Graph cuts in computer vision: Difference between revisions

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As applied in the field of [[computer vision]], '''[[graph cut optimization]]''' can be employed to [[Polynomial time|efficiently]] solve a wide variety of low-level computer vision problems (''early vision''<ref>Adelson, Edward H., and James R. Bergen (1991), "[http://persci.mit.edu/pub_pdfs/elements91.pdf The plenoptic function and the elements of early vision]", Computational models of visual processing 1.2 (1991).</ref>), such as image [[smoothing]], the stereo [[correspondence problem]], [[image segmentation]], and many other computer vision problems that can be formulated in terms of [[energy minimization]]. Many of these energy minimization problems can be approximated by solving a [[maximum flow problem]] in a [[Graph (discrete mathematics)|graph]]<ref>Boykov, Y., Veksler, O., and Zabih, R. (2001), "Fast[ http://www.cs.cornell.edu/rdz/Papers/BVZ-pami01-final.pdfFast approximate energy minimization via graph cuts]," ''IEEE Transactions on Pattern Analysis and Machine Intelligence,'' 23(11): 1222-1239.</ref> (and thus, by the [[max-flow min-cut theorem]], define a minimal [[cut (graph theory)|cut]] of the graph). Under most formulations of such problems in computer vision, the minimum energy solution corresponds to the [[MAP estimate|maximum a posteriori estimate]] 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 [[graph partitioning]] algorithms).

"Binary" problems (such as denoising a [[binary image]]) 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 [[grayscale]] image) cannot be solved exactly, but solutions produced are usually near the [[global optimum]].

== History ==

The theory of [[Cut (graph theory)|graph cuts]] used as [[graph cut optimization|an optimization method]] was first applied in [[computer vision]] in the seminal paper by Greig, Porteous and Seheult<ref name="D.M. Greig, B.T 1989">D.M. Greig, B.T. Porteous and A.H. Seheult (1989), ''[https://classes.cs.uchicago.edu/archive/2006/fall/35040-1/gps.pdf Exact maximum a posteriori estimation for binary images]'', Journal of the Royal Statistical Society, Series B, '''51''', 271–279.</ref> of [[Durham University]]. In the [[Bayesian statistics|Bayesian]] statistical context of [[smoothing]] noisy (or corrupted) images, they showed how the [[MAP estimate|maximum a posteriori estimate]] of a [[binary image]] can be obtained ''exactly'' by maximizing the [[Flow network|flow]] through an associated image network, involving the introduction of a ''source'' and ''sink''. The problem was therefore shown to be efficiently solvable. Prior to this result, ''approximate'' techniques such as [[simulated annealing]] (as proposed by the [[Donald Geman|Geman brothers]]<ref>D. Geman and S. Geman (1984), ''[http://www.dam.brown.edu/people/documents/stochasticrelaxation.pdf Stochastic relaxation, Gibbs distributions and the Bayesian restoration of images]'', IEEE Trans. Pattern Anal. Mach. Intell., '''6''', 721–741.</ref>), or [[iterated conditional modes]] (a type of [[greedy algorithm]] as suggested by [[Julian Besag]])<ref>J.E. Besag (1986), ''On the statistical analysis of dirty pictures (with discussion)'', [[Journal of the Royal Statistical Society]] Series B, '''48''', 259–302</ref> were used to solve such image smoothing problems.

Although the general <math>k</math>[[Graph coloring|-colour problem]] remains unsolved for <math>k > 2,</math> the approach of Greig, Porteous and Seheult<ref name="D.M. Greig, B.T 1989"/> has turned out<ref>Y. Boykov, O. Veksler and R. Zabih (1998), "[http://www.cs.cornell.edu/~rdz/Papers/BVZ-cvpr98.pdf Markov Random Fields with Efficient Approximations]", ''International Conference on Computer Vision and Pattern Recognition (CVPR)''.</ref><ref name="boykov2001fast">Y. Boykov, O. Veksler and R. Zabih (2001), "[http://www.cs.cornell.edu/~rdz/Papers/BVZ-pami01-final.pdf Fast approximate energy minimisation via graph cuts]", ''IEEE Transactions on Pattern Analysis and Machine Intelligence'', '''29''', 1222–1239.</ref> to have wide applicability in general computer vision problems. Greig, Porteous and Seheult approaches are often applied iteratively to a sequence of binary problems, usually yielding near optimal solutions.

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Graph cuts methods have become popular alternatives to the level set-based approaches for optimizing the ___location of a contour (see<ref>Leo Grady and Christopher Alvino (2009), "[http://leogrady.net/wp-content/uploads/2017/01/grady2009combinatorial.pdf The Piecewise Smooth Mumford-Shah Functional on an Arbitrary Graph]", IEEE Trans. on Image Processing, pp. 2547–2561</ref> for an extensive comparison). However, graph cut approaches have been criticized in the literature for several issues:

* 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<ref>Yuri Boykov and Vladimir Kolmogorov (2003), "[http://www.csd.uwo.ca/~yuri/Papers/iccv03.pdf Computing Geodesics and Minimal Surfaces via Graph Cuts]", Proc. of ICCV</ref> or by formulating the max-flow problem in continuous space.<ref>Ben Appleton and Hugues Talbot (2006), "[https://ai.google/research/pubs/pub32799.pdf Globally Minimal Surfaces by Continuous Maximal Flows]", IEEE Transactions on Pattern Analysis and Machine Intelligence, pp. 106–118</ref>

* Shrinking bias: Since graph cuts finds a minimum cut, the algorithm can be biased toward producing a small contour.<ref>Ali Kemal Sinop and Leo Grady, "[http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.93.6438&rep=rep1&type=pdf A Seeded Image Segmentation Framework Unifying Graph Cuts and Random Walker Which Yields A New Algorithm]", Proc. of ICCV, 2007</ref> For example, the algorithm is not well-suited for segmentation of thin objects like blood vessels (see<ref>Vladimir Kolmogorov and Yuri Boykov (2005), "[http://pub.ist.ac.at/~vnk/papers/KB-ICCV05.pdf What Metrics Can Be Approximated by Geo-Cuts, or Global Optimization of Length/Area and Flux]", Proc. of ICCV pp. 564–571</ref> for a proposed fix).

* 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 problems.<ref name="boykov2001fast" />

* Memory: the memory usage of graph cuts increase quickly as the image size increase. As an illustration, the Boykov-Kolmogorov max-flow algorithm v2.2 allocates <math>24n+14m</math> bytes (<math>n</math> and <math>m</math> 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.<ref>Nicolas Lermé, François Malgouyres and Lucas Létocart (2010), "[http://lipn.fr/~lerme/docs/reducing_graphs_in_graph_cut_segmentation.pdf Reducing graphs in graph cut segmentation]", Proc. of ICIP, pp. 3045–3048</ref><ref>Herve Lombaert, Yiyong Sun, Leo Grady, Chenyang Xu (2005), "[http://step.polymtl.ca/~rv101/bandedgraphcuts.pdf A Multilevel Banded Graph Cuts Method for Fast Image Segmentation]", Proc. of ICCV, pp. 259–265</ref><ref>Yin Li, Jian Sun, Chi-Keung Tang, and Heung-Yeung Shum (2004), "[http://research.microsoft.com/pubs/69040/lazysnapping_siggraph04.pdf Lazy Snapping]", ACM Transactions on Graphics, pp. 303–308</ref>

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{{Wikibooks|Algorithm Implementation|Graphs/Maximum flow/Boykov & Kolmogorov}}

The Boykov-Kolmogorov algorithm <ref>Yuri Boykov, Vladimir Kolmogorov: [http://discovery.ucl.ac.uk/13383/1/13383.pdf An Experimental Comparison of Min-Cut/Max-Flow Algorithms for Energy Minimization in Vision]. IEEE Trans. Pattern Anal. Mach. Intell. 26(9): 1124–1137 (2004)</ref> is an efficient way to compute the max-flow for computer vision related graph.

=== Implementation (approximation) ===

The Sim Cut algorithm <ref>P.J. Yim: "[https://patentimages.storage.googleapis.com/2b/1e/e9/5834a9cc3312a0/US9214029.pdf Method and System for Image Segmentation]," United States Patent US8929636, January 6, 2016</ref> approximates the graph cut by the solution of the set of simultaneous non-linear equations based on the analog electrical model of the flow network.<ref>I.T. Frisch, "On Electrical analogs for flow networks," Proceedings of IEEE, 57:2, pp. 209-210, 1969</ref> Acceleration of the algorithm is possible through parallel computing.

== Software ==