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International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B3. Istanbul 2004 
  
Correspondence between set of features at the left and right 
images is now established in the 7-dimensional parameter 
space. Features are compared by distance: 
  
S n H2 06:395) 2806597 (3) 
: k=0 max(/; (x; V7)» i? (x; mn ) 
where iH (1.9) 7 value of k-th invariant in x;, y; of the left 
image 
2 2 sth inUsti: e x 0 af fho 
Il (xxt value of k-th invariant in Xp Yj of the 
right image 
Let i=1,N, and j21,Ng , where N,, Ng are the numbers of 
candidates on the left and right images respectively. Point j is 
considered as conjugate for i, if 
S, » min(S,,), ket Nij (4) 
At given stage all conjugate points are tied in N pairs using 
criteria (4), N=min(N,, Ng). 
2.4 Features examination 
In order to verify that conjugate pairs of point features was tied 
properly, the additional information about relative coordinates 
of points positions was used. In short, space distributions of 
features at left and right images should be similar, the 
distribution itself can be described as a set of distances. 
Consider the set of points Aj, A,,..., Aj..., Ay in the plane 
image, Figure 6. 
Figure 6. Point features distribution 
Distances between points can be arranged in the form of NxN 
matrix ||A;|| as follows: 
  
A 4. 4 — A 
4 1 On, — M hs 
A, 0 . n By (5) 
A, 0 . f. 
Ay 0 
  
     
where hrs = Euclidian distance 
   
(xe Que Y 
between points A; and A, 
Xi, y; ^ image coordinates of point A; 
Xk, Yk = image coordinates of point Ay 
To verify N pairs of conjugate points, matrix llAiHl for the left 
image and AG for the right image should be compared. For 
qualitative estimation of erroneous tying variable &; is 
introduced 
Si ri^ -T i (6) 
Analysis of the histogram of variable 6j; enables to estimate the 
threshold A to reject features according to the criteria stated 
below. Note that point with number (i) has N-1 connections 
with others, appropriate distances in matrix ||A;| are: ri; r»;,..., 
Iii, l'iis1,..., Tin. Accordingly, the set of differences associated 
with conjugate pair with number (1) is 
  
877 (81, 05... 85, 1,4... Bin} (7) 
with [8l 7 minii $a Bus Bienes Bind: = norm of 
vector à; 
Pair of conjugate points is accepted if ||Bj|€ A, otherwise it is 
rejected. Verification procedure is performed for every i from | 
to N. Essential, that verification criterion based on analysis of 
matrix (5) is invariant to rotation of images. 
To make the algorithm more effective, the pyramids of images 
were used. Initial approximation for sets of points is found at 
the highest pyramid level and then defined more exactly at next 
levels using cross-correlation. The example of performance of 
the algorithms above for video frames is presented in Figure 7 
  
Figure 7. Accepted conjugate points, video camera frames 
2.5 Invariance to rotation 
At current stage it’s worth trying to optimize the dimension and 
composition of parameter space, keeping in mind that features 
should be invariant to images rotation. Really the use of 7 
invariants is more reliable, but requires considerable 
computational time. The results presented in Table 8 enable to 
conclude that optimal number of invariants, taken into 
consideration is 4. The worst percentage of success is taken 
place near a=45", as it could be expected because the discrete 
resampling error is maximum at this angle.