Segmentation and Clustering

Today’s Readings

Forsyth & Ponce, Chapter 7

(plus lots of optional references in the slides)

From images to objects

What Defines an Object?

Subjective problem, but has been well-studied

Gestalt Laws seek to formalize this

proximity, similarity, continuation, closure, common fate

see notes by Steve Joordens, U. Toronto

Extracting objects

How could this be done?

Image Segmentation

Many approaches proposed

cues: color, regions, contours

automatic vs. user-guided

no clear winner

we’ll consider several approaches today

Intelligent Scissors (demo)

Intelligent Scissors [Mortensen 95]

Approach answers a basic question

Q: how to find a path from seed to mouse that follows object boundary as closely as possible?

Intelligent Scissors

Basic Idea

Define edge score for each pixel

edge pixels have low cost

Find lowest cost path from seed to mouse

Path Search (basic idea)

Graph Search Algorithm

Computes minimum cost path from seed to all other pixels

How does this really work?

Treat the image as a graph

Defining the costs

Treat the image as a graph

Defining the costs

c can be computed using a cross-correlation filter

assume it is centered at p

Also typically scale c by its length

set c = (max-|filter response|)

where max = maximum |filter response| over all pixels in the image

Dijkstra’s shortest path algorithm

Dijkstra’s shortest path algorithm

Properties

It computes the minimum cost path from the seed to every node in the graph. This set of minimum paths is represented as a tree

Running time, with N pixels:

O(N²) time if you use an active list

O(N log N) if you use an active priority queue (heap)

takes fraction of a second for a typical (640x480) image

Once this tree is computed once, we can extract the optimal path from any point to the seed in O(N) time.

it runs in real time as the mouse moves

What happens when the user specifies a new seed?

Segmentation by min (s-t) cut [Boykov 2001]

Graph

node for each pixel, link between pixels

specify a few pixels as foreground and background

create an infinite cost link from each bg pixel to the “t” node

create an infinite cost link from each fg pixel to the “s” node

compute min cut that separates s from t

how to define link cost between neighboring pixels?

Grabcut [Rother et al., SIGGRAPH 2004]

Is user-input required?

Our visual system is proof that automatic methods are possible

classical image segmentation methods are automatic

Argument for user-directed methods?

only user knows desired scale/object of interest

Automatic graph cut [Shi & Malik]

Fully-connected graph

node for every pixel

link between every pair of pixels, p,q

cost c_pq for each link

c_pq measures similarity

similarity is inversely proportional to difference in color and position

Segmentation by Graph Cuts

Break Graph into Segments

Delete links that cross between segments

Easiest to break links that have low cost (similarity)

similar pixels should be in the same segments

dissimilar pixels should be in different segments

Cuts in a graph

Link Cut

set of links whose removal makes a graph disconnected

cost of a cut:

But min cut is not always the best cut...

Cuts in a graph

Interpretation as a Dynamical System

Treat the links as springs and shake the system

elasticity proportional to cost

vibration “modes” correspond to segments

can compute these by solving an eigenvector problem

http://www.cis.upenn.edu/~jshi/papers/pami_ncut.pdf

Color Image Segmentation

Extension to Soft Segmentation

Histogram-based segmentation

Goal

Break the image into K regions (segments)

Solve this by reducing the number of colors to K and mapping each pixel to the closest color

Clustering

How to choose the representative colors?

This is a clustering problem!

Break it down into subproblems

Suppose I tell you the cluster centers c_i

Q: how to determine which points to associate with each c_i?

K-means clustering

K-means clustering algorithm

Randomly initialize the cluster centers, c₁, ..., c_K

Given cluster centers, determine points in each cluster

For each point p, find the closest c_i. Put p into cluster i

Given points in each cluster, solve for c_i

Set c_i to be the mean of points in cluster i

If c_i have changed, repeat Step 2

Java demo: http://home.dei.polimi.it/matteucc/Clustering/tutorial_html/AppletKM.html

Properties

Will always converge to some solution

Can be a “local minimum”

does not always find the global minimum of objective function:

K-Means++

Probabilistic clustering

Basic questions

what’s the probability that a point x is in cluster m?

what’s the shape of each cluster?

K-means doesn’t answer these questions

Basic idea

instead of treating the data as a bunch of points, assume that they are all generated by sampling a continuous function

This function is called a generative model

defined by a vector of parameters θ

Mixture of Gaussians

One generative model is a mixture of Gaussians (MOG)

K Gaussian blobs with means μ_b covariance matrices V_b, dimension d

blob b defined by:

blob b is selected with probability

the likelihood of observing x is a weighted mixture of Gaussians

where

Expectation maximization (EM)

Goal

find blob parameters θ that maximize the likelihood function:

Approach:

E step: given current guess of blobs, compute ownership of each point

M step: given ownership probabilities, update blobs to maximize likelihood function

repeat until convergence

EM details

E-step

compute probability that point x is in blob i, given current guess of θ

M-step

compute probability that blob b is selected

mean of blob b

covariance of blob b

EM demo

http://lcn.epfl.ch/tutorial/english/gaussian/html/index.html

Applications of EM

Turns out this is useful for all sorts of problems

any clustering problem

any model estimation problem

missing data problems

finding outliers

segmentation problems

segmentation based on color

segmentation based on motion

foreground/background separation

...

Problems with EM

Local minima

k-means is NP-hard even with k=2

Need to know number of segments

solutions: AIC, BIC, Dirichlet process mixture

Need to choose generative model

Finding Modes in a Histogram

How Many Modes Are There?

Easy to see, hard to compute

Mean Shift [Comaniciu & Meer]

Iterative Mode Search

Initialize random seed, and window W

Calculate center of gravity (the “mean”) of W:

Translate the search window to the mean

Repeat Step 2 until convergence

Mean-Shift

Approach

Initialize a window around each point

See where it shifts—this determines which segment it’s in

Multiple points will shift to the same segment

Mean-shift for image segmentation

Useful to take into account spatial information

instead of (R, G, B), run in (R, G, B, x, y) space

D. Comaniciu, P. Meer, Mean shift analysis and applications, 7th International Conference on Computer Vision, Kerkyra, Greece, September 1999, 1197-1203.

http://www.caip.rutgers.edu/riul/research/papers/pdf/spatmsft.pdf

Choosing Exemplars (Medoids)

Taxonomy of Segmentation Methods

References

Mortensen and Barrett, “Intelligent Scissors for Image Composition,” Proc. SIGGRAPH 1995.

Boykov and Jolly, “Interactive Graph Cuts for Optimal Boundary & Region Segmentation of Objects in N-D images,” Proc. ICCV, 2001.

Shi and Malik, “Normalized Cuts and Image Segmentation,” Proc. CVPR 1997.

Comaniciu and Meer, “Mean shift analysis and applications,” Proc. ICCV 1999.


	Today’s Readings
		Forsyth & Ponce, Chapter 7
		(plus lots of optional references in the slides)


What Defines an Object?
	Subjective problem, but has been well-studied
	Gestalt Laws seek to formalize this
		proximity, similarity, continuation, closure, common fate
		see notes by Steve Joordens, U. Toronto


	Many approaches proposed
		cues: color, regions, contours
		automatic vs. user-guided
		no clear winner
		we’ll consider several approaches today


	Approach answers a basic question
		Q: how to find a path from seed to mouse that follows object boundary as closely as possible?


Basic Idea
	Define edge score for each pixel
		edge pixels have low cost
	Find lowest cost path from seed to mouse


	Graph Search Algorithm
		Computes minimum cost path from seed to all other pixels


c can be computed using a cross-correlation filter
	assume it is centered at p

Also typically scale c by its length
	set c = (max-\|filter response\|)
		where max = maximum \|filter response\| over all pixels in the image


Properties
	It computes the minimum cost path from the seed to every node in the graph. This set of minimum paths is represented as a tree
	Running time, with N pixels:
		O(N²) time if you use an active list
		O(N log N) if you use an active priority queue (heap)
		takes fraction of a second for a typical (640x480) image
	Once this tree is computed once, we can extract the optimal path from any point to the seed in O(N) time.
		it runs in real time as the mouse moves
	What happens when the user specifies a new seed?


Graph
	node for each pixel, link between pixels
	specify a few pixels as foreground and background
		create an infinite cost link from each bg pixel to the “t” node
		create an infinite cost link from each fg pixel to the “s” node
	compute min cut that separates s from t
	how to define link cost between neighboring pixels?


	Our visual system is proof that automatic methods are possible
		classical image segmentation methods are automatic





	Argument for user-directed methods?
		only user knows desired scale/object of interest


Fully-connected graph
	node for every pixel
	link between every pair of pixels, p,q
	cost c_pq for each link
		c_pq measures similarity
			similarity is inversely proportional to difference in color and position


Break Graph into Segments
	Delete links that cross between segments
	Easiest to break links that have low cost (similarity)
		similar pixels should be in the same segments
		dissimilar pixels should be in different segments


	Link Cut
		set of links whose removal makes a graph disconnected
		cost of a cut:


Treat the links as springs and shake the system
	elasticity proportional to cost
	vibration “modes” correspond to segments
		can compute these by solving an eigenvector problem
		http://www.cis.upenn.edu/~jshi/papers/pami_ncut.pdf


	Goal
		Break the image into K regions (segments)
		Solve this by reducing the number of colors to K and mapping each pixel to the closest color


	How to choose the representative colors?
		This is a clustering problem!


	Suppose I tell you the cluster centers c_i
		Q: how to determine which points to associate with each c_i?


K-means clustering algorithm
	Randomly initialize the cluster centers, c₁, ..., c_K
	Given cluster centers, determine points in each cluster
		For each point p, find the closest c_i. Put p into cluster i
	Given points in each cluster, solve for c_i
		Set c_i to be the mean of points in cluster i
	If c_i have changed, repeat Step 2

Java demo: http://home.dei.polimi.it/matteucc/Clustering/tutorial_html/AppletKM.html

Properties
	Will always converge to some solution
	Can be a “local minimum”
		does not always find the global minimum of objective function:


Basic questions
	what’s the probability that a point x is in cluster m?
	what’s the shape of each cluster?
K-means doesn’t answer these questions

Basic idea
	instead of treating the data as a bunch of points, assume that they are all generated by sampling a continuous function
	This function is called a generative model
		defined by a vector of parameters θ


One generative model is a mixture of Gaussians (MOG)
	K Gaussian blobs with means μ_b covariance matrices V_b, dimension d
		blob b defined by:

	blob b is selected with probability
	the likelihood of observing x is a weighted mixture of Gaussians


	where


	Goal
		find blob parameters θ that maximize the likelihood function:

	Approach:
		E step: given current guess of blobs, compute ownership of each point
		M step: given ownership probabilities, update blobs to maximize likelihood function
		repeat until convergence


	E-step
		compute probability that point x is in blob i, given current guess of θ



	M-step
		compute probability that blob b is selected


		mean of blob b


		covariance of blob b






	http://lcn.epfl.ch/tutorial/english/gaussian/html/index.html


Turns out this is useful for all sorts of problems
	any clustering problem
	any model estimation problem
	missing data problems
	finding outliers
	segmentation problems
		segmentation based on color
		segmentation based on motion
		foreground/background separation
	...


	Local minima
		k-means is NP-hard even with k=2


	Need to know number of segments
		solutions: AIC, BIC, Dirichlet process mixture


	Need to choose generative model


	Iterative Mode Search
		Initialize random seed, and window W
		Calculate center of gravity (the “mean”) of W:
		Translate the search window to the mean
		Repeat Step 2 until convergence


	Approach
		Initialize a window around each point
		See where it shifts—this determines which segment it’s in
		Multiple points will shift to the same segment


Useful to take into account spatial information
	instead of (R, G, B), run in (R, G, B, x, y) space
	D. Comaniciu, P. Meer, Mean shift analysis and applications, 7th International Conference on Computer Vision, Kerkyra, Greece, September 1999, 1197-1203.
		http://www.caip.rutgers.edu/riul/research/papers/pdf/spatmsft.pdf



		Mortensen and Barrett, “Intelligent Scissors for Image Composition,” Proc. SIGGRAPH 1995.
		Boykov and Jolly, “Interactive Graph Cuts for Optimal Boundary & Region Segmentation of Objects in N-D images,” Proc. ICCV, 2001.
		Shi and Malik, “Normalized Cuts and Image Segmentation,” Proc. CVPR 1997.
		Comaniciu and Meer, “Mean shift analysis and applications,” Proc. ICCV 1999.