Graph Cut
Digital Visual Effects Yung-Yu Chuang
with slides by Fredo Durand, Ramesh Raskar
Graph cut
Graph cut
• Interactive image segmentation using graph cut
• Binary label: foreground vs. background
• User labels some pixels
– similar to trimap, usually sparser
• Exploit
– Statistics of known Fg & Bg – Smoothness of label
• Turn into discrete graph optimization
– Graph cut (min cut / max flow)
F
B F
F F F B
B B
Energy function
• Labeling: one value per pixel, F or B
• Energy(labeling) = data + smoothness
– Very general situation – Will be minimized
• Data: for each pixel
– Probability that this color belongs to F (resp. B) – Similar in spirit to Bayesian matting
• Smoothness (aka regularization):
per neighboring pixel pair
– Penalty for having different label – Penalty is downweighted if the two
pixel colors are very different – Similar in spirit to bilateral filter
One labeling (ok, not best)
Data
Smoothness
Data term
• A.k.a regional term
(because integrated over full region)
• D(L)=
i -log h[Li](Ci)• Where i is a pixel
Li is the label at i (F or B), Ci is the pixel value
h[Li] is the histogram of the observed Fg (resp Bg)
• Note the minus sign
Hard constraints
• The user has provided some labels
• The quick and dirty way to include
constraints into optimization is to replace the data term by a huge penalty if not respected.
• D(L_i)=0 if respected
• D(L_i)=K if not respected
– e.g. K=- #pixels
Smoothness term
• a.k.a boundary term, a.k.a. regularization
• S(L)=
{j, i} in N B(Ci,Cj) (Li-Lj)• Where i,j are neighbors
– e.g. 8-neighborhood
(but I show 4 for simplicity)
• (Li-Lj) is 0 if Li=Lj, 1 otherwise
• B(Ci,Cj) is high when Ci and Cj are similar, low if there is a discontinuity between those two pixels
– e.g. exp(-||Ci-Cj||2/22) – where can be a constant
or the local variance
• Note positive sign
Optimization
• E(L)=D(L)+ S(L)
• is a black-magic constant
• Find the labeling that minimizes E
• In this case, how many possibilities?
– 29 (512)
– We can try them all!
– What about megapixel images?
Labeling as a graph problem
• Each pixel = node
• Add two nodes F & B
• Labeling: link each pixel to either F or B
F
B
Desired result
Data term
• Put one edge between each pixel and F & G
• Weight of edge = minus data term
– Don’t forget huge weight for hard constraints – Careful with sign
B
F
Smoothness term
• Add an edge between each neighbor pair
• Weight = smoothness term
B
F
Min cut
• Energy optimization equivalent to min cut
• Cut: remove edges to disconnect F from B
• Minimum: minimize sum of cut edge weight
B
F
cutMin cut <=> labeling
• In order to be a cut:
– For each pixel, either the F or G edge has to be cut
• In order to be minimal
– Only one edge label per pixel can be cut (otherwise could
be added)
B
F
cutEnergy minimization via graph cuts
Labels (disparities)
d1 d2 d3
edge weight
edge weight
) , ,
(x y d3 D
) , (d1 d1 V
• Graph Cost
– Matching cost between images – Neighborhood matching term
– Goal: figure out which labels are connected to which pixels
d1 d2 d3
Energy minimization via graph cuts
Energy minimization via graph cuts
d1 d2 d3
• Graph Cut
– Delete enough edges so that
• each pixel is (transitively) connected to exactly one label node
– Cost of a cut: sum of deleted edge weights
– Finding min cost cut equivalent to finding global minimum of energy function
Computing a multiway cut
• With 2 labels: classical min-cut problem
– Solvable by standard flow algorithms
• polynomial time in theory, nearly linear in practice
– More than 2 terminals: NP-hard
[Dahlhaus et al., STOC ‘92]
• Efficient approximation algorithms exist
– Within a factor of 2 of optimal
– Computes local minimum in a strong sense
• even very large moves will not improve the energy – Yuri Boykov, Olga Veksler and Ramin Zabih,
Fast Approximate Energy Minimization via Graph Cuts, International Conference on Computer Vision, September 1999.
Move examples
Starting point
Red-blue swap move
Green expansion move
The swap move algorithm
1. Start with an arbitrary labeling
2. Cycle through every label pair (A,B) in some order
2.1 Find the lowest E labeling within a single AB-swap 2.2 Go there if E is lower than the current labeling
3. If E did not decrease in the cycle, we’re done Otherwise, go to step 2
Original graph
A B
AB subgraph
(run min-cut on this graph) B
A
The expansion move algorithm
1. Start with an arbitrary labeling
2. Cycle through every label A in some order
2.1 Find the lowest E labeling within a single A-expansion 2.2 Go there if it E is lower than the current labeling
3. If E did not decrease in the cycle, we’re done Otherwise, go to step 2
GrabCut GrabCut
Interactive Foreground Extraction using Iter Interactive Foreground Extraction using Iter
ated Graph Cuts ated Graph Cuts
Carsten Rother Carsten Rother
Vladimir Kolmogorov Vladimir Kolmogorov
Andrew Blake Andrew Blake
Microsoft Research Cambridge-UK
Microsoft Research Cambridge-UK
Demo
• video
Interactive Digital Photomontage
Aseem Agarwala, Mira Dontcheva, Maneesh Agrawala, Steven Drucker, Alex Colburn, Brian Curless, David Salesin, Michael Cohen, “Interactive Digital Photomontage”, SIGGRAPH 2004
• Combining multiple photos
• Find seams using graph cuts
• Combine gradients and integrate
actual photomontage
set of originals perceived
Source images Brush strokes Computed labeling
Composite
Brush strokes Computed labeling
Interactive Digital Photomontage
• Extended depth of field
Interactive Digital Photomontage
• Relighting
Interactive Digital Photomontage
Demo
• video