
— -— 1 
nja| Bia | 7"; B, | - (n; cos0; — n; cos0,')| 4; 
Yin Vi V; 
(15) 
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B5. Vienna 1996