In: Paparoditis N., Pierrot-Deseilligny M.. Mallet C.. Tournaire O. (Eds). IAPRS. Vol. XXXVI11. Part ЗА - Saint-Mandé, France. September 1-3. 2010 
The distributions of £ of the circle shown in Figure 7 are more 
complicated than those of £ of the square shown in Figure 8. 
Furthermore, although the difference of d between two 
distributions of e of the circle shown in Figure 7 (a) and (b) is 
merely 13/128 pixel, two distributions would have a great 
difference in contrast to two distributions of £ of the square 
shown in Figure 8 (a) and (b). We concluded that the wide 
variation of the distribution of (£ x , £ v ) according to d would 
bring V having small irregularities of in the one-pixel cycle of d. 
It is assumed that V can be expressed in the similar formula to 
Equation (4) with the denominator of d. The approximation 
formula obtained by the least squares method using 2305 sets of 
(d, V) at 1/128 pixel intervals from 2 to 20 pixels in d is as 
follows: 
0.15697-0.3706^(1-^) n 
where <% /4 is the fractional part of (S+ 3/4), that is to say, ¿>3 4 = 
fra( 5 + 3/4). 
The blue, red. and green lines in Figure 6 show fy by the 
numerical integration, by the approximation, and the 
approximation errors (_^T -Jy\ respectively. We studied 
the validity of the approximation formula (11) at 1/128 pixel 
intervals from 2 to 100 pixels in d. The maximum absolute 
approximation error was 0.03312 pixel ( fy = 0.24281, 
fy = 0.20969) at d = 2 + 53/128. An absolute 
V App. 
approximation error by Equation (11) is less than 0.025 pixel 
when d > 5 and less than 0.018 pixel when d > 10. The 
accuracy of the approximation (11) of V by BCM is obviously 
lower than that of the approximation (10) of V by WCM. The 
low accuracy of the approximation (11) would be caused by the 
wide variation of the distribution of (e x , £ v ) according to d as 
mentioned previously. Nevertheless, we recognized that the 
approximation formula (11) is effective for ordinary use. 
The results by the numerical integration show that sampling in 
digitization should cause the measurement accuracy by BCM to 
oscillate on a one-pixel cycle in diameter as well as by WCM. 
Moreover, we suggested that the variance of measurement 
errors by WCM is expressed by the combination of the inverse 
proportion to the cube of the diameter and the oscillation on a 
one-pixel cycle in diameter. On the other hand, the variance of 
measurement errors by BCM should approximate to the 
combination of the inverse proportion to the diameter and the 
oscillation on a one-pixel cycle in diameter. 
We are planning to investigate the effect of quantization in 
creating a digital image on the measurement accuracy of the 
center location of a circle by both WCM and BCM. 
Furthermore, the influence of image noises on the measurement 
accuracy will be studied. 
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4. CONCLUSIONS 
Although general expressions representing the measurement 
accuracy of the center location of a circle by the two centroid 
methods WCM and BCM are unable to be obtained analytically, 
we succeeded in obtaining the variances of measurement errors 
by numerical integration and the effective approximation 
formulae of those. Additionally we conducted the analyses on 
the measurement accuracy of the center location of a line 
segment and a square by WCM and BCM. 
From the results of these analyses, we concluded that sampling 
in creating a digital image would cause the measurement 
accuracy of the center location of a circle by both WCM and 
BCM to oscillate on a one-pixel cycle in diameter. 
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