Graph cut, convex relaxation and continuous max-flow problem - PowerPoint PPT Presentation

α -expansion and α − β swap ◮ Related to garph cut, α -expansion and α − β swap are mostly popular. ◮ Approximations are made and upper bounded has been given. ◮ Boykov-Veksler-Zahib (1999).

Multiphase problems – Approach I Each point x ∈ Ω is labelled by i = 1 , 2 , · · · n . 3 u ( x ) = i , 2.5 2 1.5 ◮ One label function is enough 1 0 20 for any n phases. 40 ◮ More generall 60 100 90 80 80 70 60 50 u ( x ) = ℓ i , i = 1 , 2 , · · · n . 40 30 20 100 10 0

Multiphase problems – Approach II Each point x ∈ Ω is labelled by a vector function: u ( x ) = ( u 1 (2) , u 2 ( x ) , · · · u d ( x )) .

Multiphase problems – Approach II Each point x ∈ Ω is labelled by a vector function: u ( x ) = ( u 1 (2) , u 2 ( x ) , · · · u d ( x )) . ◮ Multiphase: Total number of phases n = 2 d . (Chan-Vese) u i ( x ) ∈ { 0 , 1 } .

Multiphase problems – Approach II Each point x ∈ Ω is labelled by a vector function: u ( x ) = ( u 1 (2) , u 2 ( x ) , · · · u d ( x )) . ◮ Multiphase: Total number of phases n = 2 d . (Chan-Vese) u i ( x ) ∈ { 0 , 1 } . ◮ More than binary labels: Total number of phases n = B d . u i ( x ) ∈ { 0 , 1 , 2 , · · · B } .

Multiphase problems – Approach III We need to identify n 1 0.8 0.6 characteristic functions 0.4 0.2 ψ i ( x ) , i = 1 , 2 · · · n : 0 0 20 40 60 n 100 90 80 80 60 70 50 40 30 100 10 20 0 � ψ i ( x ) ∈ { 0 , 1 } , ψ i ( x ) = 1 . 1 i =1 0.8 0.6 0.4 0.2 0 ◮ Relation between Approach I 0 20 40 and III: 60 100 80 90 80 70 60 50 40 20 30 100 10 0 u ( x ) = i , i = 1 , 2 , · · · n . 1 0.8 0.6 0.4 n 0.2 0 � 0 u ( x ) = i ψ i ( x ) . 20 40 60 i =1 90 100 80 80 70 60 50 30 40 20 100 10 0

Multiphase problems Multpihase problem (I) Special graph cut for Chan-Vese approach

CV Graph construction (Bae-Tai EMMCVPR2009) One pixel two pixels ◮ Associate two vertices to each grid point ( v p , 1 and v p , 2 ) ◮ For any cut ( V s , V t ) If v p , i ∈ V s then φ i = 1 for i = 1 , 2 ◮ If v p , i ∈ V t then φ i = 0 for i = 1 , 2 ◮ ◮ Figure left: graph corresponding to one grid point p ◮ Figure right: graph corresponding to two grid points p and q Red: Data edges, constituting E data ( φ 1 , φ 2 ) ◮ ◮ Blue: Regularization edges with weight w pq

Cuts for the CV-graph(Bae-Tai, EMMCVPR2009)

Minimization by graph cut Graph construction ◮ Linear system for finding edge weights = | c 2 − u 0  p | β A ( p ) + B ( p )   = | c 3 − u 0 p | β C ( p ) + D ( p )  = | c 1 − u 0 p | β A ( p ) + E ( p ) + D ( p )   = | c 4 − u 0 p | β B ( p ) + F ( p ) + C ( p )  such that E ( p ) , F ( p ) ≥ 0 ◮ For each p , E data ( φ 1 p , φ 2 p ) interaction between two binary variables. p Linear system has solution iff E data ( φ 1 p , φ 2 p ) is submodular. p

Global minimizer – conditions Graph construction ◮ Restriction E ( p ) , F ( p ) ≥ 0 implies p | β + | c 4 − u 0 p | β = A ( p ) + B ( p ) + C ( p ) + D ( p ) + E ( p ) + F ( p ) | c 1 − u 0 p | β + | c 3 − u 0 ≥ A ( p ) + B ( p ) + C ( p ) + D ( p ) = | c 2 − u 0 p | β . ◮ Therefore it is sufficient that | c 2 − I | β + | c 3 − I | β ≤ | c 1 − I | β + | c 4 − I | β , ∀ I ∈ [0 , L ] , ◮ At most three edges are required for a general submodular function of two binary variables (Kolmogorov et. al.)

Global minimizer – Guarantees Submodularity condition | c 2 − I | β + | c 3 − I | β ≤ | c 1 − I | β + | c 4 − I | β , ∀ I ∈ [0 , L ] , ◮ Proposition 1: Let 0 ≤ c 1 < c 2 < c 3 < c 4 . Condition is satisfied for all I ∈ [ c 2 − c 1 , c 4 − c 3 ] . 2 2 ◮ Proposition 2: Let 0 ≤ c 1 < c 2 < c 3 < c 4 . There exists a B ∈ N such that condition is satisfied for any β ≥ B .

CV-graph – negative weights ◮ There are infinite many solution for A , B , C , D , E , F for each pixel. ◮ We can guarantee A > 0 , B > 0 , C > 0 , D > 0. If one of E , F is negative, there is a modified graph. ◮ Some arts: sort c i as c 1 < c 2 < c 3 < c 4 , then choose f 1 ( p ) = | c 2 − u 0 p | β , f 2 ( p ) = | c 3 − u 0 p | β , f 3 ( p ) = | c 1 − u 0 p | β , f 4 ( p ) = | c 4 − u 0 p | β .

Numerical experiments Experiment 1 Figure : Experiment 3: (a) Input image, (b) ground truth, (c) gradient descent, (d) our approach, (e) alpha expansion, (f) alpha-beta swap.

Numerical experiments Experiment 2 Figure : Experiment 3: (a) Input image, (b) ground truth, (c) gradient descent, (d) our approach, (e) alpha expansion, (f) alpha-beta swap.

Numerical experiments Experiment 3 ◮ L 2 data term ( β = 2) ◮ Right: Input image. ◮ Left: Output.

Numerical experiments Experiment 4, non-submodular minimization ◮ L 1 data term ( β = 1) ◮ Right: Input image. ◮ Left: Set of pixels where residual criterion was not satisfied (empty set).

Numerical experiments Experiment 4, non-submodular minimization ◮ L 1 data term ( β = 1) ◮ Right: Input image. ◮ Left: Output (global solution).

multiphase Chan-Vese model Exact convex formulation for the Multiphase Chan-Vese model by Continuous max-flow/min-cuts

Multiphase level set representation of CV model � � |∇ H ( φ 1 ) | + α |∇ H ( φ 2 ) | + E data ( φ 1 , φ 2 ) , min α φ 1 ,φ 2 , { c i } 4 Ω Ω i =1 where � E data ( φ 1 , φ 2 ) = { H ( φ 1 ) H ( φ 2 ) | c 2 − u 0 | β + H ( φ 1 )(1 − H ( φ 2 )) | c 1 − u 0 | β Ω +(1 − H ( φ 1 )) H ( φ 2 ) | c 4 − u 0 | β +(1 − H ( φ 1 ))(1 − H ( φ 2 )) | c 3 − u 0 | β } dx . Ω 1 = { x ∈ Ω s.t. φ 1 ( x ) > 0 , φ 2 ( x ) < 0 } Ω 2 = { x ∈ Ω s.t. φ 1 ( x ) > 0 , φ 2 ( x ) > 0 } Ω 3 = { x ∈ Ω s.t. φ 1 ( x ) < 0 , φ 2 ( x ) < 0 } Ω 4 = { x ∈ Ω s.t. φ 1 ( x ) < 0 , φ 2 ( x ) > 0 }

Binary formulation of multiphase Chan-Vese model Wish to obtain global optimization framework for � � |∇ φ 1 | dx + α |∇ φ 2 | dx + E data ( φ 1 , φ 2 ) , φ 1 ,φ 2 ∈{ 0 , 1 } α min Ω Ω with � { φ 1 φ 2 | c 2 − u 0 | β + φ 1 (1 − φ 2 ) | c 1 − u 0 | β E data ( φ 1 , φ 2 ) = Ω +(1 − φ 1 ) φ 2 | c 4 − u 0 | β + (1 − φ 1 )(1 − φ 2 ) | c 3 − u 0 | β } dx . Phase 1: φ 1 = 1 , φ 2 = 0 Phase 2: φ 1 = 1 , φ 2 = 1 Phase 3: φ 1 = 0 , φ 2 = 0 Phase 4: φ 1 = 0 , φ 2 = 1

Binary formulation of multiphase Chan-Vese model Wish to obtain global optimization framework for � � |∇ φ 1 | dx + α |∇ φ 2 | dx + E data ( φ 1 , φ 2 ) , φ 1 ,φ 2 ∈{ 0 , 1 } α min Ω Ω with � { φ 1 φ 2 | c 2 − u 0 | β + φ 1 (1 − φ 2 ) | c 1 − u 0 | β E data ( φ 1 , φ 2 ) = Ω +(1 − φ 1 ) φ 2 | c 4 − u 0 | β + (1 − φ 1 )(1 − φ 2 ) | c 3 − u 0 | β } dx . Phase 1: φ 1 = 1 , φ 2 = 0 Phase 2: φ 1 = 1 , φ 2 = 1 Phase 3: φ 1 = 0 , φ 2 = 0 Phase 4: φ 1 = 0 , φ 2 = 1 Can this non-convex problem be equivalent to a convex model???

Binary formulation of multiphase Chan-Vese model Wish to obtain global optimization framework for � � |∇ φ 1 | dx + α |∇ φ 2 | dx + E data ( φ 1 , φ 2 ) , φ 1 ,φ 2 ∈{ 0 , 1 } α min Ω Ω with � { φ 1 φ 2 | c 2 − u 0 | β + φ 1 (1 − φ 2 ) | c 1 − u 0 | β E data ( φ 1 , φ 2 ) = Ω +(1 − φ 1 ) φ 2 | c 4 − u 0 | β + (1 − φ 1 )(1 − φ 2 ) | c 3 − u 0 | β } dx . Phase 1: φ 1 = 1 , φ 2 = 0 Phase 2: φ 1 = 1 , φ 2 = 1 Phase 3: φ 1 = 0 , φ 2 = 0 Phase 4: φ 1 = 0 , φ 2 = 1 Can this non-convex problem be equivalent to a convex model??? YES!!!

Binary formulation of multiphase Chan-Vese model Wish to obtain global optimization framework for � � |∇ φ 1 | dx + α |∇ φ 2 | dx + E data ( φ 1 , φ 2 ) , φ 1 ,φ 2 ∈{ 0 , 1 } α min Ω Ω with � { φ 1 φ 2 | c 2 − u 0 | β + φ 1 (1 − φ 2 ) | c 1 − u 0 | β E data ( φ 1 , φ 2 ) = Ω +(1 − φ 1 ) φ 2 | c 4 − u 0 | β + (1 − φ 1 )(1 − φ 2 ) | c 3 − u 0 | β } dx . Phase 1: φ 1 = 1 , φ 2 = 0 Phase 2: φ 1 = 1 , φ 2 = 1 Phase 3: φ 1 = 0 , φ 2 = 0 Phase 4: φ 1 = 0 , φ 2 = 1 Can this non-convex problem be equivalent to a convex model??? YES!!! Why ???

Continuous max-flow formulation � p 1 s ( x ) + p 2 sup s ( x ) dx s , p i Ω p i t , p 12 , q i ; i =1 , 2 subject to p 1 s ( x ) ≤ C ( x ) , p 2 s ( x ) ≤ D ( x ) , p 1 t ( x ) ≤ A ( x ) , p 2 t ≤ B ( x ) , | q i ( x ) | ≤ α − F ( x ) ≤ p 12 ( x ) ≤ E ( x ) , div q 1 ( x ) − p 1 s ( x ) + p 1 t ( x ) + p 12 ( x ) = 0 div q 2 ( x ) − p 2 s ( x ) + p 2 t ( x ) − p 12 ( x ) = 0

Lagrange multipliers λ 1 and λ 2 for flow conservation constraints. Lagrangian functional: � (1 − λ 1 ( x )) p 1 s ( x ) + (1 − λ 2 ( x )) p 2 t , p 12 , q i ; i =1 , 2 inf max s ( x ) dx s , p i λ 1 ,λ 2 p i Ω � λ 1 ( x ) p 1 t ( x ) + λ 2 ( x ) p 2 t ( x ) + ( λ 1 ( x ) − λ 2 ( x )) p 12 ( x ) dx + Ω � � λ 1 ( x ) div q 1 ( x ) + λ 2 ( x ) div q 2 ( x ) . + Ω Ω subject to p 1 s ( x ) ≤ C ( x ) , p 2 s ( x ) ≤ D ( x ) , p 1 t ( x ) ≤ A ( x ) , p 2 t ≤ B ( x ) , | q i ( x ) | ≤ α − F ( x ) ≤ p 12 ( x ) ≤ E ( x ) ,

Maximizing Lagrangian for all flows results in � (1 − λ 1 ( x )) C ( x )+(1 − λ 2 ( x )) D ( x )+ λ 1 ( x ) A ( x )+ λ 2 ( x ) B ( x ) dx min λ 1 ,λ 2 Ω � + max { λ 1 ( x ) − λ 2 ( x ) , 0 } E ( x ) dx − min { λ 1 ( x ) − λ 2 ( x ) , 0 } F ( x ) dx Ω � � |∇ λ 1 ( x ) | dx + α |∇ λ 2 ( x ) | dx . + α Ω Ω subject to λ 1 ( x ) , λ 2 ( x ) ∈ [0 , 1] , ∀ x ∈ Ω. = | c 2 − u 0 ( x ) | β  A ( x ) + B ( x )   = | c 3 − u 0 ( x ) | β C ( x ) + D ( x )  = | c 1 − u 0 ( x ) | β A ( x ) + E ( x ) + D ( x )   = | c 4 − u 0 ( x ) | β B ( x ) + F ( x ) + C ( x )  ◮ Convex, iff E ( x ) , F ( x ) ≥ 0 ◮ Theorem: Thresholding optimal λ 1 ( x ) and λ 2 ( x ) will give a binary global solution to multiphase Chan-Vese model

Corollaries ◮ No approximation: the global minimizer of the max-flow (convex CV) is the global minimizer of the original non-convex CV model.

Corollaries ◮ No approximation: the global minimizer of the max-flow (convex CV) is the global minimizer of the original non-convex CV model. ◮ The global minimizer is guaranteed binary ! (not true for many other convex relaxations). ◮ Why ??

Corollaries ◮ No approximation: the global minimizer of the max-flow (convex CV) is the global minimizer of the original non-convex CV model. ◮ The global minimizer is guaranteed binary ! (not true for many other convex relaxations). ◮ Why ?? � R ( u ) = |∇ u 1 | + |∇ u 2 | . Ω

Corollaries ◮ No approximation: the global minimizer of the max-flow (convex CV) is the global minimizer of the original non-convex CV model. ◮ The global minimizer is guaranteed binary ! (not true for many other convex relaxations). ◮ Why ?? � R ( u ) = |∇ u 1 | + |∇ u 2 | . Ω ◮ We can also regularize the length of the interface, then Thresholded solution is not guaranteed to be exact.

Multiphase problems ◮ A new tight relaxation with product of labels (more than binary) has been given in Goldluecke-Cremers ECCV(2010).

Multiphase problems ◮ A new tight relaxation with product of labels (more than binary) has been given in Goldluecke-Cremers ECCV(2010). ◮ The formulation can be deduced from Tight relaxation as well.

Multiphase problems ◮ A new tight relaxation with product of labels (more than binary) has been given in Goldluecke-Cremers ECCV(2010). ◮ The formulation can be deduced from Tight relaxation as well. No approximation for two-phase and four-phase Chan-Vese model (A collaboration between Bae, Lellman).

Multiphase problems ◮ A new tight relaxation with product of labels (more than binary) has been given in Goldluecke-Cremers ECCV(2010). ◮ The formulation can be deduced from Tight relaxation as well. No approximation for two-phase and four-phase Chan-Vese model (A collaboration between Bae, Lellman). More than four-phase, cannot guarantee global binary solution.

Multiphase problems ◮ A new tight relaxation with product of labels (more than binary) has been given in Goldluecke-Cremers ECCV(2010). ◮ The formulation can be deduced from Tight relaxation as well. No approximation for two-phase and four-phase Chan-Vese model (A collaboration between Bae, Lellman). More than four-phase, cannot guarantee global binary solution. ◮ Other multiphase relaxations: J. Lellmann-Kappes-Yuan-Becker-Schn¨ orr (2008), Lellmann-et-al(2009, 2010), Brown-Chan-Bresson (2011), Goldstein-Bresson-Osher (2009), Chambolle-Cremers-Pock (2009, 2012).

Multiphase problems Multiphase problem (II) Layered Graph 1 1 Boykov-Kolmogorov (PAMI 2001), Ishikawa (PAMI 2003), Darbon-Segle(JMIV, 2006), Bae-Tai (SSVM 2009)

Multiphase problems To identify n phases, we need one label function, but n labels. 3 2.5 2 1.5 1 0 20 40 60 100 90 80 80 70 60 50 40 30 20 100 10 0

Multiphase problem Figure : Need multi-labels φ ( x ) = i in Ω i , i = 1 , 2 , 3 , 4 .

Increase dimension – only need two phases |∇ φ | = |∇ u | . Figure : Just need one label: Increase the dimension, we just need u ( x , φ ) = 0 or 1.

1D signal and multiphase Figure : Left: Example cut on the graph G corresponding to a 1d image of 6 grid points. Right: Values of φ corresponding to the cut

Historical review ◮ This graph was proposed in Ishikaka (PAMI 2003).

Historical review ◮ This graph was proposed in Ishikaka (PAMI 2003). ◮ Darbon-Sigelle (JMIV, 2006), Chambolle (2006), Hochbaum (2001) has used this graph for TV minimization and related problems.

Historical review ◮ This graph was proposed in Ishikaka (PAMI 2003). ◮ Darbon-Sigelle (JMIV, 2006), Chambolle (2006), Hochbaum (2001) has used this graph for TV minimization and related problems. ◮ Using this kind of regularization, segmentation is essentially an generalization of the Quantized ROF model.

Historical review ◮ This graph was proposed in Ishikaka (PAMI 2003). ◮ Darbon-Sigelle (JMIV, 2006), Chambolle (2006), Hochbaum (2001) has used this graph for TV minimization and related problems. ◮ Using this kind of regularization, segmentation is essentially an generalization of the Quantized ROF model. ◮ Lie-Lysaker-T. (2004, 2005) is a formulation of this model with finite number of labels in a continuous domain x ∈ Ω.

Historical review ◮ This graph was proposed in Ishikaka (PAMI 2003). ◮ Darbon-Sigelle (JMIV, 2006), Chambolle (2006), Hochbaum (2001) has used this graph for TV minimization and related problems. ◮ Using this kind of regularization, segmentation is essentially an generalization of the Quantized ROF T. Pock and D. Cremers and H. model. Bischof and A. Chambolle (2010): ◮ Lie-Lysaker-T. (2004, 2005) is a gives a convex relaxation in case formulation of this model with finite both image domain and the labels are continuous. number of labels in a continuous domain x ∈ Ω.

Continuous max-flow and cut This part is based on: Bae-Yuan-T.-Boykov: CAM-10-62 (2010): a fast continuous max-flow approach to non-convex multilabeling problems.

Multiphases Costs: ρ ( u ( p ) , p ) , C ( p , q ) , i = 1 , 2 , 3 .

Discrete min-cut � � min ρ ( u v , v ) + C ( u , v ) | u v − u w | . v ∈ P ( u , v ) ∈N

Discrete max-flow � max p 1 ( v ) v ∈ P p i ( v ) ≤ ρ ( ℓ i , v ) , i = 1 , 2 , · · · n , | q i ( v , w ) | ≤ C ( v , w ) .

Continuous min-cut and max-flow Continuous min-cut: � � min ρ ( u ( x ) , x ) dx + C ( x ) |∇ u | dx . u ∈ U Ω Ω U = { u : Ω �→ { ℓ 1 , ℓ 2 , · · · ℓ n }} .

Continuous min-cut and max-flow Continuous min-cut: � � min ρ ( u ( x ) , x ) dx + C ( x ) |∇ u | dx . u ∈ U Ω Ω U = { u : Ω �→ { ℓ 1 , ℓ 2 , · · · ℓ n }} . Continuous max-flow � max p 1 ( x ) dx Ω

Continuous min-cut and max-flow Continuous min-cut: � � min ρ ( u ( x ) , x ) dx + C ( x ) |∇ u | dx . u ∈ U Ω Ω U = { u : Ω �→ { ℓ 1 , ℓ 2 , · · · ℓ n }} . Continuous max-flow � max p 1 ( x ) dx Ω p i ( x ) ≤ ρ ( ℓ i , x ) , i = 1 , 2 , · · · n ,

Continuous min-cut and max-flow Continuous min-cut: � � min ρ ( u ( x ) , x ) dx + C ( x ) |∇ u | dx . u ∈ U Ω Ω U = { u : Ω �→ { ℓ 1 , ℓ 2 , · · · ℓ n }} . Continuous max-flow � max p 1 ( x ) dx Ω p i ( x ) ≤ ρ ( ℓ i , x ) , i = 1 , 2 , · · · n , | q i ( x ) | ≤ C ( x ) ,

Continuous min-cut and max-flow Continuous min-cut: � � min ρ ( u ( x ) , x ) dx + C ( x ) |∇ u | dx . u ∈ U Ω Ω U = { u : Ω �→ { ℓ 1 , ℓ 2 , · · · ℓ n }} . Continuous max-flow � max p 1 ( x ) dx Ω p i ( x ) ≤ ρ ( ℓ i , x ) , i = 1 , 2 , · · · n , | q i ( x ) | ≤ C ( x ) , (div q i − p i + p i +1 )( x ) = 0 , q i · n = 0 .

Equivalence Theorem: The continuous min-cut and max-flow problems are dual to each other. A ”threshold” of any solutions of the ”convex” min-cut problem is a global minimizer for the ”non-convex” min-cut problem.

Algorithm Algorithm: Primal-dual algorithm is tested and is fast. Primal variables: The flow variables. Dual variables: The cut u which turn out to the Lagrangian of the ”flow conservation” constraints.

Infinite number of labels For the number of labels, instead of: U = { u : Ω �→ { ℓ 1 , ℓ 2 , · · · ℓ n }} . we use ”infinite number of labels”: U = { u : Ω �→ [ ℓ min , ℓ max ] } . This is exactly the same problem considered in: T. Pock and D. Cremers and H. Bischof and A. Chambolle (2010).

Continuous labels As the number of labels goes to the limit of infinity, the max-flow problem with the flow constraints turns into: � sup p ( ℓ min , x ) dx p , q Ω s.t. p ( ℓ, x ) ≤ ρ ( ℓ, x ) , | q ( ℓ, x ) | ≤ α, ∀ x ∈ Ω , ∀ ℓ ∈ [ ℓ min , ℓ max ] a.e. x ∈ Ω , ℓ ∈ [ ℓ min , ℓ max ] . div x q ( ℓ, x ) + ∂ ℓ p ( ℓ, x ) = 0 ,

Continuous labels As the number of labels goes to the limit of infinity, the max-flow problem with the flow constraints turns into: � sup p ( ℓ min , x ) dx p , q Ω s.t. p ( ℓ, x ) ≤ ρ ( ℓ, x ) , | q ( ℓ, x ) | ≤ α, ∀ x ∈ Ω , ∀ ℓ ∈ [ ℓ min , ℓ max ] a.e. x ∈ Ω , ℓ ∈ [ ℓ min , ℓ max ] . div x q ( ℓ, x ) + ∂ ℓ p ( ℓ, x ) = 0 , The convex min-cut problem (the dual problem to the max-flow) is: � ℓ max � � � min α |∇ x λ | − ρ ( ℓ, x ) ∂ ℓ λ ( ℓ, x ) dxd ℓ λ ( ℓ, x ) ∈ [0 , 1] Ω ℓ min � + (1 − λ ( ℓ min , x )) ρ ( ℓ min , x ) + λ ( ℓ max , x ) ρ ( ℓ max , x ) dx Ω subject to ∂ ℓ λ ( ℓ, x ) ≤ 0 , λ ( ℓ min , x ) ≤ 1 , λ ( ℓ max , x ) ≥ 0 , ∀ x ∈ Ω , ∀ ℓ ∈ [ ℓ min , ℓ max ] (3)

Continuous labels The convex min-cut problem (the dual problem to the max-flow) is: � ℓ max � � � min α |∇ x λ | − ρ ( ℓ, x ) ∂ ℓ λ ( ℓ, x ) dxd ℓ λ ( ℓ, x ) ∈ [0 , 1] ℓ min Ω � + (1 − λ ( ℓ min , x )) ρ ( ℓ min , x ) + λ ( ℓ max , x ) ρ ( ℓ max , x ) dx Ω subject to ∂ ℓ λ ( ℓ, x ) ≤ 0 , λ ( ℓ min , x ) ≤ 1 , λ ( ℓ max , x ) ≥ 0 ,

Continuous labels The convex min-cut problem (the dual problem to the max-flow) is: � ℓ max � � � min α |∇ x λ | − ρ ( ℓ, x ) ∂ ℓ λ ( ℓ, x ) dxd ℓ λ ( ℓ, x ) ∈ [0 , 1] ℓ min Ω � + (1 − λ ( ℓ min , x )) ρ ( ℓ min , x ) + λ ( ℓ max , x ) ρ ( ℓ max , x ) dx Ω subject to ∂ ℓ λ ( ℓ, x ) ≤ 0 , λ ( ℓ min , x ) ≤ 1 , λ ( ℓ max , x ) ≥ 0 , The following is the model from Poct et al (2010): (Note the difference) � ℓ max � � � min α |∇ x λ | + ρ ( ℓ, x ) | ∂ ℓ λ ( ℓ, x ) | dxd ℓ . λ ( ℓ, x ) ∈{ 0 , 1 } Ω ℓ min subject to λ ( ℓ min , x ) = 1 , λ ( ℓ max , x ) = 0 .

Algorithm Algorithm: Primal-dual algorithm is tested and is fast. Primal variables: The flow variables. Dual variables: The cut u which turn out to the Lagrangian of the ”flow conservation” constraints.

Multiphase problems Multiphase problem (III) Graph for characteristic functions 1 1 Yuan-Bae-T.-Boykov (ECCV’10)

Multi-partitioning problem Multi-partitioning problem (Pott’s model) n n � � � � min f i dx + g ( x ) ds , { Ω i } Ω i ∂ Ω i i =1 i =1 ∪ n ∩ n such that i =1 Ω i = Ω , i =1 Ω i = ∅

Multi-partitioning problem Multi-partitioning problem (Pott’s model) n n � � � � min f i dx + g ( x ) ds , { Ω i } Ω i ∂ Ω i i =1 i =1 ∪ n ∩ n such that i =1 Ω i = Ω , i =1 Ω i = ∅ Pott’s model in terms of characteristic functions n n n � � � � � min u i ( x ) f i ( x ) dx + g ( x ) |∇ u i | dx , s.t. u i ( x ) = 1 u i ( x ) ∈{ 0 , 1 } Ω Ω i =1 i =1 i =1