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Graph-based Segmentation
Computer VisionCS 543 / ECE 549
University of Illinois
Derek Hoiem
02/25/10
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
i
Images as graphs
• Fully-connected graph– node for every pixel– link between every pair of pixels, p,q– similarity wij for each link
j
wij
c
Source: Seitz
Similarity matrix
Increasing sigma
Segmentation by Graph Cuts
• 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
w
A B C
Source: Seitz
Cuts in a graph
• Link Cut– set of links whose removal makes a graph disconnected– cost of a cut:
AB
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
AB
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 cuts1. 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
http://www.cs.berkeley.edu/~malik/papers/SM-ncut.pdfDetails:
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
edgesji
jii
i datayydataydataEnergy,
21 ),;,(),;(),;( y
Node yi: pixel label
Edge: constrained pairs
Cost to assign a label to each pixel
Cost to assign a pair of labels to connected pixels
Markov Random Fields
• Example: “label smoothing” gridUnary potential
0 10 0 K1 K 0
Pairwise Potential
0: -logP(yi = 0 ; data)1: -logP(yi = 1 ; data)
edgesji
jii
i datayydataydataEnergy,
21 ),;,(),;(),;( y
Solving MRFs with graph cuts
edgesji
jii
i datayydataydataEnergy,
21 ),;,(),;(),;( y
Source (Label 0)
Sink (Label 1)
Cost to assign to 0
Cost to assign to 1
Cost to split nodes
Solving MRFs with graph cuts
edgesji
jii
i datayydataydataEnergy,
21 ),;,(),;(),;( y
Source (Label 0)
Sink (Label 1)
Cost to assign to 0
Cost to assign to 1
Cost to split nodes
Grab cuts and graph cutsGrab cuts and graph cuts Grab cuts and graph cutsGrab cuts and graph cuts
User Input
Result
Magic Wand (198?)
Intelligent ScissorsMortensen and Barrett (1995)
GrabCut
Regions Boundary Regions & Boundary
Source: Rother
Colour ModelColour Model Colour ModelColour Model
Gaussian Mixture Model (typically 5-8 components)
Foreground &Background
Background
Foreground
BackgroundG
R
G
RIterated graph cut
Source: Rother
Graph cutsGraph cuts Boykov and Jolly (2001)Boykov and Jolly (2001)
Graph cutsGraph cuts Boykov and Jolly (2001)Boykov and Jolly (2001)
ImageImage Min CutMin Cut
Cut: separating source and sink; Energy: collection of edges
Min Cut: Global minimal enegry in polynomial time
Foreground Foreground (source)(source)
BackgroundBackground(sink)(sink)
Source: Rother
Graph cuts segmentation1. Define graph
– usually 4-connected or 8-connected2. Define unary potentials
– Color histogram or mixture of Gaussians for background and foreground
3. Define pairwise potentials
4. Apply graph cuts5. Return to 2, using current labels to compute
foreground, background models
2
2
21 2
)()(exp),(_
ycxc
kkyxpotentialedge
));((
));((log)(_
background
foreground
xcP
xcPxpotentialunary
Moderately straightforward Moderately straightforward
examples examples
Moderately straightforward Moderately straightforward
examples examples
… GrabCut completes automatically
GrabCut – Interactive Foreground Extraction 10
Difficult ExamplesDifficult Examples Difficult ExamplesDifficult Examples
Camouflage &
Low ContrastHarder CaseFine structure
Initial Rectangle
InitialResult
GrabCut – Interactive Foreground Extraction 11
Using graph cuts for recognition
TextonBoost (Shotton et al. 2009 IJCV)
Using graph cuts for recognition
TextonBoost (Shotton et al. 2009 IJCV)
Unary Potentials
Alpha Expansion Graph Cuts
Limits of graph cuts
• Associative: edge potentials penalize different labels
• If not associative, can sometimes clip potentials
• Approximate for multilabel– Alpha-expansion or alpha-beta swaps
Must satisfy
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
Further reading and resources
• 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 threshold link
• Generalized Hough transform = points vote for shape parameters
• Straight line detector = canny + gradient orientations orientation binning linking 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.pdfhttp://www-clmc.usc.edu/~adsouza/notes/mix_gauss.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