Graph Cut

Graph Cut

Digital Visual Effects g Yung-Yu Chuang

with slides by Fredo Durand, Ramesh Raskar

Graph cut

Graph cut

• Interactive image segmentation using graph cut

Bi l b l f d b k d

• Binary label: foreground vs. background

• User labels some pixels

– similar to trimap, usually sparser

• Exploit

F F B

p

– Statistics of known Fg & Bg – Smoothness of label

F F

F F B

B Smoothness of label

• Turn into discrete graph optimization

Graph cut (min cut / max flow)

B F B

– Graph cut (min cut / max flow)

Energy function

• Labeling: one value per pixel, F or B

• Energy(labeling) = data + smoothness

• Energy(labeling) data + smoothness

– Very general situation

– Will be minimized One labeling

• Data: for each pixel

– Probability that this color belongs to F (resp. B)

g (ok, not best)

– Similar in spirit to Bayesian matting

• Smoothness (aka regularization):

i hb i i l i Data

per neighboring pixel pair

– Penalty for having different label – Penalty is downweighted if the two

Data

– Penalty is downweighted if the two pixel colors are very different

– Similar in spirit to bilateral filter

Smoothness Smoothness

Data term

• A.k.a regional term

(because integrated over full region) (because integrated over full region)

• D(L)= ^{( )} 

^{-log h[L g [}

^{](C ](}

^{) )}

• Where i is a pixel

L

is the label at i (F or B), L

is the label at i (F or B), C

is the pixel value

h[L

] is the histogram of the observed Fg h[L

] is the histogram of the observed Fg (resp Bg)

• Note the minus sign

• Note the minus sign

Hard constraints

• The user has provided some labels

Th i k d di i l d

• The quick and dirty way to include

constraints into optimization is to replace the d t t b h lt if t t d 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=- #pixelsg p

Smoothness term

• a.k.a boundary term, a.k.a. regularization S(L)  B(C C ) (L L )

• S(L)= 

B(C

,C

) (L

-L

)

• Where i,j are neighbors

– e.g. 8-neighborhood

(but I show 4 for simplicity)

(L L ) is 0 if L L 1 otherwise

• (L

-L

) is 0 if L

=L

, 1 otherwise

• B(C

,C

) is high when C

and C

are similar, low if there is a discontinuity between those two pixels there is a discontinuity between those two pixels

– e.g. exp(-||C_i-C_j||²/2²) where can be a constant – where  can be a constant

or the local variance

• Note positive sign

• Note positive sign

Optimization

• E(L)=D(L)+ S(L)

 i bl k i

•  is a black-magic constant

• Find the labeling that minimizes E

• In this case, how many possibilities?

– 22 (512)⁹ (512)

– We can try them all!

– What about megapixel images?What about megapixel images?

Labeling as a graph problem

• Each pixel = node

Add d F & B

• Add two nodes F & B

• Labeling: link each pixel to either F or B

F

B

Data term

• Put one edge between each pixel and F & G W i h f d i d

• Weight of edge = minus data term

– Don’t forget huge weight for hard constraints – Careful with sign

F

B

Smoothness term

• Add an edge between each neighbor pair W i h h

• Weight = smoothness term

F

B

Min cut

• Energy optimization equivalent to min cut

C d di F f B

• Cut: remove edges to disconnect F from B

• Minimum: minimize sum of cut edge weight

F

B

Min cut <=> labeling

• In order to be a cut:

F h i l ith th F G d h t b t – 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

F

(otherwise could be added)

B

Energy minimization via graph cuts

edge weight

) , ,

(x y d₃ D

d₃

d₂

d₁

Labels edge weight

Labels

(disparities) ^V(d1,d1)

Energy minimization via graph cuts

d d d₃

d₁ d₂ d₁

• Graph Cost

– Matching cost between images – Neighborhood matching term

– Goal: figure out which labels are connected to g which pixels

Energy minimization via graph cuts

d d d₃

d₁ d₂ d₁

• Graph Cut

– Delete enough edges so that

• each pixel is (transitively) connected to exactly one label node

label node

– Cost of a cut: sum of deleted edge weights

Finding min cost cut equivalent to finding global – 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 – More than 2 terminals: NP-hard

[Dahlhaus et al., STOC ‘92]

• Efficient approximation algorithms exist

• Efficient approximation algorithms exist

– Within a factor of 2 of optimal

– Computes local minimum in a strong senseComputes 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 Minimization via Graph Cuts, International Conference on Computer Vision, September 1999.

Move examples

Red-blue swap move

Starting point

Green expansion move

The swap move algorithm

1. Start with an arbitrary labeling

2. Cycle through every label pair (A,B) in some ordery g y p ( , )

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

B B

O i i l h

A

AB b h

A

Original graph AB subgraph

(run min-cut on this graph)

The expansion move algorithm

1. Start with an arbitrary labeling

2 Cycle through every label A in some order 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 labelingg

3. If E did not decrease in the cycle, we’re done Otherwise, go to step 2

GrabCut GrabCut

Interacti e Foregro nd E traction Interacti e Foregro nd E traction Interactive Foreground Extraction Interactive Foreground Extraction using Iterated Graph Cuts

using Iterated Graph Cuts Carsten Rother

Carsten Rother Carsten Rother Carsten Rother

Vladimir Kolmogorov Vladimir Kolmogorov Andrew Blake

Andrew Blake Andrew Blake Andrew Blake

Microsoft Research Cambridge

Microsoft Research Cambridge--UK UK

Demo

• video

Interactive Digital Photomontage

• Combining multiple photos

• Combining multiple photos

• Find seams using graph cuts

• Combine gradients and integrate

• Combine gradients and integrate

Aseem Agarwala, Mira Dontcheva, Maneesh Agrawala, Steven Drucker, Alex Colburn, Brian Curless, David Salesin, Michael Cohen, “Interactive Digital Photomontage”, SIGGRAPH 2004

actual photomontage

set of originals perceived

Source images Brush strokes Computed labeling

Composite

Brush strokes Computed labeling

Brush strokes Computed labeling

Interactive Digital Photomontage

• Extended

depth of

depth of

field

Interactive Digital Photomontage

• Relighting

Interactive Digital Photomontage

Demo

• video