the intersection of four pixel. It is easy to compute the 
gradient k' of line with Roberts operator 
2k-1 k>1 
| k/(2-Ikl) -1<k<1 (15) 
2k+1 k<-1. 
If the edge does not pass the intersection, the error is 
much larger. The error exists still, with Sobel operator 
k O<k<1/3 
k'= { 7k2+6k-1 1/3<k<1. (16) 
-9k222k-1 
That is said, when the line direction is replaced by the 
gradient direction, there is the model error that can't be 
neglected. So the methods using gradient direction, such 
as Hough Transfomation, can not obtain high accuracy . 
2.2 The mathematical model of the new method 
The corner is the intersection of two straight line. If two 
edge line forming corner are accurately determined, the 
corner coordinate can be obtained by solved the cross 
point. It is well known that the intensily curve of an ideal 
edge is a knife-edge curve: 
x 
(x) ^ sx) dx (17) 
where s(x) is the line-spread function. 
So the gradient of image : 
Ag(x) m. (x) dx = s( 18 
X ee sQOdx-s(x 
g ni s(x) (18) 
dx X 
Considering the different of the intensity of knife-edge 
curve, a conclusion may be obtained: the gradient of an 
ideal edge's image is proportion to the line-spread 
function. An ideal line-spread function is Gauss function: 
1 
s(x, y)-— —- exp[-k(x cos 0 + y sin 6 - p] (19) 
T. 
So the gradient of the image can be expressed: 
A g(x,y)=a exp[-k(x cos 0 y sin 6 -py?] (20) 
This is the adjustment's function model. Regarding the 
magnitude of gradient as observed value, we can obtain 
an error equation 
v(x.y)-Coda * C1dk +Cpdp+C 3d0+C4 (21) 
where 
Cy=exp[-kp (x cos 6p + y sin 6 -pp)?] 
C1=-ap Cp (x cos 6p + y sin 6p -pg)? 
C2=2ap kg Cp (x cos 69 + y sin 6p -pp) 
C3=C2 (x sin 6p - y cos 69) 
Cé=ap€i - guy) 
ag, ky, py and 6j are the initial parameters approximations. 
If Roberts gradient is used, so 
80 
  
  
À g(i.j)=sqri[(gi+1j+1-81,j)*Hgi+1,j-8ij+1061 (22) 
dA g= cos 8 dgij + sin B dei+1 j 
sin B dgi je. + cos 8 dgi+1,j+1 (23) 
Where £ is the gradient angle 
If noise variance is m? 
mA, 2—cos? B m?--sin? 8 m2--sin? 8 m?4cos? 8 m? 
som? (24) 
It shows that the weights of observed values are equal. 
After the error equation is normalized and the normal 
equation is solved iteratively, straight line parameters 
(p.0) can be accurately obtained. 
2.3 Initial value 
The parameters py and 6g can be obtained by using 
Hough Transformation. Because parameter a is the 
maximal gradient, thus 
ag-max[Ag(x.y)] (25) 
and 
In À g(xy,yg) - In ag 
kg- (26) 
(x cos 0 g* y sin 0p -pp)? 
  
where (xy. yy) is a point near the line. 
2.4 Gross error 
In order to reject the gross error, iterative process of 
weight functions is used. So the gross error can be 
automatically got away. In our study, wcight function is : 
1 00%<op? or op?/vi?»1 
wij={ (27) 
og//vi? otherwise 
2.5 The window of accurate location 
In order to make full use of line message and get away 
other message, the criterion of window selection is that 
the window is longer along the line and it is not wider 
along the normal direction of the line. The points near the 
corner should be also rejected. Otherwise, they will 
influence the accuracy of location because of the 
interference of two line each other (See Fig 1). 
  
  
  
  
  
2.6 Ca 
Alter 
coordi 
XC 
yc 
where 
straigh 
3.1 In! 
Stand: 
where 
Invers 
covari 
siright 
The c 
(p1, 01 
The d 
where 
Fx 
F 
from 
So in