2 approximate rotation angles from the equations of each 
harmonic. Let 0, and 0, denote the rotation angles solved 
when k = 1 and k = 2 respectively, then At can be deter- 
mined by using the following algorithm: 
if |0, — 02| close to0, At=0; 
if |9, — 02| close tor, At =. 
5. EXPERIMENTS 
5.1 Matching Closed Lines 
Fig. 3 shows a digitized, closed line (A). The line 
B is a candidate to match line A. The starting points are 
indicated by solid circles. In order to check the computed 
transformation parameters we copied line A to line B by 
the following transformation: 
translation Az = 150 
Ay = 100 
scale S 0.6 
rotation 0  — 45? 
After the transformation, we shifted the starting point to 
the 10th node and added Gaussian noise (u. 0,0 z 3). 
  
  
  
  
Fig. 3. Example of matching closed lines. 
According to the matching process described in sec- 
tion 4, line B is transformed to fit line A. Through the 
matching process, the mean-square error (intrinsic mea- 
sure) as well as the transformation parameters and phase 
shift (extrinsic measure) are calculated. The mean-square 
error is expected to be about 3, and the calculated trans- 
formation parameters should be the inverse-transformation 
parameters used to copy line B. 
The following results are obtained: 
Intrinsic measure -mean-square error — 3.44 
Extrinsic measure-translation Az = —149.3 
Ay =-99.9 
scale 5 = 1.65 
rotation 0 — —44.9? 
phase shift At = —86.4° 
474 
The intrinsic measure is close to the number we expected. 
Line B’ in Fig. 3 is the transformed line B using the esti- 
mated transformation parameters. Note that the area be- 
tween line A and B’ is minimized by the proposed matching 
process. 
5.2 Matching Open Lines 
Here, we repeat the procedure for open lines. Line D 
in Fig. 4 is a copy of line C, obtained with the following 
transformation: 
translation Az = —150 
Ay — —100 
scale S 05 
rotation 0 = 180° 
Before the transformation we added Gaussian noise (n = 
0,7 —3) to C. 
  
  
  
  
Fig. 4. Example of matching open lines. 
We performed two experiments. First, the starting 
points of line C and D are at the same end. Second, we 
changed the starting point of line D to the other end. In 
both cases we obtained the same results. Line D’ shown in 
Fig. 4 is the transformed version of line D. The matching 
results are listed as follows 
Intrinsic measure - mean-square error = 2.6 
Extrinsic measure- translation Az = 150.5 
Ay -— 99.8 
scale S = 1.976 
rotation 0 =—179.4° 
They correspondent with the transformation applied to gen- 
erate line D. 
CONCLUSION 
The mean-square error obtained in our matching ap- 
proach is an ideal intrinsic measure to the goodness of 
the match. This quantity is obviously more transparent 
than the use of a table of deviation on shape invariants 
[Granlund,1972; Lin and Hwang, 1989]. 
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