# Lecture12 - Graph-based Segmentation

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```					                               02/25/10

Graph-based Segmentation

Computer Vision
CS 543 / ECE 549
University of Illinois

Derek Hoiem
Last class

• Gestalt cues and principles of organization

• Mean-shift segmentation
– Good general-purpose segmentation method
– Generally useful clustering, tracking technique

• Watershed segmentation
– Good for hierarchical segmentation
– Use in combination with boundary prediction
Today’s class

• Treating the image as a graph
– Normalized cuts segmentation
– MRFs Graph cuts segmentation

• Recap

• Go over HW2 instructions
Images as graphs

i
wij
c
j

• Fully-connected graph
– node for every pixel
– link between every pair of pixels, p,q
– similarity wij for each link
Source: Seitz
Similarity matrix

Increasing sigma
Segmentation by Graph Cuts

w

A        B       C

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

Source: Seitz
Cuts in a graph

B
A

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

One idea: Find minimum cut
• gives you a segmentation
• fast algorithms exist for doing this

Source: Seitz
But min cut is not always the best cut...
Cuts in a graph

B
A

Normalized Cut
• a cut penalizes large segments
• fix by normalizing for size of segments

• volume(A) = sum of costs of all edges that touch A

Source: Seitz
Recursive normalized cuts
1. Given an image or image sequence, set up a weighted
graph: G=(V, E)
–    Vertex for each pixel
–    Edge weight for nearby pairs of pixels

2. Solve for eigenvectors with the smallest eigenvalues:
(D − W)y = λDy
–    Use the eigenvector with the second smallest eigenvalue
to bipartition the graph
–    Note: this is an approximation

4. Recursively repartition the segmented parts if
necessary

Details:   http://www.cs.berkeley.edu/~malik/papers/SM-ncut.pdf
Normalized cuts results
Normalized cuts: Pro and con
•    Pros
–   Generic framework, can be used with many
different features and affinity formulations
–   Provides regular segments
•    Cons
–   Need to chose number of segments
–   High storage requirement and time complexity
–   Bias towards partitioning into equal segments
• Usage
–   Use for oversegmentation when you want
regular segments
Graph cuts segmentation
Markov Random Fields
Node yi: pixel label

Edge: constrained
pairs

Cost to assign a label to        Cost to assign a pair of labels to
each pixel                       connected pixels

Energy(y; , data)   1 ( yi ; , data)           2   ( yi , y j ; , data)
i              i , jedges
Markov Random Fields
Unary potential
• Example: “label smoothing” grid                               0: -logP(yi = 0 ; data)
1: -logP(yi = 1 ; data)

Pairwise Potential
0    1
0 0    K
1 K    0

Energy(y; , data)   1 ( yi ; , data)           2   ( yi , y j ; , data)
i                i , jedges
Solving MRFs with graph cuts
Source (Label 0)

Cost to assign to 0

Cost to split nodes

Cost to assign to 1
Sink (Label 1)

Energy(y; , data)   1 ( yi ; , data)           2   ( yi , y j ; , data)
i           i , jedges
Solving MRFs with graph cuts
Source (Label 0)

Cost to assign to 0

Cost to split nodes

Cost to assign to 1
Sink (Label 1)

Energy(y; , data)   1 ( yi ; , data)           2   ( yi , y j ; , data)
i           i , jedges
Grab cuts and graph cuts

Magic Wand     Intelligent Scissors         GrabCut
(198?)     Mortensen and Barrett (1995)

User
Input

Result

Regions            Boundary           Regions & Boundary

Source: Rother
Colour Model

R                                     R
Foreground &             Iterated
Background               graph cut       Foreground

Background   G                          Background
G

Gaussian Mixture Model (typically 5-8 components)

Source: Rother
Graph cuts
Boykov and Jolly (2001)
Foreground
(source)
Image
Min Cut

Background
(sink)
Cut: separating source and sink; Energy: collection of edges
Min Cut: Global minimal enegry in polynomial time

Source: Rother
Graph cuts segmentation
1. Define graph
– usually 4-connected or 8-connected
2. Define unary potentials
– Color histogram or mixture of Gaussians for
background and foreground
 P(c( x); foreground ) 
unary _ potential( x)   log                         
 P(c( x);              
           background ) 
3. Define pairwise potentials
  c( x)  c( y ) 2 
                    
edge _ potential( x, y)  k1  k2 exp                    

       2 2         

4. Apply graph cuts
foreground, background models
Moderately straightforward
examples

… GrabCut completes automatically
GrabCut – Interactive Foreground Extraction   10
Difficult Examples

Camouflage &
Fine structure             Harder Case
Low Contrast

Initial
Rectangle

Initial
Result

GrabCut – Interactive Foreground Extraction        11
Using graph cuts for recognition

TextonBoost (Shotton et al. 2009 IJCV)
Using graph cuts for recognition

Unary Potentials

Alpha Expansion
Graph Cuts

TextonBoost (Shotton et al. 2009 IJCV)
Limits of graph cuts

• Associative: edge potentials penalize different labels
Must satisfy

• If not associative, can sometimes clip potentials

• Approximate for multilabel
– Alpha-expansion or alpha-beta swaps
Graph cuts: Pros and Cons
• Pros
– Very fast inference
– Can incorporate recognition or high-level priors
– Applies to a wide range of problems (stereo,
image labeling, recognition)
• Cons
– Not always applicable (associative only)
– Need unary terms (not used for generic
segmentation)
• Use whenever applicable

• Normalized cuts and image segmentation (Shi and Malik)
http://www.cs.berkeley.edu/~malik/papers/SM-ncut.pdf

• N-cut implementation
http://www.seas.upenn.edu/~timothee/software/ncut/ncut.html

• Graph cuts
– http://www.cs.cornell.edu/~rdz/graphcuts.html
– Classic paper: What Energy Functions can be Minimized via Graph
Cuts? (Kolmogorov and Zabih, ECCV '02/PAMI '04)
Recap of Grouping and Fitting
Line detection and Hough transform
• Canny edge detector =
smooth  derivative  thin 

• Generalized Hough transform =
points vote for shape parameters

• Straight line detector =
check for straightness
Robust fitting and registration
Key algorithm
• RANSAC
Clustering
Key algorithm
• Kmeans
EM and Mixture of Gaussians
Tutorials:
http://www.cs.duke.edu/courses/spring04/cps196.1/.../EM/tomasiEM.pdf
Segmentation
• Mean-shift segmentation
– Flexible clustering method, good segmentation

• Watershed segmentation
– Hierarchical segmentation from soft boundaries

• Normalized cuts
– Produces regular regions
– Slow but good for oversegmentation

• MRFs with Graph Cut
– Incorporates foreground/background/object
model and prefers to cut at image boundaries
– Good for interactive segmentation or
recognition
Next section: Recognition
• How to recognize
– Specific object instances
– Faces
– Scenes
– Object categories
– Materials

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