meanshift算法详解

Intuitive Description
Region of interest Center of mass
Mean Shift vector
Objective : Find the densest region Distribution of identical billiard balls
Intuitive Description
Region of interest Center of mass
Mean Shift vector
Objective : Find the densest region Distribution of identical billiard balls
Intuitive Description
Region of interest Center of mass
Mean Shift
Theory and Applications
Yaron Ukrainitz & Bernard Sarel
Agenda
• Mean Shift Theory • What is Mean Shift ? • Density Estimation Methods • Deriving the Mean Shift • Mean shift properties
Objective : Find the densest region Distribution of identical billiard balls
What is Mean Shift ?
A tool for: Finding modes in a set of data samples, manifesting an underlying probability density function (PDF) in RN
• Applications • Clustering • Discontinuity Preserving Smoothing • Object Contour Detection • Segmentation • Object Tracking
Mean Shift Theory
Intuitive Description
• Suitable for real data analysis • Does not assume any prior shape (e.g. elliptical) on data clusters • Can handle arbitrary feature spaces
• The window size (bandwidth selection) is not trivial
Real Modality Analysis
An example
Window tracks signify the steepest ascent directions
Mean Shift Strengths & Weaknesses
Strengths : Weaknesses :
• Application independent tool
( x-μi )2 2 i 2
PDF(x) =
c e
i i

Estimate
Assumed Underlying PDF
Real Data Samples
Kernel Density Estimation
Parzen Windows - Function Forms
1 n P ( x) K ( x - x i ) n i 1
n x - xi 2 xi g h i 1 x m ( x) 2 n x x i g h i 1
•Translate the Kernel window by m(x)
Mean Shift vector
Objective : Find the densest region Distribution of identical billiard balls
Intuitive Description
Region of interest Center of mass
Mean Shift vector
x - xi K (x - xi ) ck h
2

Size of window
n x g i i c n c n i 1 P(x) ki gi n x n i 1 n i 1 gi i 1
Data point density implies PDF value !
Assumed Underlying PDF
Real Data Samples
Non-Parametric Density Estimation
Assumed Underlying PDF
Real Data Samples
Region of interest Center of mass
Mean Shift vector
Objective : Find the densest region Distribution of identical billiard balls
Intuitive Description
Region of interest Center of mass
Adaptive Gradient Ascent
• Convergence is guaranteed for infinitesimal steps only infinitely convergent, (therefore set a lower bound)
• For Uniform Kernel ( ), convergence is achieved in a finite number of steps • Normal Kernel ( ) exhibits a smooth trajectory, but is slower than Uniform Kernel ( ).
g(x) k (x)
Computing The Mean Shift
n x g i i c n c n i 1 P(x) ki gi n x n i 1 n i 1 gi i 1
Yet another Kernel density estimation ! Simple Mean Shift procedure: • Compute mean shift vector
Data Non-parametric Density GRADIENT Estimation (Mean Shift) PDF Analysis
Non-Parametric Density Estimation
Assumption : The data points are sampled from an underlying PDF
g(x) k (x)
Mean Shift Mode Detection
What happens if we reach a saddle point ?
Perturb the mode position and check if we return back
Updated Mean Shift Procedure: • Find all modes using the Simple Mean Shift Procedure • Prune modes by perturbing them (find saddle points and plateaus) • Prune nearby – take highest mode in the window
Objective : Find the densest region Distribution of identical billiard balls
Intuitive Description
Region of interest Center of mass
Fra Baidu bibliotek
Mean Shift vector
Objective : Find the densest region Distribution of identical billiard balls
Real Modality Analysis
Tessellate the space with windows
Run the procedure in parallel
Real Modality Analysis
The blue data points were traversed by the windows towards the mode
A function of some finite number of data points x1…xn
Data
In practice one uses the forms:
K (x) c k ( xi )
i 1
d
or
K (x) ck x

Same function on each dimension
Function of vector length only
Kernel Density Estimation
Various Kernels
1 n P ( x) K ( x - x i ) n i 1
A function of some finite number of data points x1…xn
Mean Shift Properties
• Automatic convergence speed – the mean shift vector size depends on the gradient itself.
• Near maxima, the steps are small and refined
• Normal Kernel
Kernel Density Estimation
Gradient
1 n P ( x) K ( x - x i ) n i 1
Give up estimating the PDF ! Estimate ONLY the gradient
Using the Kernel form: We get :
g(x) k (x)
Computing Kernel Density The Estimation Mean Shift
Gradient
n x g i i c n c n i 1 P(x) ki gi n x n i 1 n i 1 gi i 1
Non-Parametric Density Estimation ?
Assumed Underlying PDF
Real Data Samples
Parametric Density Estimation
Assumption : The data points are sampled from an underlying PDF
PDF in feature space • Color space Non-parametric • Scale spaceDensity Estimation • Actually any feature space you can conceive Discrete PDF Representation •…
Examples:
c 1 x • Epanechnikov Kernel K E (x) 0
• Uniform Kernel

2

Data
x 1 otherwise
c x 1 KU (x) 0 otherwise
1 2 K N (x) c exp x 2