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Interactive Graph Cuts
for Optimal Boundary & Region Segmentation of Objects in N-D Images Yuri Y. Boykov and Marie-Pierre Jolly
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From [3]
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Interactive Graph Cuts for Optimal Boundary & Region - - PowerPoint PPT Presentation
Interactive Graph Cuts for Optimal Boundary & Region - - PowerPoint PPT Presentation
Interactive Graph Cuts for Optimal Boundary & Region Segmentation of Objects in N-D Images Yuri Y. Boykov and Marie-Pierre Jolly 1 From [3] 2 Vision problems as image labeling depth (stereo) object index (segmention) original
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Vision problems as image labeling
- depth (stereo)
- object index (segmention)
- original intensity (image restoration)
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Labeling problems can be cast in terms of energy minimization
Labeling of pixels Penalty for pixel labeling Interaction between neighboring pixels. Smoothing term.
E(L) =
- p∈P
Dp(Lp) +
- p,q∈N
Vp,q(Lp, Lq)
Dp : Lp → L : P → Lp
Vp,q : P x P →
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Energy minimization can be solved with graph cuts
S terminal S terminal T terminal T terminal
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- Energy function and graph construction
- Min-cut of graph minimizes energy
- Summar max-flow/min-cut algorithms
- Rest of bone segmentation example
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Some Notation
L = (L1, . . . , Lp, . . . , L|P |)
P = set of pixels p
N = set of unordered pairs (p, q) of neighbors in P
Binary vector representing a binary segmentation
G = (V, E) graph with nodes, V, and edges, E B = set of user defined background pixels O = set of user defined object pixels
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Labeling problems can be cast in terms of energy minimization
Labeling of pixels Penalty for pixel labeling Interaction between neighboring pixels. Smoothing term.
E(L) =
- p∈P
Dp(Lp) +
- p,q∈N
Vp,q(Lp, Lq)
Dp : Lp → L : P → Lp
Vp,q : P x P →
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Their Energy Function
E(L) = λ ·
- p∈P
Rp(Lp) +
- (p,q)∈N
B(p, q) · δ(Lp, Lq) Dp(Lp) becomes regional term Rp(Lp) Vp,q becomes boundary term B(p, q) · δ(Lp, Lq)
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Regional term
E(L) = λ ·
- p∈P
Rp(Lp) +
- (p,q)∈N
B(p, q) · δ(Lp, Lq)
Penalize pixel label based on local properties Negative log-likelihood of intensity
Rp(obj) = − ln Pr(Ip|O) Rp(bkq) = − ln Pr(Ip|B)
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Boundary term: penalize dissimilar neighbors
E(L) = λ ·
- p∈P
Rp(Lp) +
- (p,q)∈N
B(p, q) · δ(Lp, Lq) B(p, q) ∝ exp(−(Ip − Iq)2 2σ2 ) · 1 dist(p, q)
r p s q
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Boundary term: penalize dissimilar neighbors
B(p, q) ∝ exp(−(Ip − Iq)2 2σ2 ) · 1 dist(p, q)
r p s q
E(L) = λ ·
- p∈P
Rp(Lp) +
- (p,q)∈N
B(p, q) · |Lp − Lq|
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Graph construction: cost of n-links
Object Background
c(p, q) = B(p, q) if (p, q) ∈ N B(p, q)
T S
B(p, q) ∝ exp(−(Ip − Iq)2 2σ2 ) · 1 dist(p, q)
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Graph construction: cost of t-link (p, S)
Object Background
If p ∈ O then c(p, q) = K K K = 1 + max
p∈P
- q:(p,q)∈N
B(p, q)
T S
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Object Background
If p / ∈ O ∪ B then c(p, q) = λ · Rp(bkg)
Graph construction: cost of t-link (p, S)
T S
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Object Background
Graph construction: cost of t-link (p, S)
T S
If p ∈ B then c(p, q) = 0
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Claim: min-cut of graph minimizes energy
- Min-cut on G is a feasible cut
- Each feasible cut has a unique binary
segmentation
- Segmentation associated with min-cut that
satisfies user defined constraints minimizes the energy function
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Summary of max-flow/min-cut algorithms
- Augmenting paths (Ford and Fulkerson)
- Push-relabel (Goldberg and Tarjan)
- Their implementation (see [2])
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From [3]
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Citations
- [1] Kolmogorov, V. and Zabih, R. What energy functions can be
minimized via graph cuts? IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 26, no. 2, pp. 147-159, February 2004.
- [2] Boykov, Y. and Kolmogorov, V. An experimental comparison of min-
cut/max-flow algoritms for energy minimization in computer vision. IEEE
- Trans. Pattern Analysis and Machine Intelligence, vol. 26, no. 9, pp.
1124-1137, September 2004.
- [3] Boykov, Y., Torr, T. and Zabih, R. Tutorial on Discrete Optimization
Methods in Computer Vision. ECCV 2004
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