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Gonzalez 
International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B3. Istanbul 2004 
The total intensity of the image is given by Bo : 
We find that moments depend on the intensity or 
grey level. One important thing about moment 
features of objects is that they can be used 
regardless of location in the image and size of the 
object. Image moments include the following: 
centre of mass, variance and orientation. 
CENTRE OF MASS: 1f we regard the intensity 
or grey level B (x,y) at each point (x,y) of the 
given image (B) as the ‘’mass“ of (x,y), we can 
define the centroid (centre of mass) as well as the 
other moments of B. It is that point where all the 
mass of an object could be concentrated without 
changing the first moment of the object about any 
axis. The centre of mass is the centre of area of a 
figure, which in practice is chosen as the position 
of the figure. In the two dimensional case, the 
centre of mass is given by (B10, Bo1)- 
By, = >. 2 x B(x.y)/By Q) 
Bu = 2 2.9 B(x,)/Bw 
VARIANCE (c | : This is given by the second 
s C 
moment about the centre of mass (8 KL ) 
2 : : s 
O'. Characterises the extension or spread of the 
object in the x direction. 
From the above equations the position and 
variance of an object can be calculated using first 
and second moments only. These moments are 
invariant under linear co-ordinate transformations 
(Jain 1989 pp. 381). Thus we do not need the 
original image to obtain the first and second 
moments. Projections.of the image are sufficient. 
This is of great interest since the projections are 
more compact and suggests faster algorithms 
(Horn 1986 pp.54). Existing literature gives the 
accuracy of the simple moment centroiding 
algorithm to be between 0.05 to 0.1 pixel 
(Schildknecht 1994, Chubunichev pp.152-153). 
2.2 Point spread function (PSF) Fitting 
The basic principle in this technique is that all 
images of stars on a CCD frame have, baring 
distortion introduced by the camera optics; the 
same form but differ from one another in 
intensity or scaling ratio and position. (Teuber, 
1993). Thus fitting a suitable defined PSF to a 
series of images will give relative magnitudes 
m — zpt — 2.5 log(scaling ratio) (4) 
613 
where zpt is the magnitude assigned to the PSF. 
Mathematical curves are fitted to the real data until a 
good match is obtained. The parameters determined are 
the position and the scaling ratio. The mathematical 
equation used to model the profile of the stellar image is 
usually the Gaussian function (Buil, 1991). 
2 
I(r) =1(0)exp| — (5) 
where: 
r is the radius with respect to the centre of the 
star image 
oc 1s a parameter characterizing the stars spreading 
I(0) 1s the maximum intensity 
  
Linearising equation (10) we get 
In 7(r) 21n 1(0)+ br’ (6) 
The equation above is of the form Y = A + BX. The 
coefficients (A, B) can be estimated by linear regression 
from (X, Y) pairs. 
S X S Xy S XX S y 
d= 2 
S HS C 
Bs SxSy —nS yy 
Sy — nS yy 
and 
> 
I 
X 
Sy = SS DX; IIT mS X. Sy 
il i-l 
n 
SPEM XE. 
i=l 
Expressing r in terms of image co-ordinates gives 
2 
m2 = (x =x)" + (v= vo) (8) 
Replacing r in equation (11) by equation (13) we get. 
in 7(;)-tn (0) ds -x) (7) | © 
The above equation can be written as 
E, = GC, + EX, t OY; +ex/ +e,x} (0 
where E; = In I(r;) + In I(0), i is the pixel number and 
(x;,ÿ;) are pixel co-ordinates. The centroid, (Xo,ÿo), Of the 
image is computed after solving for the coefficients, c; 
(j=0...4) of equation (10) using the least squares 
criterion. Differentiating (10) and using the necessary 
conditions for extremum gives;