The document describes the mean shift algorithm and its application to object tracking in computer vision. Mean shift is an iterative procedure that moves data points to the average of nearby points, converging at modes of the data's probability density function. It can be used for tracking by modeling a target object's color distribution and applying mean shift to match candidate locations in subsequent frames. The algorithm maximizes the Bhattacharyya coefficient between color distributions to find the best match for the target's new location in each frame.

1. Mean Shift Tracking
CS4243 Computer Vision and Pattern Recognition
Leow Wee Kheng
Department of Computer Science
School of Computing
National University of Singapore
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2. Mean Shift
Mean Shift
Mean Shift [Che98, FH75, Sil86]
An algorithm that iteratively shifts a data point to the average of
data points in its neighborhood.
Similar to clustering.
Useful for clustering, mode seeking, probability density estimation,
tracking, etc.
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3. Mean Shift
Consider a set S of n data points xi in d-D Euclidean space X.
Let K(x) denote a kernel function that indicates how much x
contributes to the estimation of the mean.
Then, the sample mean m at x with kernel K is given by
n
K(x − xi ) xi
i=1
m(x) = n (1)
K(x − xi )
i=1
The difference m(x) − x is called mean shift.
Mean shift algorithm: iteratively move date point to its mean.
In each iteration, x ← m(x).
The algorithm stops when m(x) = x.
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4. Mean Shift
The sequence x, m(x), m(m(x)), . . . is called the trajectory of x.
If sample means are computed at multiple points, then at each
iteration, update is done simultaneously to all these points.
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5. Mean Shift Kernel
Kernel
Typically, kernel K is a function of x 2 :
K(x) = k( x 2 ) (2)
k is called the profile of K.
Properties of Profile:
1 k is nonnegative.
2 k is nonincreasing: k(x) ≥ k(y) if x < y.
3 k is piecewise continuous and
∞
k(x)dx < ∞ (3)
0
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6. Mean Shift Kernel
Examples of kernels [Che98]:
Flat kernel:
1 if x ≤ 1
K(x) = (4)
0 otherwise
Gaussian kernel:
K(x) = exp(− x 2 ) (5)
(a) Flat kernel (b) Gaussian kernel
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7. Mean Shift Density Estimation
Density Estimation
Kernel density estimation (Parzen window technique) is a popular
method for estimating probability density
[CRM00, CRM02, DH73].
For a set of n data points xi in d-D space, the kernel density
estimate with kernel K(x) (profile k(x)) and radius h is
n
˜ 1 x − xi
fK (x) = K
nhd h
i=1
(6)
n 2
1 x − xi
= k
nhd h
i=1
The quality of kernel density estimator is measured by the mean
square error between the actual density and the estimate.
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8. Mean Shift Density Estimation
The mean square error is minimized by the Epanechnikov kernel:

 1
(d + 2)(1 − x 2 ) if x ≤ 1
KE (x) = 2Cd (7)
 0 otherwise
where Cd is the volume of the unit d-D sphere, with profile

 1
(d + 2)(1 − x) if 0 ≤ x ≤ 1
kE (x) = 2Cd (8)
 0 if x > 1
A more commonly used kernel is Gaussian
1 1 2
K(x) = exp − x (9)
(2π)d 2
with profile
1 1
k(x) = exp − x (10)
(2π)d 2
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9. Mean Shift Density Estimation
Define another kernel G(x) = g( x 2 ) such that
dk(x)
g(x) = −k (x) = − (11)
dx
Important Result
Mean shift with kernel G moves x along the direction of the gradient of
˜
density estimate f with kernel K.
Define density estimate with kernel K.
But, perform mean shift with kernel G.
Then, mean shift performs gradient ascent on density estimate.
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10. Mean Shift Density Estimation
Proof:
˜
Define f with kernel G
n 2
˜ C x − xi
fG (x) ≡ g (12)
nhd h
i=1
where C is a normalization constant.
Mean shift M with kernel G is
n 2
x − xi
g xi
h
i=1
MG (x) ≡ n 2
−x (13)
x − xi
g
h
i=1
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11. Mean Shift Density Estimation
Estimate of density gradient is the gradient of density estimate
˜ fK (x) ≡ ˜
fK (x)
n 2
2 x − xi
= (x − xi ) k
nhd+2 h
i=1
n 2
(14)
2 x − xi
= (xi − x) g
nhd+2 h
i=1
2 ˜
= fG (x)MG (x)
Ch2
Then,
Ch2 ˜ fK (x)
MG (x) = (15)
˜
2 fG (x)
Thus, MG is an estimate of the normalized gradient of fK .
So, can use mean shift (Eq. 13) to obtain estimate of fK .
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12. Mean Shift Tracking
Mean Shift Tracking
Basic Ideas [CRM00]:
Model object using color probability density.
Track target object in video by matching color density.
Use mean shift to estimate color density and target location.
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13. Mean Shift Tracking Object Model
Object Model
Let xi , i = 1, . . . , n, denote pixel locations of model centered at 0.
Represent color distribution by discrete m-bin color histogram.
Let b(xi ) denote the color bin of the color at xi .
Assume size of model is normalized; so, kernel radius h = 1.
Then, the probability q of color u in the model is
n
qu = C k( xi 2 ) δ(b(xi ) − u) (16)
i=1
C is the normalization constant
n −1
2
C= k( xi ) (17)
i=1
Kernel profile k weights contribution by distance to centroid.
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14. Mean Shift Tracking Object Model
δ is the Kronecker delta function
1 if a = 0
δ(a) = (18)
0 otherwise
That is, contribute k( xi 2 ) to qu if b(xi ) = u.
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15. Mean Shift Tracking Target Candidate
Target Candidate
Let yi , i = 1, . . . , nh , denote pixel locations of target centered at y.
Then, the probability p of color u in the target is
nh 2
y − yi
pu (y) = Ch k δ(b(yi ) − u) (19)
h
i=1
Ch is the normalization constant
nh 2 −1
y − yi
Ch = k (20)
h
i=1
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16. Mean Shift Tracking Color Density Matching
Color Density Matching
Use Bhattacharyya coefficient ρ
m
ρ(p(y), q) = pu (y) qu (21)
u=1
√ √ √ √
ρ is the cosine of vectors ( p1 , . . . , pm ) and ( q1 , . . . , qm ) .
Large ρ means good color match.
For each image frame, find y that maximizes ρ.
This y is the location of the target.
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17. Mean Shift Tracking Tracking Algorithm
Tracking Algorithm
Given {qu } of model and location y of target in previous frame:
1 Initialize location of target in current frame as y.
2 Compute {pu (y)} and ρ(p(y), q).
3 Apply mean shift: Compute new location z as
nh 2
y − yi
g yi
h
i=1
z= nh 2
(22)
y − yi
g
h
i=1
4 Compute {pu (z)} and ρ(p(z), q).
5 While ρ(p(z), q) < ρ(p(y), q), do z ← 1 (y + z).
2
6 If z − y is small enough, stop. Else, set y ← z and goto (1).
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18. Mean Shift Tracking Tracking Algorithm
Step 3: In practice, a window of pixels yi is considered.
Size of window is related to h.
Step 5 is used to validate the target’s new location.
Can stop Step 5 if y and z round off to the same pixel.
Tests show that Step 5 is needed only 0.1% of the time.
Step 6: can stop algorithm if y and z round off to the same pixel.
To track object that changes size, varies radius h
(see [CRM00] for details).
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19. Mean Shift Tracking Tracking Algorithm
Example 1: Track football player no. 78 [CRM00].
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20. Mean Shift Tracking Tracking Algorithm
Example 2: Track a passenger in train station [CRM00].
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21. Reference
Reference I
Y. Cheng.
Mean shift, mode seeking, and clustering.
IEEE Trans. on Pattern Analysis and Machine Intelligence,
17(8):790–799, 1998.
D. Comaniciu, V. Ramesh, and P. Meer.
Real-time tracking of non-rigid objects using mean shift.
In IEEE Proc. on Computer Vision and Pattern Recognition, pages
673–678, 2000.
D. Comaniciu, V. Ramesh, and P. Meer.
Mean shift: A robust approach towards feature space analysis.
IEEE Trans. on Pattern Analysis and Machine Intelligence,
24(5):603–619, 2002.
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22. Reference
Reference II
R. O. Duda and P. E. Hart.
Pattern Classification and Scene Analysis.
Wiley, 1973.
K. Fukunaga and L. D. Hostetler.
The estimation of the gradient of a density function, with
applications in pattern recognition.
IEEE Trans. on Information Theory, 21:32–40, 1975.
B. W. Silverman.
Density Estimation for Statistics and Data Analysis.
Chapman and Hall, 1986.
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