if the end points of a rectangle side lie in the nodes of a grid, 
where maximum value i equal 
Ins = ERU ? (2) 
p 
JA[ is the smallest integer not smaller than A (ceiling-function). 
Take advantage the formula (2) and geometrical constructions 
it is easy to show that 
ES D ^ (3) 
if 9 & [ 0, arctg J> and 
| — 
À p 
B o qm EN (4) 
^ 
if t | « 2 
arctg — = 
s 2 2 4 
When horizontal projection a, of the rectangle side a is less than 
corresponding projection L, of the sequence (1), the width of 
non-sensitivity zone is 
B* = cos © — jo Sin ® (jo = 0, 1,...,q -2) (5) 
"3 
for 0s o S —. 
4 
4. ESTIMATIONS OF THE LOCATION ERRORS 
Let all corner points of a rectangle lie in the nodes of a grid. 
Then we have (see Lebedev and Zolotoy, 2001) 
A-2425 (6) 
where f is calculated by (3) or (4). 
In the other cases we have (fora, = Ly) 
ApT AT A (7) 
where A, is minimal distance between the corner point of a 
rectangle and its black contour points. 
Using geometrical constructions we can show that is correct the 
next estimations for Ao: 
\2 (cos @ - B) < Ay <2(cos ¢ - p). (8) 
  
   
   
   
  
   
   
   
  
  
  
   
   
  
  
   
  
   
   
  
  
   
  
  
   
  
  
  
   
  
  
   
   
   
   
  
  
  
   
  
   
  
    
   
   
    
  
    
   
    
  
   
   
   
International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B1. Istanbul 2004 
Then according to (6) and (7) we have the next estimation for 
As from (8): 
V2 cos p < À < 2c05  — ( 2-2 ) p. (9) 
Note that the function A defined by the expression (6) and the 
functions A's 7 2cosQ — 2 — Ja Jp. As = 2 cosp defined 
by the inequalities (9) are symmetric functions relative to @, = 
(1/4)n in intervals [(1/2)(n — 1), (1/2)n] (n = 1,2,...). Besides, 
these functions are periodic with the minimal period in 7/2. 
If the width of non-sensitivity zone is determined by the 
equality (5), the estimation of the corresponding error is 
  
^ * 7 N2cos^ 9 - 2josin © cos Q ^ jo sin" 9 (10) 
where index jo corresponds the side of a rectangle that oriented 
under the angle o for its lower right and upper left angles, and 
oriented under the angle q + 7/2 for its lower left and upper 
right angles. 
Estimations (6), (9), and (10) are the same for the other figures 
with mutually perpendicular boundary segments. For acute- 
angled objects these estimations are lower estimations, and for 
the obtuse-angled objects they are upper estimations. Proper 
formulas can be derived in analogous manner. 
5. PLOTS OF THE ERRORS 
The functions A» = 2coso - [o -./2 Jf. Ay = J? cos 0, 
and A=+/2 B are plotted in figure 2 (the curves 1,2 and 3 
correspondingly). We can see that maximal error (curvel) 
exceeds minimal error (curve3) approximately in 32 times. But 
the probability that all corner points of a rectangle lie in the 
nodes of a grid is hardly possible. Thus, the worst case (curvel) 
or the middle case ( curve2) are more likely. We can also see 
that the expressions for the errors estimation can be used for the 
optimization of the images reading (aerospace pictures, 
drawings, schemes, etc.). In particular, the positional accuracy 
of the rectangles (and other figures with mutually perpendicular 
boundary segments) reading can be increased in the manner of 
certain orientation of the figures on a grid. For example, if the 
object was oriented along the axes OX, OY of coordinate 
system XOY combined with the scanner coordinate system, the 
error is maximum and equal À = H5. It can be decreased 
in 23 times if the figure angle orientation will be 35? (see 
curve 3 in fig. 2). For the other cases (see curves 1 and 2 in fig. 
2) optimal angle orientation of the objects is 45°. It permits us 
to reduce the errors to step of a grid. Correspondingly, 
maximum error for a centroid of a rectangle is in this case 
À = AS /2. It is in /2 times less than for the angle orientation 
@ = 0°. The scattering of the coordinates x,, y. of a centroid will 
be also less for ¢ = 45°. For example, if these coordinates 
satisfy a uniform distribution, the variance D = 1/24 for ¢ = 45 
° and D = 1/12 for ¢ = 0°. In all cases the positional accuracy is 
increased if we estimate the corner points position by both black 
and white contour points coordinates and put them as an 
average. 
   
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