MATCHING 
Epipolar Line Conditions: 
   
search 
range 
9 textured 
surface 
  
  
  
" A ene eplpolar 
*line* 
  
  
  
  
  
  
image #1 
  
  
  
Figure 3 - Epipolar Matching 
From projective geometry for the "target" image, the 
following relationship exists: 
ne te ru (X-X9*ruy(Y-Y)*ra4(2-2,) 2 
2 Py (X=X try, (Y-Y +r (2-2) 
  
m (X-X +7 (Y-Y)+r,4(Z-Z.) 
75 (XX Yr (YY + (Z-Z) 
  
y * y, dye 
where x, y are image co-ordinates of the point of interest in 
the "target" image; x,,y, and c are the principal point position 
and principal distance of the camera, as calculated in pre- 
calibration; dx, dy are the lens distortion parameters as 
defined in (4) and (5) below; X, Y, Z are the object space co- 
ordinates of the point of interest; X,, Y, Z, are the 
perspective centre co-ordinates of the "target" image and 
r,;..T;; are the rotation matrix elements derived from the 
camera orientation angles. 
dx=xS, *ya +x(Kır 2 «+R; 4 +K,r 6)+ 
-— 4 
+P (r2+2x)+2P xy @ 
dy=xa+y(K,r +E ,r*+K r)+2P xy +P (r*+2y") (5) 
where S, is the scale factor in the x-direction and a is a 
shearing factor allowing for affine deformations. K, ,K, K, are 
radial lens distortion parameters and P,P, are decentring 
distortion parameters, 
= x-x 
P 
y-y (6) 
= 
r = x ty 
Pre-selecting Z values at regular intervals within the 
constraint of the epipolar line makes it possible to evaluate a 
= | XI 
I 
732 
series of X and Y object space co-ordinates of potential 
candidates, corresponding to the points of interest. The point 
interval is chosen to a step of 1 pixel in image space. The 
search range along the epipolar line can be limited with prior 
estimation of the object depth. 
The X/Y co-ordinates are evaluated from the collinearity 
equations (2) and (3) in the form: 
  
  
  
  
AXr, -cr, AXr 768 X RHS, = 
AYr er, AYrh ctt Y RHS, 
hence 
X AXr,76"y AXr o lia ! | RHS, 
= (8) 
Y A Yr, 673 A Yr, 760732 RHS, 
where 
AX = x-x +dx 
, (9) 
AY - y-y,tdy 
and 
RHS,- -(AXr,, 7er )Z *(AXr,, oP x, * 
c 
0 
*(AXr 760431, (AXr,s 0132, (10) 
RHS,=-(4A Yr,,-cr„)Z+(AYr, 7er, )X,* 
*(AYr, 7670, * (AYr,, 7er 2, an 
In the next step, the image co-ordinates of the candidates in 
the conjugate "search" images are calculated using the 
collinearity equations (2) and (3). 
A window surrounding the image position of each of the 
candidates along the epipolar line is used to determine the 
correlation (Ryy) between the "target" image patch and the 
respective "search" image patches. 
  
2. TL £8068) 
XY = = 
iz (g,-£ yx (g,-g 
where g, and g, represent the grey-scale values on the "target" 
and "search" windows respectively and g, and g the average 
grey-scale values for the "target" and "search" windows 
respectively. 
(12) 
This process is executed for the full extent of the search depth 
range and the maximum correlation, i.e. the value closest to 
1, is chosen as corresponding to the most likely match for the 
point of interest in the search images. This position is then 
refined in a least squares, grey-scale area-based matching 
algorithm with imposed geometric constraints. 
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B4. Vienna 1996