"ance. September 1-3, 2010 
ln: Paparoditis N., Pierrot-Deseilligny M.. Mallet C.. Tournaire O. (Eds). 1APRS. Vol. XXXVIII. Part ЗА - Saint-Mandé, France. September 1-3. 2010 
I is the difference of w, r 
?t and w tJ = 0 if gij < gl 
g T at the most ordinary 
"s of the center location 
tnd a square by BCM, 
cally, were obtained by 
sûrement errors of the 
; continuous versus the 
measurement errors by 
rule. On the contrary, 
not continuous versus 
those variances by the 
îe side of a square, the 
128 pixel intervals from 
n. 
racy 
stimation of the center 
J a circle by WCM and 
or numerically, proved 
both WCM and BCM 
mre without bias. 
:er location of a figure 
ind V (= V x + V,) of ( s x , 
square and a circle are 
tean squares of errors 
l line segment and y/v 
circle in the figures as 
:USSION 
the length d was placed 
))(0<s< 1). 
.hown in Table 1. Here 
return the integer and 
У- 
Si 
(1 -s) 
_J 
a(,v + d) 
îe segment 
the measurement error 
nent by WCM derived 
int(s + £/) i 
+ 2 ( 2 ) 
The variance V x of e x can be obtained analytically and be 
expressed by Equation (3). 
V .IflilzfL (3) 
' 12 d’- 
where <5 is the fractional part of d, that is to say, S= fra(d). 
Equation (3) indicates that V x oscillates on a one-pixel cycle in 
length and the local maxima of V x in the one-pixel cycle are 
inversely proportional to the square of d. Moreover it shows 
that V x has the local minima 0 in the one-pixel cycle when S= 0. 
Figure 1 shows the RMSE ^Jy’ from d- 2 to c/= 20. 
Figure 1. RMSE ^Jy~ of line segment 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 length. 
3.1.2 BCM: Table 2 shows the measurement error s x of the 
center location of the line segment by BCM using the threshold 
gj = 1/2 derived from g, shown in Table 1. Here S is the 
fractional part of d. 
£x 
0 < s < - 
2 
— <S< 1 
2 
0 < s < 
:h 
-( 
-Э 
2 1 2 y 
<5<- 
2 
1 
2 
H 
1 
2 <?< 
(H 
>- 
H) 
-I 
Hi 
(K 
<5<1 
3 
2 
( S' 
s+— 
{ 2, 
S= fra(cf) 
Table 2. Measurement error e x of line segment by WCM 
Length (pixel) 
Figure 2. RMSE of line segment by BCM 
The result of the analysis demonstrates that sampling in 
digitization causes the measurement accuracy by BCM to 
oscillate on a one-pixel cycle in length as well as by WCM. 
3.2 Square 
It was assumed that a square with the side d was placed as each 
side was parallel to the x-axis ory-axis with its center on (,v + 
d/2, i + d/2) (0<j<1,0</< 1). 
The gray value gij of the square is shown in Table 3. Here 
int(.v) and fra(x) are the functions to return the integer and 
fractional parts of the value x respectively. 
Su 
i = 0 
1 <i<(S- 1) 
i = S 
7 = 0 
(i-o 
■s'O - 0 
1 
к 
VI 
VI 
U-S) 
1 
s' 
7= T 
(1 -5) t' 
i' 
s't' 
S = int(i + d),T - int(/ + d), s' = fra(.v + d), t' = fra(t + d) 
Table 3. Grey value giJ of square 
3.2.1 WCM: Equation (5) expresses the measurement error 
(s x , £ v ) of the center location of the square by WCM derived 
from gy shown in Table 3. 
{2(s + i/)-int(s + £/)-ljint(.v + £/) 1 
2d + 2 
{2(/ + <7)-int(H-c/)-l}int(t + d) 1 
2d + 2 
(5) 
The variances (V x , V x ) and V of (e x , can be obtained 
analytically and be expressed by Equation (6). 
The variance V x of s x can be obtained analytically and be 
expressed by Equation (4). 
V. = 
1 -3^(1 -¿>) 
12 
(4) 
1 S 2 (l-S) 2 
12 d 2 
V = V x + V y =2V x 
\s 2 {\-s) 2 
6 d 2 
Equation (4) indicates that V x oscillates on a one-pixel cycle in 
length and is independent of the integer part of d. Figure 2 
shows the RMSE from d= 2 to d= 5. 
where Ô is the fractional part of d. 
(6) 
33