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Overview of graph cuts. Outline. Introduction S-t Graph cuts Extension to multi-label problems Compare simulated annealing and alpha-expansion algorithm. Introduction. Discrete energy minimization methods that can be applied to Markov Random Fields (MRF) with binary labels or multi-labels. - PowerPoint PPT Presentation
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2010/5/172010/5/17 11
Overview of graph cutsOverview of graph cuts
2010/5/172010/5/17 22
OutlineOutlineIntroductionIntroduction
S-t Graph cuts S-t Graph cuts
Extension to multi-label problemsExtension to multi-label problems
Compare simulated annealing and alpha-Compare simulated annealing and alpha-expansion algorithmexpansion algorithm
2010/5/172010/5/17 33
IntroductionIntroductionDiscrete energy minimization methods Discrete energy minimization methods that can be applied to Markov Random that can be applied to Markov Random Fields (MRF) with binary labels or multi-Fields (MRF) with binary labels or multi-labels.labels.
2010/5/172010/5/17 44
OutlineOutlineIntroductionIntroduction
S-t Graph cutsS-t Graph cuts
Extensions to multi-label problemsExtensions to multi-label problems
Compare simulated annealing and alpha-Compare simulated annealing and alpha-expansion algorithmexpansion algorithm
2010/5/172010/5/17 55
Max flow / Min cut Max flow / Min cut Flow networkFlow network
Maximize amount of flows from source to sinkMaximize amount of flows from source to sink
Equal to minimum capacity removed from the Equal to minimum capacity removed from the network that no flow can pass from the source to the network that no flow can pass from the source to the sink sink
ts
Max-flow/Min-cut method : Augmenting paths (Ford Fulkerson Algorithm)
2010/5/172010/5/17 66
A subset of edges such that source and sink A subset of edges such that source and sink become separatedbecome separated
G(C)=<V,E-C>G(C)=<V,E-C>
the cost of a cut :the cost of a cut :
Minimum cut : a cut whose cost is the least over Minimum cut : a cut whose cost is the least over all cutsall cuts
S-t Graph CutS-t Graph Cut
C E
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How to separate a graph to two How to separate a graph to two class?class?
Two pixels p1 and p2 corresponds to two Two pixels p1 and p2 corresponds to two class s and t.class s and t.
Pixels p in the Graph classify by Pixels p in the Graph classify by subtracting p with two pixels p1,p2. subtracting p with two pixels p1,p2. d1=(p-p1), d2 = (p-p2)d1=(p-p1), d2 = (p-p2)
If d1 is closer zero than d2, p is class s.If d1 is closer zero than d2, p is class s.
Absolute of d1 and d2 Absolute of d1 and d2
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Noise in the boundary of two classNoise in the boundary of two class
The classified graph may have the noise The classified graph may have the noise occurs nearing the pixel (p1+p2)/2occurs nearing the pixel (p1+p2)/2
Adding another constrain (smoothing) to Adding another constrain (smoothing) to prevent this problem.prevent this problem.
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energy functionenergy function
Npq
qppqp
pp ffwfDfE )()()(
},{ tsf p
t-links n-links
Boundary termRegional term
pqw
n-links
t
s a cut C(s)pD
(t)pD
t-lin
k
t-link
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S-t Graph cuts for optimal S-t Graph cuts for optimal boundary detectionboundary detection
n-links
t
s a cut Chard constraint
hard constraint
Minimum cost cut can be computed in polynomial
time
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Global minimized for binary Global minimized for binary energy functionenergy function
Characterization of Characterization of binarybinary energies that can be energies that can be globally minimized by globally minimized by s-ts-t graph cuts graph cuts
p,q(f ) D (f ) V (f , f )p p p qp pq N
E
f { , }p s t
E(f) can be minimized by s-t graph cuts
V( , ) V( , ) V( , ) V( , )s s t t s t t s
t-links n-links
Boundary termRegional term
(regular function)
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Regular FRegular F22 functions: functions:
What Energy Functions Can Be What Energy Functions Can Be Minimized via Graph Cuts?Minimized via Graph Cuts?
2010/5/172010/5/17 1313
OutlineOutlineIntroductionIntroduction
S-t Graph cutsS-t Graph cuts
Extensions to multi-label problemsExtensions to multi-label problems
Compare simulated annealing and alpha-Compare simulated annealing and alpha-expansion algorithmexpansion algorithm
2010/5/172010/5/17 1414
Multi way Graph cut algorithmMulti way Graph cut algorithmNP-hard problem(3 or more labels)NP-hard problem(3 or more labels) two labels can be solved via two labels can be solved via s-ts-t cuts (Greig et. al 1989) cuts (Greig et. al 1989)
Two approximation algorithmsTwo approximation algorithms(Boykov(Boykov et.al 1998,20et.al 1998,2001)01)
Basic idea : Basic idea : break multi-way cut computation into a sbreak multi-way cut computation into a sequence of binary s-t cuts.equence of binary s-t cuts. Alpha-expansionAlpha-expansion Each label competes with the other labels for space in the imaEach label competes with the other labels for space in the ima
gege Alpha-beta swapAlpha-beta swap Define a move which allows to change pixels from alpha to beDefine a move which allows to change pixels from alpha to be
ta and beta to alphata and beta to alpha
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other labelsa
Alpha-expansion moveAlpha-expansion moveBreak multi-way cut computation into a sequence of binary s-t cuts
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Alpha-expansion algorithmAlpha-expansion algorithm
-expansion
algorithm
(|L| iterations)
Stop when no expansion move would decrease energy
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Alpha-expansion algorithmAlpha-expansion algorithmGuaranteed approximation ratio by the algorithm:Guaranteed approximation ratio by the algorithm:Produces a labeling Produces a labeling ff such that , such that ,
where where f*f* is the global minimum is the global minimum
and and *2* fEkfEfE
:,min
:,max
V
Vk
Prove in : efficient graph-based energy minimization methods in computer vision
2010/5/172010/5/17 1818
alphaalpha-expansion moves-expansion moves
initial solution
-expansion
-expansion
-expansion
-expansion
-expansion
-expansion
-expansion
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Alpha-Beta swap algorithmAlpha-Beta swap algorithm
- -swap
algorithm
Handles more general energy function
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MovesMoves
αexpansionα-βswapInitial labeling
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MetricMetricSemi-metric Semi-metric – –
If V also satisfies the triangle inequalityIf V also satisfies the triangle inequality
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Alpha-expansion : MetricAlpha-expansion : MetricAlpha-expansion satisfy the regular functionAlpha-expansion satisfy the regular function
Alpha-beta swapAlpha-beta swap
Prove in: what energy functions can be minimized via graph cuts?
2010/5/172010/5/17 2323
Different types of InteractionDifferent types of Interaction V V
V(dL)
dL=Lp-Lq
Potts model
“discontinuity preserving”Interactions V
V(dL)
dL=Lp-Lq
“Convex”Interactions V
V(dL)
dL=Lp-Lq
V(dL)
dL=Lp-Lq
“linear”
model
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convexconvex vs. vs. discontinuity-preservingdiscontinuity-preserving””
“linear” V
truncated “linear” V
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The use of Alpha-expansion and The use of Alpha-expansion and alpha-beta swap alpha-beta swap
Three energy function, each with a quadratic DThree energy function, each with a quadratic Dpp..– E1 = DE1 = Dpp + min(K,|f + min(K,|fpp-f-fqq||22))– E2 uses the Potts modelE2 uses the Potts model– E3 = DE3 = Dpp + min(K,|f + min(K,|fpp-f-fqq|)|)
E1 : semi-metric (use ) E1 : semi-metric (use ) E2,E3 : metric (can use both) E2,E3 : metric (can use both)
p qV= (f f )
- -s wap
2010/5/172010/5/17 2626
OutlineOutlineIntroductionIntroduction
S-t Graph cutsS-t Graph cuts
Extensions to multi-label problemsExtensions to multi-label problems
Compare simulated annealing and alpha-Compare simulated annealing and alpha-expansion algorithmexpansion algorithm
2010/5/172010/5/17 2727
Single “one-pixel” move (Simulated annealing)
Single alpha-expansion move
Only one pixel change its label at a time
Large number of pixels can change their labels simultaneously
Computationally intensive O(2^n)
(s-t cuts)
2010/5/172010/5/17 2828
參考文獻參考文獻Graph Cuts in Vision and Graphics: Graph Cuts in Vision and Graphics: Theories and ApplicationTheories and Application
Fast Approximate Energy Minimization via Fast Approximate Energy Minimization via Graph Cuts , 2001Graph Cuts , 2001
What energy functions can be minimized What energy functions can be minimized via graph cuts?via graph cuts?