In: Paparoditis N., Pierrot-Deseilligny M.. Mallet C.. Tournaire O. (Eds), 1APRS. Vol. XXXVIII. Part ЗА - Saint-Mandé, France. September 1-3. 2010 
Equation (6) indicates that V oscillates on a one-pixel cycle in 
side similarly to V x of the line segment with the length d and the 
local maxima of F in the one-pixel cycle are inversely 
proportional to the square of d. Moreover, it shows that F has 
the local minima 0 when S = 0 in the one-pixel cycle in the 
same way as V x of the line segment. Figure 3 shows the RMSE 
>Jv from d- 2 to d = 20. 
Figure 3. RMSE yfv of square by WCM 
The result of the analysis demonstrates that sampling in 
digitization causes the measurement accuracy by WCM to 
oscillate on a one-pixel cycle in side. 
3.2.2 BCM: The region division of (5, 1) according to the 
thresholding result of the pixel containing each vertex of the 
square is much complicated and it takes a large space to show 
the formulae of the measurement error (¿^., £,,) of the center 
location of the square by BCM using the threshold g T = 1/2. 
Furthermore, the variances (V x , F v ) and Fof (s r , s y ) are unable 
to be obtained analytically. Therefore, we decided to omit 
showing the formulae of (e x , s v ) in this paper. 
F were obtained by numerical integration and Figure 3 shows 
the RMSE yjv from d = 2 to d= 20. 
Figure 4. RMSE yfv of square by BCM 
The results by the numerical integration show that F oscillates 
on a one-pixel cycle in side similarly to F v of the line segment. 
However, those show that the local maxima and local minima 
of F in the one-pixel cycle increase as d increases in contrast to 
the fact that F v of the line segment is independent of the integer 
part of the length d. The increases of the local maxima and 
local minima of F with an increase in d could be explained in 
terms that the thresholding result of the pixel containing each 
vertex of the square would make the measurement accuracy 
higher and the influence of these four vertex pixels would 
decrease as d increases. 
If <5=0, that is to say, d is an integer, F can be obtained 
analytically and be expressed by Equation (7) The red broken 
line in Figure 4 shows Jv calculated by using Equation (7). 
V = 
+ 
-"“ + log„2 
|-41og,2 
J(d+ir 
i_L_ + l 
J(rf+i) 6 
(7) 
When d » 1, F becomes nearly independent of the integer part 
of d similarly to V x of the line segment with the length d and 
could be approximated to Equation (8). 
V = V +V =2 F 
]-3d{]-S) 
6 
(8) 
The results by the numerical integration show that sampling in 
digitization would cause the measurement accuracy by BCM to 
oscillate on a one-pixel cycle in side as well as by WCM. 
3.3 Circle 
It was assumed that a circle with the diameter d was placed as 
its center was located on (s + d/2. t + d/2) (0 < s < 1, 0 < t < 1). 
Table 4 shows the gray value g,y of the circle when d = 2 and 1 = 
0. The general expression of g, 7 is complicated with sine and 
inverse sine functions in the same way as Table 4 shows. 
Consequently, the measurement error (s x , £,,) by either WCM or 
BCM is expressed by the combination of sine and inverse sine 
functions, and the variances (V x , F,,) and Fof (s x , s v ) are unable 
to be obtained analytically. 
i 
gu a 
= 0,1) 
0 
(H<) 
-■^-sin2^, 
1 
—(#i +0 2 ) +—(sin26 , l +sin2#,) 
2 '4 
2 
1 
:h<) 
— ~sin 2 0 2 
0\ = sin '(5), 02 = sin '(1 - s) 
Table 4. Grey value g, 7 of circle when d= 2 and t - 0 
3.3.1 WCM: The variances (V x , V y ) and F of the 
measurement errors (s x , of the center location of the circle 
by WCM were obtained by numerical integration. Figure 5 
shows the RMSE Jv from d = 2 to d= 20. 
.* 0.015 
-0.010 
2 0.005 
* o.ooo|V 
-0.005 
By numerical integration 
By approximation 
Approximation error 
Diameter (pixel) 
Figure 5. RMSE yfÿ of circle by WCM