IntroductionKKernel density Kernel choices |Peak finding/Mean-shiftICam-shift EXample 1 Show graphically of mean-shift Starting from x C f1(x)= and the circle nho2x=r Search shown Radius Find the peak of S XO the pde Repeat the task if oo°o your first selection .o 33 9 is at x PDF of mosquitoes in CU x1 Camshift vod
Introduction | Kernel density | Kernel choices | Peak finding | Mean-shift | Cam-shift Example 1 Show graphically of mean-shift • Starting from x and the circle shown • Find the peak of the PDF. • Repeat the task if your first selection is at x’ = − = n i i h d h x x K nh C f x 1 ( ) ˆ Camshift v9d 16 x1 x2 PDF of mosquitoes in CU x x’ Search Radius = Sr
Introduction I Kernel density (Kernel choices Peak finding I Mean-shift I Cam-shift Kernel choices How about the kernels function“K”(內核)? Building blocks Camshift vod 17
Introduction | Kernel density | Kernel choices | Peak finding | Mean-shift | Cam-shift Kernel choices How about the kernels function “K” (內核 )? Building blocks Camshift v9d 17
Introduction Kernel density I Kernel choices Peak finding Mean-shift Cam-shift Define Kernel and profile A. Kernels Kernel DEFINTION 1. Let X be the n-dimensional Euclidean space, R: The variable x of a Denote the ith component ofxE X by xp. The norm ofxEX kernel is a point in is a nonnegative number ll such that k-l=2- .The the n-dimensional space inner product of x and y in X is(x,y)=2x,yiAfunction Profile K: X-, is said to be a kernel if there exists a profile The variable for a k:[0,]→R, such that profile is a 2-norm K(x)=( value(length of a d vector in the n- 1)k is nonnegative. dimensional space 2)k is nonincreasing: k(a)2k(b)if a< b 3) k is piecewise continuous and(<∞ From: Cheng, Yizong(August 1995). Mean Shift, Mode Seeking and Clustering.IEEE Transactions on Pattern Analysis and Machine intelligence(IEEE)17 (8): 790- 799.c0:104109/34400568. 18
Introduction | Kernel density | Kernel choices | Peak finding | Mean-shift | Cam-shift Define Kernel and Profile • Kernel – The variable x of a kernel is a point in the n-dimensional space • Profile – The variable for a profile is a 2-norm value (length of a vector in the ndimensional space) ||x||2 Camshift v9d 18 From:Cheng, Yizong (August 1995). "Mean Shift, Mode Seeking, and Clustering". IEEE Transactions on Pattern Analysis and Machine Intelligence (IEEE) 17 (8): 790– 799. doi:10.1109/34.400568
Introduction I Kernel density (Kernel choices Peak finding I Mean-shift I Cam-shift Kernel choices(內核) K(wdw=l Different radial symmetric kernel K(w)=K(w) for all values of w Epanechnikov h K c-12)1|1 otherwise 00095 Uniform 0 otherwise Normal 0.008 (Gaussian) k、()=ce3h 0002 Camshift vod
Introduction | Kernel density | Kernel choices | Peak finding | Mean-shift | Cam-shift Kernel choices (內核) • Different radial symmetric kernel ( ) − = 0 otherwise 1 || || || || 1 ( ) 2 c w w KE w 2 ( ) ( ) for all values of w ( ) 1 h x x w K w K w . K w dw − i = − = = + − Camshift v9d 19 Epanechnikov Uniform Normal (Gaussian) 2 || || 2 1 ( ) w N K w c e − = = 0 otherwise || || 1 ( ) c w KU w
Introduction Kernel density I Kernel choices Peak finding Mean-shift Cam-shift The use of the epanechnikov Kel kernel function By definition(x)=56∑kal) Radial Symmetric Kernel X- where Keral()is a kernel function, where w K k h Here, Keral(wp)=x(wl )is a radial symmetric function, Le, use the square distance between x and x normalized by h as the parameter use Epanechnikov k()=k,()as the kernel function (like h=radius of a circle). ∑ where k(w) lw|≤1 otherwise Camshift vod
Introduction | Kernel density | Kernel choices | Peak finding | Mean-shift | Cam-shift The use of the Epanechnikov (KE) kernel function Camshift v9d 20 ( ) ( ) ( ) ( ) − = − = = = − = = = − = = = = = = 0 otherwise 1 || || || || 1 where ( ) ( ) , If use Epanechnikov as the kernelfunction ( ) Here, ( ) is a radialsymmetric function , where ( ) is a kernelfunction, where By definition ( ) , 2 1 , 1 2 1 2 , 2 1 , c w w k w h x x k w nh c P x k( ) k ( ) h x x k nh c k w nh c P xKeral w k w h x x Keral w w Keral w nh c P x E n i i d E k,d n k E n i i d k,d n i d k,d h K i n i d k,d n k (like radius of a circle). by as the parameter between and normalized I.e, use the square distance , Radial SymmetricKernel , 2 = − − h h x x h x x k h x x K i i i