2.1 The relationship between original and rectified images 
Figure 1 illustrates the geometric configuration of distortion 
rectification. P/anel' is an original image (called distorted 
image) and planel is a rectified image (called a normal image). 
§; denotes the first projective centers. B; is the length of the 
baseline. The point p,' in the original image and the point. p, 
in the normal image are the projections of the point P in object 
space. 
  
    
   
G 4 8p 8 P 
Fig. 1 Geometry of image sequence rectification. 
Plane] 
P > 
Fig. 2 The relationship between the original and the normal 
images. 
As illustrated in Figure 1, the coordinate relation for the point 
D' from the image plane coordinate system to the camera 
coordinate system can be expressed by 
U m, 
p^ (1) 
Vo 1=| Mo) my My -R Y, 
Ho m. X 
Wy Hh Nyy Wins m 
where ",»Y»VWp» are coordinates of the point P'j in the 
camera coordinate system, (,j-123 j=123) are 
i, , $5 : 4) 
components of the rotate matrix R, for the first image, f 
denotes the image sensor constant, x , y , denote the 
ph? ^ p^ 
coordinates of the point, P'; in the image plane, plane! * 
Figure 2 illustrates the relationship between the original 
(distorted) image and the normal (rectified) image for the point 
P. The relation can be expressed by 
deu, EB ap Tune TIE niu 
Zw myx, +m,y, — " 
) zw al pa pn Ta) à 
Pi Pu Zw : my xX, +msyy, —my,f 
where “Vy denote the coordinates of the point P, in the 
normal image plane, planel. Similarly, we can get an n-th 
equation describing n-th rectified and original images. 
  
  
(au cH ae (4a) 
Pr pa Z,. S P5, t5, 2 o. zr 
2-0 SE Wnt Ward Ged) 
Ps Pins rox vr f 
Zur, 31^ p, 327p, 33. 
Because the orientation parameters of the camera are not 
known, we cannot solve Equations 2, 3 and 4 directly. In other 
words, we cannot determine the relationship between the 
original image and the normal image directly. However, using 
the EPI analysis technique, all conjugate epipolar lines in the 
image sequences should lie in an identical epipolar plane. Thus, 
we can build a relative relationship among the normal image 
sequences (if the original image sequences have been rectified). 
First the v coordinates of all conjugate points in the normal 
image sequences are constrained to be equal (see Figure 5 in 
Zhou et al., 1999). Thus, we can have 
Bax, +Myy, Mal = IX, ry, TÉ (5) 
mx, +M2Y 5, = FaXy ty, a d 
Equation 5 means that all rectified conjugate points in image 
sequences lie in an identical plane, and all epipolar lines are co- 
planar. Thus, the trajectories in the "new" EPIs (called rectified 
EPI) generated by the rectified image sequences can meet the 
EPI analysis technique's requirement, i.e. all trajectories in the 
EPIs are straight lines when the camera moves along a 
constrained straight-line path and viewing direction is 
orthogonal to the flying direction (Zhou et al., 1999). In order 
to simplify the derivation, we chose the first equation to discuss 
the computing process. 
MyX, +My, Ebo. MAX yy y fast (6) 
  
myx, +My,y,. zs f aX, tüÜysy, nf 
We rewrite Equation 6 
hx, x, +Lx, Yr, Pe thy, Xy Thy 7, (7) 
cfe fiy, frs, tf 0 
where: ; = Maya Ha d,-mHn, mah: d m,ng mn 
1 2191 WA 2 21732 31 22 3 21433 317725 
LH na Uu; Hong Hun [mola Hn 
I Haong HH demam Ha. dy HH, His. (8) 
Observing Equation 7, it is not difficult to find that it is a 
defective equation. Hattori and Myint (1995) and Niini (1990) 
presented a method for solving this type of equation. We may 
rewrite Equation 7 into the form 
ql, xy, 8b y, Lf +L, X41, 3, HL, +L, + (9) 
where: L, =41(f 4), L=LIf-k), L, zl L, zl), 
LB), E. zB) Le L=(f-L)k, q7 yy, 7 Xy 
Equation 9 describes the relative relationship of neighbor two 
images because the left side of Equation 9, q describes the y 
parallel of two images. If the q is equal to zero, the relative 
orientation is determined. Thereby, we can solve the parameters 
of I ~[, from Equation 9 using least equation estimation 
(LSM) when more than 8 conjugate points in the left and right 
images are measured. 
Even though the parameters implicitly express the relationship 
between the original image and normal image, we still cannot 
rectify the original image employing only implicit parameters 
L, ~ L, because they are function of the six rotation angles in 
two rotation matrixes R and R . We developed the following 
algorithm for direct rectification: 
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