1202:
Metrication artifacts: When an image is represented by a 4-connected lattice, graph cuts methods can exhibit unwanted "blockiness" artifacts. Various methods have been proposed for addressing this issue, such as using additional edges or by formulating the max-flow problem in continuous
240:
the approach of Greig, Porteous and
Seheult has turned out to have wide applicability in general computer vision problems. Greig, Porteous and Seheult's approaches are often applied iteratively to a sequence of binary problems, usually yielding near optimal solutions.
74:
of a solution. Although many computer vision algorithms involve cutting a graph (e.g., normalized cuts), the term "graph cuts" is applied specifically to those models which employ a max-flow/min-cut optimization (other graph cutting algorithms may be considered as
1210:
Multiple labels: Graph cuts is only able to find a global optimum for binary labeling (i.e., two labels) problems, such as foreground/background image segmentation. Extensions have been proposed that can find approximate solutions for multilabel graph cuts
1197:
Graph cuts methods have become popular alternatives to the level set-based approaches for optimizing the location of a contour (see for an extensive comparison). However, graph cut approaches have been criticized in the literature for several issues:
538:
1206:
Shrinking bias: Since graph cuts finds a minimum cut, the algorithm can be biased toward producing a small contour. For example, the algorithm is not well-suited for segmentation of thin objects like blood vessels (see for a proposed
705:
1313:
consists of a directed graph with edges labeled with capacities, and there are two distinct nodes: the source and the sink. Intuitively, it is easy to see that the maximum flow is determined by the bottleneck.
384:
algorithm. In this way, the Power
Watershed may be viewed as a generalization of graph cuts that provides a straightforward connection with other energy optimization segmentation/clustering algorithms.
1286:
are respectively the number of nodes and edges in the graph). Nevertheless, some amount of work has been recently done in this direction for reducing the graphs before the maximum-flow computation.
772:
1180:: Associate a penalty to disagreeing pixels by evaluating the difference between their segmentation label (crude measure of the length of the boundaries). See Boykov and Kolmogorov ICCV 2003
252:
from over a graph, constrained by user seeds (or unary terms) set to 0 or 1, in which the minimization of the indicator function over the graph is optimized with respect to an exponent
1350:β An implementation of the maxflow algorithm described in "An Experimental Comparison of Min-Cut/Max-Flow Algorithms for Energy Minimization in Computer Vision" by Vladimir Kolmogorov
1161:
981:
864:
448:
590:
1029:
928:
488:
1167:
In practice, pixels are defined as neighbors if they are adjacent either horizontally, vertically or diagonally (4 way connectivity or 8 way connectivity for 2D images).
483:
1330:
The Sim Cut algorithm approximates the minimum graph cut. The algorithm implements a solution by simulation of an electrical network. This is the approach suggested by
378:
90:) can be solved exactly using this approach; problems where pixels can be labeled with more than two different labels (such as stereo correspondence, or denoising of a
1244:
238:
1733:
1214:
Memory: the memory usage of graph cuts increases quickly as the image size increases. As an illustration, the Boykov-Kolmogorov max-flow algorithm v2.2 allocates
348:
322:
296:
1284:
1264:
887:
270:
207:
1035:
This term can be modeled using different local (e.g. texons) or global (e.g. histograms, GMMs, Adaboost likelihood) approaches that are described below.
1604:
1465:
1045:
We use intensities of pixels marked as seeds to get histograms for object (foreground) and background intensity distributions: P(I|O) and P(I|B).
597:
1331:
1093:
71:
1575:
1186:: If the color is very different, it might be a good place to put a boundary. See Lafferty et al. 2001; Kumar and Hebert 2003
70:
of the graph). Under most formulations of such problems in computer vision, the minimum energy solution corresponds to the
1723:
806:
Allows to re-run the algorithm much faster after modifying the problem (e.g. after new seeds have been added by a user).
381:
59:
712:
1738:
1660:
1170:
Costs can be based on local intensity gradient, Laplacian zero-crossing, gradient direction, color mixture model,...
174:
123:
1085:
Compute non-parametric statistics of the model-interior texons, either on intensity or on Gabor filter responses.
1322:
The Boykov-Kolmogorov algorithm is an efficient way to compute the max-flow for computer vision-related graphs.
1728:
1306:
1183:
63:
248:. proposed a general image segmentation framework, called the "Power Watershed", that minimized a real-valued
1536:
351:
1601:
1526:β, IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol. 33, No. 7, pp. 1384-1399, July 2011
1372:; an algorithm for computing an approximate solution of the minimum s-t cut in a massively parallel manner.
1296:
107:
47:
39:
22:
1113:
933:
818:
1576:
A Seeded Image
Segmentation Framework Unifying Graph Cuts and Random Walker Which Yields A New Algorithm
402:
118:. Allan Seheult and Bruce Porteous were members of Durham's lauded statistics group of the time, led by
1634:
548:
1523:
533:{\displaystyle S\in \{0{\text{ for background}},1{\text{ for foreground/object to be detected}}\}^{N}}
1443:
993:
892:
55:
1707:
I.T. Frisch, "On
Electrical analogs for flow networks," Proceedings of IEEE, 57:2, pp. 209-210, 1969
1424:
1074:
A texon (or texton) is a set of pixels that has certain characteristics and is repeated in an image.
1177:
166:
134:
51:
1335:
1058:
We usually use two distributions: one for background modelling and another for foreground pixels.
249:
130:
notable as the first ever female member of staff of the Durham
Mathematical Sciences Department.
103:
67:
43:
455:
357:
1690:
115:
1502:
1217:
214:
94:
image) cannot be solved exactly, but solutions produced are usually near the global optimum.
1682:
1589:
What
Metrics Can Be Approximated by Geo-Cuts, or Global Optimization of Length/Area and Flux
1403:
1310:
178:
1648:
An
Experimental Comparison of Min-Cut/Max-Flow Algorithms for Energy Minimization in Vision
1608:
542:
111:
76:
35:
27:
17:
327:
301:
275:
1098:
1048:
Then, we use these histograms to set the regional penalties as negative log-likelihoods.
1482:
1269:
1249:
872:
255:
192:
189:
127:
783:
Standard Graph cuts: optimize energy function over the segmentation (unknown S value).
1717:
1686:
1621:
161:. The problem was therefore shown to be efficiently solvable. Prior to this result,
182:
170:
150:
142:
138:
119:
87:
1061:
Use a
Gaussian mixture model (with 5β8 components) to model those 2 distributions.
137:
statistical context of smoothing noisy (or corrupted) images, they showed how the
1588:
1444:
Stochastic relaxation, Gibbs distributions and the
Bayesian restoration of images
1549:
1390:
1647:
1562:
1565:", IEEE Transactions on Pattern Analysis and Machine Intelligence, pp. 106β118
1309:
we can solve energy minimization by maximizing the flow over the network. The
54:. Many of these energy minimization problems can be approximated by solving a
1694:
50:, and many other computer vision problems that can be formulated in terms of
709:
Optimization: The segmentation can be estimated as a global minimum over S:
91:
83:
1673:
Cederbaum, I. (1962-08-01). "On optimal operation of communication nets".
1487:
International
Conference on Computer Vision and Pattern Recognition (CVPR)
1347:
1362:β fast multi-core max-flow/min-cut solver optimized for grid-like graphs
700:{\displaystyle E(x,S,C,\lambda )=E_{\rm {color}}+E_{\rm {coherence}}}
153:
through an associated image network, involving the introduction of a
1163:β binary term describing the coherence between neighborhood pixels.
1537:
The Piecewise Smooth Mumford-Shah Functional on an Arbitrary Graph
1650:. IEEE Trans. Pattern Anal. Mach. Intell. 26(9): 1124β1137 (2004)
1622:
A Multilevel Banded Graph Cuts Method for Fast Image Segmentation
1353:
1633:
Yin Li, Jian Sun, Chi-Keung Tang, and Heung-Yeung Shum (2004), "
592:
where C is the color parameter and Ξ» is the coherence parameter.
1600:
Nicolas Lermé, François Malgouyres and Lucas Létocart (2010), "
1524:
Power Watersheds: A Unifying Graph-Based Optimization Framework
1522:
Camille Couprie, Leo Grady, Laurent Najman and Hugues Talbot, "
1462:
On the statistical analysis of dirty pictures (with discussion)
1408:
IEEE Transactions on Pattern Analysis and Machine Intelligence,
1507:
IEEE Transactions on Pattern Analysis and Machine Intelligence
1365:
791:
First step optimizes over the color parameters using K-means.
1620:
Herve Lombaert, Yiyong Sun, Leo Grady, Chenyang Xu (2005), "
1302:
Minimization is done using a standard minimum cut algorithm.
30:
solve a wide variety of low-level computer vision problems (
1393:", Computational models of visual processing 1.2 (1991).
799:
These 2 steps are repeated recursively until convergence.
1082:
Determine a good natural scale for the texture elements.
1550:
Computing Geodesics and Minimal Surfaces via Graph Cuts
1425:
Exact maximum a posteriori estimation for binary images
1391:
The plenoptic function and the elements of early vision
298:, the Power Watershed is optimized by graph cuts, when
114:
in the seminal paper by Greig, Porteous and Seheult of
1428:, Journal of the Royal Statistical Society, Series B,
1418:
1416:
1359:
1031:β unary term describing the likelihood of each color.
1563:
Globally Minimal Surfaces by Continuous Maximal Flows
1334:. Acceleration of the algorithm is possible through
1272:
1252:
1220:
1116:
996:
936:
895:
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551:
491:
458:
405:
360:
330:
304:
278:
258:
217:
195:
794:
Second step performs the usual graph cuts algorithm.
324:
the Power Watershed is optimized by shortest paths,
185:) were used to solve such image smoothing problems.
1663:," United States Patent US8929636, January 6, 2016
1503:
Fast approximate energy minimisation via graph cuts
1422:D.M. Greig, B.T. Porteous and A.H. Seheult (1989),
1404:
Fast approximate energy minimization via graph cuts
1099:
Contour and Texture Analysis for Image Segmentation
1094:
Deformable-model based Textured Object Segmentation
1483:Markov Random Fields with Efficient Approximations
1278:
1258:
1238:
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1023:
975:
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858:
766:
699:
584:
532:
477:
442:
372:
342:
316:
290:
264:
232:
201:
1539:", IEEE Trans. on Image Processing, pp. 2547β2561
1389:Adelson, Edward H., and James R. Bergen (1991), "
767:{\displaystyle {\arg \min }_{S}E(x,S,C,\lambda )}
1497:
1495:
1402:Boykov, Y., Veksler, O., and Zabih, R. (2001), "
1064:Goal: Try to pull apart those two distributions.
822:
724:
1356:β some graph cut libraries and MATLAB wrappers
1173:Different energy functions have been defined:
1587:Vladimir Kolmogorov and Yuri Boykov (2005), "
1548:Yuri Boykov and Vladimir Kolmogorov (2003), "
1481:Y. Boykov, O. Veksler and R. Zabih (1998), "
8:
1637:", ACM Transactions on Graphics, pp. 303β308
1501:Y. Boykov, O. Veksler and R. Zabih (2001), "
521:
498:
431:
412:
1447:, IEEE Trans. Pattern Anal. Mach. Intell.,
485:(soft segmentation). For hard segmentation
452:Output: Segmentation (also called opacity)
1535:Leo Grady and Christopher Alvino (2009), "
517: for foreground/object to be detected
1602:Reducing graphs in graph cut segmentation
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1661:Method and System for Image Segmentation
1561:Ben Appleton and Hugues Talbot (2006), "
1466:Journal of the Royal Statistical Society
987:Likelihood / Color model / Regional term
1382:
1348:http://pub.ist.ac.at/~vnk/software.html
1107:Prior / Coherence model / Boundary term
1734:Computational problems in graph theory
889:is composed of two different models (
7:
1156:{\displaystyle E_{\rm {coherence}}}
976:{\displaystyle E_{\rm {coherence}}}
859:{\displaystyle \Pr(x\mid S)=K^{-E}}
1646:Yuri Boykov, Vladimir Kolmogorov:
1147:
1144:
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443:{\displaystyle x\in \{R,G,B\}^{N}}
367:
14:
1675:Journal of the Franklin Institute
585:{\displaystyle E(x,S,C,\lambda )}
1574:Ali Kemal Sinop and Leo Grady, "
1332:Cederbaum's maximum flow theorem
1611:", Proc. of ICIP, pp. 3045β3048
1024:{\displaystyle E_{\rm {color}}}
923:{\displaystyle E_{\rm {color}}}
126:, with the optimisation expert
1441:D. Geman and S. Geman (1984),
1354:http://vision.csd.uwo.ca/code/
1326:Implementation (approximation)
837:
825:
761:
737:
628:
604:
579:
555:
1:
1624:", Proc. of ICCV, pp. 259β265
389:Binary segmentation of images
139:maximum a posteriori estimate
72:maximum a posteriori estimate
1687:10.1016/0016-0032(62)90401-5
1591:", Proc. of ICCV pp. 564β571
1053:GMM (Gaussian mixture model)
1368:β An implementation of the
82:"Binary" problems (such as
16:As applied in the field of
1755:
1366:http://virtualscalpel.com/
1294:
478:{\displaystyle S\in R^{N}}
175:iterated conditional modes
373:{\displaystyle p=\infty }
1307:max-flow min-cut theorem
1184:Conditional random field
64:max-flow min-cut theorem
1239:{\displaystyle 24n+14m}
352:random walker algorithm
233:{\displaystyle k>2,}
1578:", Proc. of ICCV, 2007
1318:Implementation (exact)
1297:Graph cut optimization
1280:
1260:
1240:
1157:
1025:
977:
924:
883:
860:
768:
701:
586:
534:
479:
444:
374:
344:
318:
292:
266:
234:
203:
108:an optimization method
48:object co-segmentation
40:correspondence problem
23:graph cut optimization
1281:
1261:
1241:
1158:
1026:
978:
925:
884:
861:
769:
702:
587:
535:
480:
445:
375:
345:
319:
293:
267:
235:
211:remains unsolved for
204:
188:Although the general
110:was first applied in
1270:
1250:
1218:
1114:
994:
934:
893:
873:
819:
786:Iterated Graph cuts:
713:
598:
549:
506: for background
489:
456:
403:
380:is optimized by the
358:
350:is optimized by the
328:
302:
276:
256:
244:In 2011, C. Couprie
215:
193:
169:(as proposed by the
56:maximum flow problem
1724:Bayesian statistics
1460:J.E. Besag (1986),
1360:http://gridcut.com/
1178:Markov random field
804:Dynamic graph cuts:
343:{\displaystyle p=2}
317:{\displaystyle p=0}
291:{\displaystyle p=1}
167:simulated annealing
165:techniques such as
66:, define a minimal
52:energy minimization
26:can be employed to
1739:Image segmentation
1607:2012-03-27 at the
1410:23(11): 1222-1239.
1336:parallel computing
1276:
1256:
1236:
1153:
1021:
973:
920:
879:
856:
764:
697:
582:
530:
475:
440:
370:
340:
314:
288:
262:
250:indicator function
230:
199:
149:by maximizing the
77:graph partitioning
62:(and thus, by the
44:image segmentation
1279:{\displaystyle m}
1259:{\displaystyle n}
882:{\displaystyle E}
869:where the energy
518:
507:
265:{\displaystyle p}
202:{\displaystyle k}
116:Durham University
1746:
1708:
1705:
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1572:
1566:
1559:
1553:
1552:", Proc. of ICCV
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1311:max-flow problem
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1277:
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778:Existing methods
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733:
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179:greedy algorithm
145:can be obtained
1754:
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1729:Computer vision
1714:
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1609:Wayback Machine
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811:Energy function
805:
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543:Energy function
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209:-colour problem
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112:computer vision
100:
36:image smoothing
18:computer vision
12:
11:
5:
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1681:(2): 130β141.
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171:Geman brothers
128:Margaret Greig
102:The theory of
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1635:Lazy Snapping
1630:
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183:Julian Besag
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143:binary image
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120:Julian Besag
101:
88:binary image
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32:early vision
31:
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1659:P.J. Yim: "
1305:Due to the
177:(a type of
163:approximate
124:Peter Green
34:), such as
28:efficiently
1718:Categories
1468:Series B,
1451:, 721β741.
1432:, 271β279.
1377:References
1295:See also:
1090:Examples:
104:graph cuts
1695:0016-0032
1472:, 259β302
1291:Algorithm
1211:problems.
1193:Criticism
1176:Standard
1040:Histogram
849:−
832:∣
759:λ
722:
626:λ
577:λ
496:∈
463:∈
410:∈
382:watershed
368:∞
92:grayscale
84:denoising
1605:Archived
1342:Software
394:Notation
272:. When
135:Bayesian
106:used as
1370:Sim Cut
1246:bytes (
399:Image:
147:exactly
133:In the
98:History
1693:
1203:space.
1077:Steps:
173:), or
155:source
1207:fix).
1069:Texon
246:et al
141:of a
60:graph
58:in a
1691:ISSN
1266:and
930:and
354:and
222:>
159:sink
157:and
151:flow
122:and
1683:doi
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68:cut
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