The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences. Vol. XXXVII. Part B7. Beijing 2008 
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where K H (x) = \H[' 12 K(H~ l,2 x) (5) 
Considering the complexity of the estimation, the bandwidth 
matrix H in (5) is usually chosen as proportional to the 
identity matrix H = h 2 1 , and then only one bandwidth h 
must be provided. The kernel density estimator can be 
expressed as (6). 
kernel and the center if the kernel x, noted with Wl h G , called 
the mean shift vector. 
From (10), (9) becomes 
Kernel K(x) is a class of bounded functions satisfying some 
conditions(Comaniciu and Meer, 2002; M.P.Wand and 
M.C.Jones, 1995) and a special function k(x) called profile 
function is defined satisfying 
m k,a( x ) 
u> c ?m 
2 /«w 
(11) 
K(x) = C k k(H) 
Where C k is normalization positive constant which makes K(x) 
integrate to 1. 
When a constant bandwidth h in formulae (6) is extended to a 
bandwidth function h(Xj) (hi for short) associated with the 
sample point x h an adaptive sample point estimator is got (8). 
/w=z:, 
rc/z(x.) c 
K 
^ X, - X ^ 
7 7 
(8) 
For profile function k(x) , if k\x) exists for all X G [0,oo) , 
define g(x) = —k'(x) , and kernel G(x) = (|M| ) . 
Then the density gradient estimator of (8)is obtained: 
The expression (11) illustrates that, at location X , the mean 
shift vector evaluated with kernel G is proportional to the 
density gradient estimate obtained with kernel K (Comaniciu 
and Meer,2002; Ozertem et al.,2008) and 
when m h q (x) —> 0 , Vf K (x) —> 0 .So the mean shift 
vector always points toward the direction of maximum increase 
in the density, finding the point where the estimate 
v/*(*) = o is equal to finding the root of 
M h G = 0.According to (10), an iteration formulation can be 
given: 
Z M 1 
/=1 frd+2 X iS 
y J+ 1 
f 
2 \ 
1 
V 
1 — 
V 
J 
Z n 1 
/'=1 fj c/+2 ^ 
f 
x i~yj 
2 ^ 
V 
h, 
) 
7 = 1,2,- 
(12) 
The density estimator of X with kernel G , noted as f G (x) . 
The second square bracket term is the difference between the 
weighted mean of data points fallen in the bandwidth range of 
When the kernel K has a convex and monotonically decreasing 
profile, it can be proved that the sequences 
{yj }y'=l,2.... converge, and the proof is given in the literature 
(Comaniciu and Meer,2002; Zhi ,W., and Zi,C.2007; Xiang ,L. 
et al.,2005). It also has been proved that the introduction of 
variable bandwidth function doesn’t change the convergence of 
Mean shift iteration. The detail proof can be found in the 
APPENDIX section of (Comaniciu et al.,2001). It is more 
important that a variable bandwidth mean shift algorithm with 
pilot density estimate has shown its superiority over the fixed 
bandwidth procedure(Comaniciu et al.,2001; Georgescu et al.). 
In our segmentation algorithm, a pilot density estimate via 
nearest neighbours as used in (Georgescu et al.)is employed. 
Let x i>k be the k-nearest neighbour of the central point x h the Lj 
norm is used as distance metric and the distance is taken as the