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Talk Outline appearance based tracking patch similarity using histogram tracking by mean shift experiments, discussion Mean shift Tomáš Svoboda, svoboda@cmp.felk.cvut.cz Czech Technical University in Prague, Center for Machine Perception http://cmp.felk.cvut.cz Last update: April 8, 3 Please note that the lecture will be accompanied be several sketches and derivations on the blackboard and few live-interactive demos in Matlab Appearance based tracking 2/2 2 2 illustration from [] Histogram based representation 3/2 2 2
Patch comparison 4/2 2 2 2 2 8 7 6 4 histogram difference assume normalized histograms, i.e m u= p u = /2 d = ρ[p, q] where ρ[p, q] is the Bhattacharyya coefficient ρ[p, q] = m pu q u u= 2 2 8 7 6 4 2 2 Similarity measured by the Bhattacharyya coefficient 6/2 4 4 Similarity surface of the frame 6 4 6 7.9.8.7.6..4.3.2. The object is the 4 plate and the model is histogram of image intensities. s(y) = m pu (y)q u u= where p(y) is the histogram of at position y and q is the histogram of the template.
Problem: finding modes in probability density 7/2 4 4 Similarity surface of the frame 6 4 6 7.9.8.7.6..4.3.2. the complete enumeration of similarity surface can be costly, can we do it faster and more elegantly? 8/2 Density Gradient Estimation Mean shift procedure 9/2 3 3 Figure borrowed from [4]
Meanshift segmentation of colours - color distribution /2 4 4 Figure from [2] Meanshift segmentation of colours - color modes seeking /2 Figure from [2] Mean shift segmentation - and space 2/2 u, v are here spatial pixel coordinates different normalization for and spatial coordinates
Multivariate kernel density estimator 3/2 Normal kernel, KN (x) = exp( 2 x 2 ) Epanechnikov kernel, KE(x) = x 2.8..6.4..2........ Given n data points x i in d-dimensional space R d. f h,k (x) = nh d looking for extremum of fh,k(x) gradient fh,k(x) = n ( ) x xi K h i= Kernels 4/2 Normal kernel, K N (x) = exp( 2 x 2 ) Epanechnikov kernel, K E (x) = x 2.8..6.4..2........ Can be seen as membership function. Remind Kernel density estimation (Parzen method). Remember convolution? 6 6 Taken from http://en.wikipedia.org/wiki/kernel_density_estimation Mean-shift iterations /2 Assuming a reasonable differentiable kernel K, iterate till convergence: y k+ = n i= x ig( y k x i 2 ) n i= g( y k x i 2 ) g is the derivative of kernel profile.
Mean-shift tracking - ratio histogram Ratio histogram: ( ) qu r u = min, p u 6/2 where q is the histogram of the target and p is the histogram of the current frame. w i = r b(xi ) (just binning) Image intensities (or colors) are tranformed into weights, w i, by back projection of the ratio histogram. Mean-shift iterations: y k+ = n i= w ix i g( y k x i 2 ) n i= w ig( y k x i 2 ) Mean-shift tracking - Bhattacharya coeficient model, coordinates x i centered at : 7/2 n q u = C k( x i 2 )δ(b(x i ) u) i= target candidate centered at y: n h p u (y) = C h i= k ( y x i h 2) δ(b(x i ) u) ms tracking - object and its model 8/2 7 7 Figure from []
ms tracking - iterations 9/2 8 8 Figure from [] Mean-shift originally from [3]. References /2 [] Robert Collins. CSE/EE486 Computer Vision I. slides, web page. http://www.cse.psu.edu/~rcollins/cse486/. Robert kindly gave general permission to reuse the material. [2] Dorin Comaniciu and Peter Meer. Mean shift: A robust approach toward feature space analysis. IEEE Transactions on Pattern Analysis and Machine Analysis, 24():63 69, May 2. [3] Keinosuke Fukunaga and Larry D. Hostetler. The estimation of the gradient of a density function, with appilcations in pattern recognition. IEEE Transactions on Information Theory, 2():32 4, January 97. [4] Milan Šonka, Václav Hlaváč, and Roger Boyle. Image Processing, Analysis and Machine Vision. Thomson, 3rd edition, 7. [] Tomáš Svoboda, Jan Kybic, and Václav Hlaváč. Image Processing, Analysis and Machine Vision. A MATLAB Companion. Thomson, 7. Accompanying www site http://visionbook.felk.cvut.cz. End 2/2
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