If more than one contour pixel appears in any direction then the 
corresponding radius takes as distance the nearest contour pixel 
to the central pixel. If no contour pixel is found in one particular 
direction a value equal to 7 is assigned to the corresponding 
radius (as radius 4 in figure 1(b)). 
With the aim of providing a low cost real time tracking system 
for industrial applications a specific image processing board for 
an industrial PC has been implemented [Aranda,96]. This 
specific processor supplies to the host the polar descriptions of 
the local features included into the tracking windows. Two 
boards (one for each camera) are required in order to perform 
stereo tracking of the selected local features. 
The host uses this set of polar descriptors to recognize and to 
locate the tracked targets while tracking them in an image 
sequence and also to perform stereo matching. Recognition is 
performed by looking for the minimum of the next distance 
function: 
E= 
I 
3 
(Pur (k) "if (k)) 
k=0 
Where py( 6) is the polar description of the tracked local feature 
which acts as a model, and p;(6) is the polar description 
associated to every i local feature in the tracking window. 
Once the target included in a tracking window is recognized the 
host relocates its corresponding tracking window to a new 
searching position in the next image frame. This process is 
repeated for every target in every tracking window for both left 
and right images. Image processing performed by the specific 
processors and target recognition computed by the host by 
software, are overlapped in time obtaining a total computation 
time of 20 ms (video rate). 
As polar transform has been reduced to only eight radial 
samples, the distance function F, used in tracking and stereo 
matching of the local features, is highly affected by localization 
error included in the radial measures. In next sections the 
expected error on this distance function will be presented. 
2. DISCRETIZATION ERROR 
Let be Ax i Ay the resolution errors due to the sampling in the 
two coordinates of the image; they define the pixel dimensions. 
Squared pixels are achieved by adjusting sampling period on the 
image processing boards, so is assumed that Ax = Ay. In the 
following only dimension X is considered. Results are the same 
for dimension Y. 
The exact position of a certain image point (i.e. a contour pixel) 
along dimension X, namely x, is measured by a discrete value 
xn» Which may be different from x, since a discretization error is 
produced. In fact, given x,, (measured position) for a certain 
image point, it is known that the real position (x) satisfies: x € 
[ x, - 2/2, x, t AX2 
The discretization error (£x) is defined as the difference between 
the real and the measured position: £x 2 x - x,, and its value is 
contained in the interval £x € [- Ax/2, - Av2 ]. Therefore, ex 
is upper-bounded by £x, = Ax/2. 
However, £x,4, is not a good measure for the localization error 
made in the image acquisition, since it is very infrequent to 
make so big errors. It is preferable to characterize the 
672 
discretization error by means of its expectation (or mean value) 
and its standard deviation. 
Since there is not any a priori information for the distribution of 
the real position (x) in the interval [ x, - Ax/2, x,, - Ax/2 ], in 
this paper x will be considered as a random variable with a 
uniform distribution between the limits of the pixel (see figure 
2). 
1/Ax 
  
  
  
  
  
A 
v 
X m-AX/A2 Xm X mt AX/2 
x — U [Xm - Ax/2, Xm + Ax/2 ] 
Figure 2. Uniform distribution for the real position x 
Using this uniform distribution, the probability that x is situated 
in an interval of longitude dx around xm is dx/Ax, if dx € [ x,, - 
AX/2, X, -- Ax/2 ], and 0 otherwise. The expectation of the 
discretization error £x — x - x, is then calculated as: 
1 pontAx/2 1 Ax/2 
u= EJEx] = A NACE OI: = hd =0 
The variance of ex is given by: 
Ex] = E[Ex? ]- E[£x] 
se]- [7 
+ 2 
ny good 
1 
Xi) OX = — 
(x — Am) .dx | 12 
—Ax/ 
The resolution error Ax is / pixel. So the standard deviation of 
the discretization error is, finally: 
O (Ex ) = Vo (ex) = Ax /VI2 = 0,288 pixels. 
It can be seen that this value is much smaller as the maximum 
which was previously calculated (about the half of it): 
EXmax = 472 = 0,5 pixels. 
3. ERROR ON POLAR TRANSFORM 
In this section the effect of the discretization error on the 
longitude of the radii of the polar transformation of the image is 
analyzed. Two possibilities will be distinguished: when the radii 
are on the image coordinate axes directions (horizontal and 
vertical radii) and diagonal cases. Again, horizontal and vertical 
radii need similar treatment, since squared pixels are considered 
(Ax = Ay); only the horizontal case is detailed here. 
For horizontal radii, their longitude (7) is given by the relative 
position of the contour pixel with respect to the central pixel of 
the image transformation window. Then r = x -xc, being x the 
exact position of the contour on dimension X and xc the exact 
position of the transformation window center. 
However, it is only possible to measure a discrete longitude of 
the radii, which is given by the expression r,, = x, -xc , where 
X 18 the discrete position of the image point on the X axis. 
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