IntroductionKKernel density Kernel choices I Peak finding IMean-shiftICam-shift EXample 1 Show graphically of mean-shift Starting from x C f1(x)= ∑ Kaxi and the circle Search shown Radius Find the peak of S XO the pde Repeat the task if oo°o your first selection o● is at x PDF of mosquitoes in CU x1 Camshift v 0.a 16
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 v.0.a 16 x1 x2 PDF of mosquitoes in CU x x’ Search Radius = Sr
Introduction I Kernel density (Kernel choices Peak finding I Mean-shift / Cam-shift Kernel choices How about the kernels function“K(內核)? Building blocks Camshift v 0.a 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 v.0.a 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. eEE Transactions on pattern Analysis and Machine Intelligence(IEEE17(8): 790 799.doi:101109/34400568
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 v.0.a 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 / Cam-shift Kernel choices(内核) K(w)w=l Different radial symmetric kerne K(w)=K(w) for all values of w Epanechnikov h K(v)= c( w 12) w ks1 otherwise 00095 Uniform lv|≤1 K() U 0 otherwise Normal 0.008 (Gaussian) N(w)=c'ethvIP K 0002 Camshift v 0.a
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 v.0.a 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 P&(x)=kd 2 Keral(w), nh Radial Symmetric Kernel where Keral(w )is a kernel function, where w x-xi/K h h Here, Keral( w)=k(bw is a radial symmetric function, I.e, use the square distance (=u心)=∑ x-x between x and x normalized y h as the parameter If use Epanechnikov k()=k() as the kernel function(like h=radius of a circle). ∑ x-xi lw|≤1 where re otherwise Camshift v 0.a
Introduction | Kernel density | Kernel choices | Peak finding | Mean-shift | Cam-shift The use of the Epanechnikov (KE) kernel function Camshift v.0.a 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 , RadialSymmetricKernel , 2 = − − h h x x h x x k h x x K i i i