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An Introduction to Graph-Cut
By: Paul Scovanner
An Introduction to Graph-Cut
Overview
Min-cut
Normalized-Cut
Applications
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An Introduction to Graph-Cut: Algorithm for Image Segmentation and Minimum-Cut, Study notes of Computer Science
An introduction to graph-cut, an algorithm used for finding globally optimal segmentation solutions, also known as min-cut or normalized-cut. The concept of cuts in graphs, the problem with min-cut, and the solution with normalized-cut. Applications of graph-cut include image segmentation, stereo, 3d reconstruction, and more. The document also discusses the labeling problem and the energy minimization concept.
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Pre 2010
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An Introduction to Graph-Cut
By: Paul Scovanner
An Introduction to Graph-Cut
Overview Min-cut Normalized-Cut Applications
An Introduction to Graph-Cut
Graph-cut is an algorithm that finds a globally optimal segmentation solution. Also know as Min-cut. Equivalent to Max-flow. [1]
[1] Wu and Leahy: An Optimal Graph Theoretic Approach to Data Clustering:…
What is a “cut”?
A graph G = (V,E) can be partitioned into two disjoint sets, by simply removing edges connecting the two parts.
The degree of dissimilarity between these two pieces can be computed as total weight of the edges that have been removed. In graph theoretic language it is called the cut : [2].
A , B , A ∪ B = V , A ∩ B = 0
∑ ∈ ∈
u AvB
cut AB wu v
,
[2] Shi and Malik: Normalized cuts and image segmentation.
Finding the Minimum-cut
Finding the Minimum-cut
Finding the Minimum-cut
Finding the Minimum-cut
Normalized-cut
Instead use normalized cut (Ncut).
assoc BV
cut AB
assoc AV
cut AB
Ncut AB = +
∑ ∈ ∈
u AtV
assoc AV wut
,
= ⎛^ +
assocBV
assocB B assoc AV
Nassoc AB assoc AA
Normalized-cut
assocBV
cut AB assocAV
Ncut AB = cut AB +
( , )
( , ) ( , ) ( , )
( , ) ( , ) assocB V
assocBV assocBB assocAV
= assoc^ AV − assocAA + −
= −⎛^ +
assocBV
assocB B assocAV
assocA A
= 2 − Nassoc ( A , B )
Given
2 5 1 (^3 )
Pixel labeling problem
Assignment cost for giving a particular label to a particular node. Written as D_._
Separation cost for assigning a particular pair of labels to neighboring nodes. Written as V_._
Find
Labeling f = (f 1 ,…,fn )
5 1
2
3 4
Such that the sum of the assignment costs and separation costs (the energy E) is small
Energy Minimization
Optimizing the labeling problem can be thought
of as minimizing some energy function.
measure of image discrepancy
measure of smoothness or
other visual constraints
What do graph cuts provide?
For less interesting V , polynomial
algorithm for global minimum!
For a particularly interesting V ,
approximation algorithm
Proof of NP hardness
For many choices of V , algorithms that find
a “strong” local minimum
Very strong experimental results
Graph Cut based Segmentation
Graph Cut based Segmentation
n-links
s
hard t^ a cut
constraint
hard constraint
User Guided Segmentation; Specifies hard constraints.
3D Applications
2D and Time
[3] Wang et al.: Interactive Video Cutout
Some Results
[5] Rother, Kolmogorov and Blake: “GrabCut“ - Interactive Foreground Extraction using Iterated Graph Cuts
Some Results
Some Results
Some Results