The expected value for &fis, as usual: 
E[ef]= E[ erl - er2 ]= E[ er1l J-E[ &r2]=0-0= 0 
and the variance for the error associated to differences between 
diagonal radii is: 
O(&f)=0"(erl-er2)=2.0"(er)=4.0*(ex)=4.1/12=0,33 pixels 
so the standard deviation is: 
o(&f)- No'(ef) = 0,577 pixels 
5. ERROR ON LOCAL FEATURES RECOGNITION 
In this section the effect of the discretization error on the 
recognition of the local features of the image is quantified. The 
recognition process is used by the proposed system in order to 
locate the tracked targets while tracking them in an image 
sequence and also to perform stereo matching. 
The distance function used to compare two polar transforms has 
an exact value (F) and a calculated value from given data (F,,): 
F - Y (10) - 7200) = Y, f*(6) 
98-0 8-0 
7 7 4 
F, 2 Y (01,0) -72,(00 - Y £20) (4) 
60-0 
8-0 
Where r/(0) and r2(0) are the exact values of each radii of the 
polar transformations to be compared, and r1,, (0) , r2,, (6) are 
the discrete values associated to them. 
Then, eF = F -F,, . This error depends on the accumulated error 
from the radii comparison due to the discretization error. 
For each element of the summation of expression (4), there can 
be defined another error term: 
ER =f fr = (Int fu” = 2.fm E (+ ed) 
From this expression, the last term can be disregarded. Then, for 
horizontals and verticals radii (0—0,2,4,6 ): 
IER 7 2. f, MAX (8) = 2. f, 1 = 2.f, pixels” 
E[cR ] z 2 . f, Elef]= 0 
XER)= (2... o (ef) (2/3). f, 0,816.f,, pixels". 
for diagonal radii (0=1,3,5,7 ): 
leR, =2 . fi. max(ef) = 2. f,,. V2 = 2V2.f,, pixels® 
E[eR]=2.f,.E[e¢f]= 0 
O(€R) =V(2. £,,)>.5( &f )=V(4/3). f,, = 1,154.f,, pixels’ 
For any radius direction, the standard &R is proportional to f,,. 
Therefore, the relative error £R/f,? decreases when fn increases. 
It has to be considered f, is only defined between values 0 and 
7. 
The total error of function F is: 
7 7 7 7 
£F-F-F,-Y 06) - Y 5,00 Y [P^ 0)- (:0)]- YER) 
0-0 0-0 0-0 0-0 
& _Slar ( EN VIC sou fe even 
E[er]-3 E[er,]-0 
o°|er|- > 0? |er, | Y»/20) 
8-0 
0-0 
= 
Il 
2/3 if 0 even 
4/3 if 8 odd 
674 
  
Q lr]-|£0 les] so wy = fo ws 
8-0 80-0 4/3 if 0 odd 
From this results, it can be observed than expected value for 
error in local feature recognition due to discretization of image 
coordinates is equal to 0. Standard deviation of this error grows 
up in the same proportion as the square root of the measured 
distance : o( €F ) = VF, . In fact, as a first approach, the 
standard deviation of this error can be upper-bounded by the 
expression: O(€F ) < 1,15.VF,, . 
That is, the greater is the real similitude between two polar 
transformations (small F,), the smaller is the error made in the 
measurement of this similitude. 
6. ERROR ON 3D DISTANCE ESTIMATION 
In this section the effect of image discretization on measuring 
the disparity between corresponding local features from both 
left and right cameras is analyzed. Disparity determines the z 
coordinate of the tracked local features. 
The 3D distance (or depth) is measured from disparity by means 
of the following expression: 
Z=AB/(xe-xd) Z,=AB/(xe,-xd,) &Z- 22 
where À is the focal distance (equal to both cameras) and B is 
the camera separation (baseline). Once cameras are calibrated, 
these quantities act as constants. Variables xe and xd are the x 
coordinates of a certain local feature of the scene in the left and 
right image respectively. 
The disparity (g) is determined by the difference between these 
values: g = xe —xd so Z= A.B/g (with g>0 since xe>xd). 
In fact, what is known is : g,, = xe,, -xd,, and Zn = A.B/g,,. 
In this case, g, can be null (equal to 0) owing to the 
discretization error. If so, the distance cannot be determined. 
Singular cases will not be treated here. 
The error associated to the disparity is already known, since it is 
the same as the error on the difference of horizontal radii 
calculated in section 4: 
€g— 8 - 8m=(Xxe -xd) - (xe,, -xd,,)=€xe - exd 
lEgmaxl = lExemaxl + \ExAmasl = AX/2 + A/2 = Ax = 1 pixel 
E[eg | = E[exe - exd] = 0 pixels 
o( eg) =1/V6 = 0,408 pixels. 
using that, the error made when calculating Z is: 
&=2-Z,= (AB/g)-(AB/g,) 
AB AB  AB& AB €g , €g 
ER &, EG t&)e. ^. gg. ^R, 
  
  
  
which depends on the Z itself. So, it is more interesting the 
relative error on Z, which corresponds to the following 
expression: &Z/Z - -€g/g, 
It is possible to quantify this relative error, using the measured 
disparity (g,, ) and its associated error (gg): 
12/2 mix! = 168 / £m 7 178p, 
E[£Z/Z] 2 -(1/g, ).E[eg ] = 0 
o(eZ/Z) = (1/25 ). O(€g) = 0,408/8,, 
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