Mean Shift Theory and Applications Reporter: Zhongping Ji.
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Transcript of Mean Shift Theory and Applications Reporter: Zhongping Ji.
![Page 1: Mean Shift Theory and Applications Reporter: Zhongping Ji.](https://reader036.fdocuments.us/reader036/viewer/2022062423/56649e725503460f94b70ea6/html5/thumbnails/1.jpg)
Mean ShiftTheory and Applications
Reporter: Zhongping Ji
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Agenda• Mean Shift Theory
• What is Mean Shift ? • Density Estimation Methods• Deriving the Mean Shift• Mean shift properties
• Applications• Clustering• Discontinuity Preserving Smoothing• Object Contour Detection• Segmentation• Object Tracking
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Papers
• Mean Shift: A Robust Approach Toward Feature Space Analysis
Authors: Dorin Comaniciu, Peter Meer (Rutgers University EECS, Member IEEE). IEEE Trans. Pattern analysis and machine intelligence 24(5), 2002
Field: application of modern statistical methods to image understanding problems
• A Topological Approach to Hierarchical Segmentation using Mean Shift
Authors: Sylvain Paris, Fredo Durand. (MIT EECS, computer science and artificial intelligence laboratory) Proceedings of the IEEE conference on Computer Vision and Pattern Recognition (CVPR'07)
Field: most aspects of image processing
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Mean Shift Theory
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Intuitive Description
Distribution of identical billiard balls
Region ofinterest
Center ofmass
Mean Shiftvector
Objective : Find the densest region
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Intuitive Description
Distribution of identical billiard balls
Region ofinterest
Center ofmass
Mean Shiftvector
Objective : Find the densest region
![Page 7: Mean Shift Theory and Applications Reporter: Zhongping Ji.](https://reader036.fdocuments.us/reader036/viewer/2022062423/56649e725503460f94b70ea6/html5/thumbnails/7.jpg)
Intuitive Description
Distribution of identical billiard balls
Region ofinterest
Center ofmass
Mean Shiftvector
Objective : Find the densest region
![Page 8: Mean Shift Theory and Applications Reporter: Zhongping Ji.](https://reader036.fdocuments.us/reader036/viewer/2022062423/56649e725503460f94b70ea6/html5/thumbnails/8.jpg)
Intuitive Description
Distribution of identical billiard balls
Region ofinterest
Center ofmass
Mean Shiftvector
Objective : Find the densest region
![Page 9: Mean Shift Theory and Applications Reporter: Zhongping Ji.](https://reader036.fdocuments.us/reader036/viewer/2022062423/56649e725503460f94b70ea6/html5/thumbnails/9.jpg)
Intuitive Description
Distribution of identical billiard balls
Region ofinterest
Center ofmass
Mean Shiftvector
Objective : Find the densest region
![Page 10: Mean Shift Theory and Applications Reporter: Zhongping Ji.](https://reader036.fdocuments.us/reader036/viewer/2022062423/56649e725503460f94b70ea6/html5/thumbnails/10.jpg)
Intuitive Description
Distribution of identical billiard balls
Region ofinterest
Center ofmass
Mean Shiftvector
Objective : Find the densest region
![Page 11: Mean Shift Theory and Applications Reporter: Zhongping Ji.](https://reader036.fdocuments.us/reader036/viewer/2022062423/56649e725503460f94b70ea6/html5/thumbnails/11.jpg)
Intuitive Description
Distribution of identical billiard balls
Region ofinterest
Center ofmass
Objective : Find the densest region
![Page 12: Mean Shift Theory and Applications Reporter: Zhongping Ji.](https://reader036.fdocuments.us/reader036/viewer/2022062423/56649e725503460f94b70ea6/html5/thumbnails/12.jpg)
What is Mean Shift ?
Non-parametricDensity Estimation
Non-parametricDensity GRADIENT Estimation
(Mean Shift)
Data
Discrete PDF Representation
PDF Analysis
PDF in feature space• Color space• Scale space• Actually any feature space you can conceive• …
A tool for:Finding modes in a set of data samples, manifesting an underlying probability density function (PDF) in RN
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Non-Parametric Density Estimation
Assumption : The data points are sampled from an underlying PDF
Assumed Underlying PDF Real Data Samples
Data point density implies PDF value !
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Assumed Underlying PDF Real Data Samples
Non-Parametric Density Estimation
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Assumed Underlying PDF Real Data Samples
?Non-Parametric Density Estimation
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Parametric Density Estimation
Assumption : The data points are sampled from an underlying PDF
Assumed Underlying PDF
2
2
( )
2
i
PDF( ) = i
iic e
x-μ
x
Estimate
Real Data Samples
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Kernel Density Estimation Parzen Windows - Function Forms
1
1( ) ( )
n
ii
P Kn
x x - x A function of some finite number of data pointsx1…xn
DataIn practice one uses the forms:
1
( ) ( )d
ii
K c k x
x or ( )K ckx x
Same function on each dimension Function of vector length only
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Kernel Density EstimationVarious Kernels
1
1( ) ( )
n
ii
P Kn
x x - x A function of some finite number of data pointsx1…xn
Examples:
• Epanechnikov Kernel
• Uniform Kernel
• Normal Kernel
21 1
( ) 0 otherwise
E
cK
x xx
1( )
0 otherwiseU
cK
xx
21( ) exp
2NK c
x x
Data
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Kernel Density EstimationGradient
1
1 ( ) ( )
n
ii
P Kn
x x - x Give up estimating the PDF !Estimate ONLY the gradient
2
( ) iiK ck
h
x - xx - x
Using theKernel form:
We get :
1
1 1
1
( )
n
i in ni
i i ni i
ii
gc c
P k gn n g
xx x
Size of window
g( ) ( )kx x
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Kernel Density EstimationGradient
1
1 1
1
( )
n
i in ni
i i ni i
ii
gc c
P k gn n g
xx x
Computing The Mean Shift
g( ) ( )kx x
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1
1 1
1
( )
n
i in ni
i i ni i
ii
gc c
P k gn n g
xx x
Computing The Mean Shift
Yet another Kernel density estimation !
Simple Mean Shift procedure:• Compute mean shift vector
•Translate the Kernel window by m(x)
2
1
2
1
( )
ni
ii
ni
i
gh
gh
x - xx
m x xx - x
g( ) ( )kx x
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Mean Shift Mode Detection
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
What happens if wereach a saddle point
?
Perturb the mode positionand check if we return back
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AdaptiveGradient Ascent
Mean Shift Properties
• Automatic convergence speed – the mean shift vector size depends on the gradient itself.
• Near maxima, the steps are small and refined
• Convergence is guaranteed for infinitesimal steps only infinitely convergent,
• For Uniform Kernel ( ), convergence is achieved in a finite number of steps
• Normal Kernel ( ) exhibits a smooth trajectory, but is slower than Uniform Kernel ( ).
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Real Modality Analysis
Tessellate the space with windows
Run the procedure in parallel
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Real Modality Analysis
The blue data points were traversed by the windows towards the mode
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Real Modality AnalysisAn example
Window tracks signify the steepest ascent directions
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Mean Shift Strengths & Weaknesses
Strengths :
• Application independent tool
• Suitable for real data analysis
• Does not assume any prior shape (e.g. elliptical) on data clusters
• Can handle arbitrary feature spaces
• Only ONE parameter to choose
• h (window size) has a physical meaning, unlike K-Means
Weaknesses :
• The window size (bandwidth selection) is not trivial
• Inappropriate window size can cause modes to be merged, or generate additional “shallow” modes Use adaptive window size
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Mean Shift Applications
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Clustering
Attraction basin : the region for which all trajectories lead to the same mode
Cluster : All data points in the attraction basin of a mode
Mean Shift : A robust Approach Toward Feature Space Analysis, by Comaniciu, Meer
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ClusteringSynthetic Examples
Simple Modal Structures
Complex Modal Structures
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ClusteringReal Example
Initial windowcenters
Modes found Modes afterpruning
Final clusters
Feature space:L*u*v representation
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ClusteringReal Example
L*u*v space representation
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ClusteringReal Example
2D (L*u) space representation
Final clusters
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Discontinuity Preserving SmoothingFeature space : Joint domain = spatial coordinates + color space
( )s r
s rs r
K C k kh h
x xx
Meaning : treat the image as data points in the spatial and gray level domain
Image Data(slice)
Mean Shiftvectors
Smoothingresult
Mean Shift : A robust Approach Toward Feature Space Analysis, by Comaniciu, Meer
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Discontinuity Preserving Smoothing
y
z
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Discontinuity Preserving SmoothingThe effect of window sizein spatial andrange spaces
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Discontinuity Preserving SmoothingExample
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Discontinuity Preserving SmoothingExample
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Object Contour DetectionRay Propagation
Vessel Detection by Mean Shift Based Ray Propagation, by Tek, Comaniciu, Williams
Accurately segment various objects (rounded in nature) in medical images
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Object Contour DetectionRay Propagation
Use displacement data to guide ray propagation
Discontinuity preserving smoothing
Displacementvectors
Vessel Detection by Mean Shift Based Ray Propagation, by Tek, Comaniciu, Williams
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Object Contour DetectionRay Propagation
( , ) ( , )Ray
s t Speed x y Nt
Speed function
Normal to the contour
( , ) ( , ) ( , )Speed x y f disp x y x y
Curvature
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Object Contour DetectionOriginal image
Gray levels along red line
Gray levels aftersmoothing
Displacement vectors Displacement vectors’derivative
( , ) ( , ) ( , )Speed x y f disp x y x y
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Object Contour DetectionExample
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Object Contour DetectionExample
Importance of smoothing by curvature
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SegmentationSegment = Cluster, or Cluster of Clusters
Algorithm:• Run Filtering (discontinuity preserving smoothing)• Cluster the clusters which are closer than window size
Image Data(slice)
Mean Shiftvectors
Segmentationresult
Smoothingresult
Mean Shift : A robust Approach Toward Feature Space Analysis, by Comaniciu, Meerhttp://www.caip.rutgers.edu/~comanici
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SegmentationExample
…when feature space is only gray levels…
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SegmentationExample
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SegmentationExample
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SegmentationExample
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SegmentationExample
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SegmentationExample
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SegmentationExample
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SegmentationExample
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Image editing
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A Topological Approach to Hierarchical Segmentation using Mean Shift
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Fast and Hierarchical
• Efficient numerical schemes to evaluate the density function and extract its modes.
• A hierarchical segmentation based on a topological analysis of mean shift.
• Do not aim for better segmentation accuracy and focus on computational efficiency and creating a hierarchy.
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FAST COMPUTATION OF THE FEATURE POINT DENSITY
Coarse grid
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EXTRACTION OF THE DENSITY MODES
• 0 label: The processed node is a local maximum. We create a new label.
• 1 label: All the ascending paths go to the same summit. We copy the label.
• 2+ labels: Boundary between two modes. We put a special marker.
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Boundaries refinement
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Hierarchical Segmentation
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Boundary Persistence
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Simplification
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Examples
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Examples
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• Monkey saddle
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Ask & Answer