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OPPA European Social Fund Prague & EU: We invest in your future.
Transcript of OPPA European Social Fund Prague & EU: We invest in your ... · OPPA European Social Fund Prague &...
OPPA European Social FundPrague & EU: We invest in your future.
Talk Outline
� appearance based tracking
� patch similarity using histogram
� tracking by mean shift
� experiments, discussion
Mean shift 1
Tomáš Svoboda, [email protected] Technical University in Prague, Center for Machine Perception
http://cmp.felk.cvut.czLast update: April 8, 2013
1Please note that the lecture will be accompanied be several sketches and derivations on the blackboardand few live-interactive demos in Matlab
2/21Appearance based tracking
2
2illustration from [1]
3/21Histogram based representation
image patch
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4/21Patch comparison
image patch
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image patch
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5/21histogram difference
assume normalized histograms, i.e∑mu=1 pu = 1
d =√1− ρ[p, q]
where ρ[p, q] is the Bhattacharyyacoefficient
ρ[p, q] =m∑
u=1
√puqu
image patch
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image patch
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6/21
Similarity measured by the Bhattacharyyacoefficient
Similarity surface of the frame 206
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The object is the “4” plate and the model is histogram of image intensities.
s(y) =m∑
u=1
√pu(y)qu
where p(y) is the histogram of image patch at position y and q is thehistogram of the template.
7/21Problem: finding modes in probability density
Similarity surface of the frame 206
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� the complete enumeration of similarity surface can be costly,
� can we do it faster and more elegantly?
8/21
Density Gradient Estimation
9/21Mean shift procedure
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3Figure borrowed from [4]
10/21
Meanshift segmentation of colours - colordistribution
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4Figure from [2]
11/21
Meanshift segmentation of colours - color modesseeking
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5Figure from [2]
12/21Mean shift segmentation - intensity and space
u, v are here spatial pixel coordinates
different normalization for intensity and spatial coordinates
13/21Multivariate kernel density estimator
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Normal kernel, KN (x) = exp(−12‖x‖2)
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Epanechnikov kernel, KE(x) = 1 −‖x‖2
Given n data points xi in d-dimensional space Rd.
f̃h,K(x) =1
nhd
n∑
i=1
K
(x− xi
h
)
� looking for extremum of fh,K(x)
� gradient ∇fh,K(x) = 0
14/21Kernels
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Normal kernel, KN (x) = exp(−12‖x‖2)
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Epanechnikov kernel, KE(x) = 1 −‖x‖2
Can be seen as membership function.Remind Kernel density estimation (Parzen method).
Remember convolution? 6
6Taken from http://en.wikipedia.org/wiki/Kernel_density_estimation
15/21Mean-shift iterations
Assuming a reasonable differentiable kernel K, iterate till convergence:
yk+1 =
∑ni=1 xig(‖yk − xi‖2)∑n
i=1 g(‖yk − xi‖2)
g is the derivative of kernel profile.
16/21Mean-shift tracking - ratio histogram
Ratio histogram:
ru = min
(qu
pu, 1
)
where q is the histogram of the targetand p is the histogram of the currentframe. wi = rb(xi) (just binning)
Image intensities (or colors) are tranformed into weights, wi, by backprojection of the ratio histogram. Mean-shift iterations:
yk+1 =
∑ni=1wixig(‖yk − xi‖2)∑n
i=1wig(‖yk − xi‖2)
17/21Mean-shift tracking - Bhattacharya coeficient
model, coordinates x∗i centered at 0:
qu = Cn∑
i=1
k(‖x∗i‖2)δ(b(x∗
i )− u)
target candidate centered at y:
pu(y) = Ch
nh∑
i=1
k
(∥∥∥∥y − xi
h
∥∥∥∥2)δ(b(xi)− u)
18/21ms tracking - object and its model
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7Figure from [5]
19/21ms tracking - iterations
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8Figure from [5]
20/21References
Mean-shift originally from [3].
[1] Robert Collins. CSE/EE486 Computer Vision I. slides, web page.http://www.cse.psu.edu/~rcollins/CSE486/. Robert kindly gave general permission to reuse thematerial.
[2] Dorin Comaniciu and Peter Meer. Mean shift: A robust approach toward feature space analysis. IEEETransactions on Pattern Analysis and Machine Analysis, 24(5):603–619, May 2002.
[3] Keinosuke Fukunaga and Larry D. Hostetler. The estimation of the gradient of a density function, withappilcations in pattern recognition. IEEE Transactions on Information Theory, 21(1):32–40, January 1975.
[4] Milan Šonka, Václav Hlaváč, and Roger Boyle. Image Processing, Analysis and Machine Vision.Thomson, 3rd edition, 2007.
[5] Tomáš Svoboda, Jan Kybic, and Václav Hlaváč. Image Processing, Analysis and Machine Vision. AMATLAB Companion. Thomson, 2007. Accompanying www site http://visionbook.felk.cvut.cz.
21/21End
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