As applied in the field of computer vision, graph cut optimization can be employed to efficiently solve a wide variety of low-level computer vision problems (early vision), such as image smoothing, the stereo correspondence problem, image segmentation, object co-segmentation, numerous military applications (eg Automatic target recognition) and many other problems that can be formulated in terms of energy minimization (eg Climate Science and Environmental modelling). Graph cut techniques are now increasingly being used in combination with more general spatial Artificial intelligence techniques (eg to enforce structure in Large language model output to sharpen tumour boundaries and similarly for various Augmented reality, Self-driving car, Robotics, Google Maps applications etc). Many of these energy minimization problems can be approximated by solving a maximum flow problem in a graph (and thus, by the max-flow min-cut theorem, define a minimal cut of the graph). Under most formulations of such problems in computer vision, the minimum energy solution corresponds to the 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 foundational theory of graph cuts in computer vision was first developed by Margaret Greig, Bruce Porteous and Allan Seheult (GPS) of Durham University in a now legendary discussion contribution to Julian Besag's 1986 paper and a more detailed follow on paper in 1989. In the Bayesian statistical context of smoothing noisy images, using a Markov random field as the image prior distribution, they showed with a mathematically beautiful proof how the maximum a posteriori estimate of a binary image can be obtained exactly by maximizing the flow through an associated image network, or graph, involving the introduction of a source and sink and Log-likelihood ratios. The problem was shown to be efficiently solvable exactly, an unexpected result as the problem was believed to be computationally intractable (NP hard). GPS also addressed the computational cost of the max-flow algorithm on large graphs, a significant concern at the time. They proposed a partitioning algorithm (see Section 4 of GPS) involving the recursive amalgamation of non-overlapping blocks, or tiles, which gave a 12X increase in speed. This approach recursively solved and amalgamated independent sub-graphs until the whole graph was solved. While contemporaries like Geman and Geman had advocated Parallel computing in the context of Simulated annealing, the GPS blocking strategy offered a deterministic structure amenable to parallelisation and anticipated modern artificial intelligence design across multiple GPUs. However, until recently, this aspect of the paper was largely ignored and subsequent research focused on Serial computer global search trees, such as the Boykov-Kolmogorov algorithm. Although the general k {\displaystyle k} -colour problem is NP hard for k > 2 , {\displaystyle k>2,} the GPS approach has turned out to have very wide applicability in general computer vision problems. This was first demonstrated by Boykov, Veksler and Zabih who, in a seminal paper published more than 10 years after the original GPS paper, and in other important works, lit the blue touch paper for the general adoption of graph cut techniques in computer vision. They showed that, for general problems, the GPS approach can be applied iteratively to sequences of binary problems, using their now ubiquitous alpha-expansion algorithm, yielding near optimal solutions. Prior to these results, approximate local optimisation techniques such as simulated annealing (as proposed by the Geman brothers) or iterated conditional modes (a type of greedy algorithm suggested by Julian Besag) were used to solve such image smoothing problems. Building on these advancements, GPS graph cut optimization was subsequently adapted for interactive image segmentation, most notably through the "GrabCut" algorithm introduced by Carsten Rother, Vladimir Kolmogorov, and Andrew Blake of Microsoft Research, Cambridge. GrabCut extended earlier interactive graph cut methods by replacing monochrome image histograms with Gaussian mixture models to estimate colour distributions, and by employing an iterative GPS energy minimisation scheme. This approach significantly simplified user interaction, requiring only a rough bounding box around the target object rather than detailed user-drawn strokes, and it quickly became a standard tool in both academic research and commercial image editing software. The GPS paper connected and bridged profound ideas from Mathematical statistics (Bayes' theorem, Markov random field), Physics (Ising model), Optimisation (Energy function) and Computer science (Network flow problem) and led the move away from approximate local and slow optimisation approaches (eg simulated annealing) to more powerful exact, or near exact, faster global optimisation techniques. It is now recognised as seminal as it was well ahead of its time and, in particular, was published years before the computing power revolution of Moore's law and GPUs. Significantly, GPS was published in a mathematical statistics (rather than a computer vision) journal, and this led to it being overlooked by the computer vision community for many years. It is unofficially known as "The Velvet Underground" paper of computer vision (ie although very few computer vision people read the paper [bought the record], those that did, most importantly Boykov, Veksler and Zabih, started new and important research [formed a band]). This is confirmed by GPS' very large amplification ratio (2nd order citations/first order citations), estimated at well in excess of 100. Despite the foundational nature of the GPS work, formal recognition from the computer vision community has predominantly gone to the researchers who followed to extend and popularise the graph cut method. For example, Boykov, Veksler and Zabih deservedly received a Helmholtz Prize from the ICCV in 2011. This prize recognises ICCV papers from 10 or more years earlier that have had a significant impact on computer vision research. In 2011, Couprie et al. proposed a general image segmentation framework, called the "Power Watershed", that minimized a real-valued indicator function from [0,1] 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 p {\displaystyle p} . When p = 1 {\displaystyle p=1} , the Power Watershed is optimized by graph cuts, when p = 0 {\displaystyle p=0} the Power Watershed is optimized by shortest paths, p = 2 {\displaystyle p=2} is optimized by the random walker algorithm and p = ∞ {\displaystyle p=\infty } is optimized by the watershed 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. == Binary segmentation of images == === Notation === Image: x ∈ { R , G , B } N {\displaystyle x\in \{R,G,B\}^{N}} Output: Segmentation (also called opacity) S ∈ R N {\displaystyle S\in R^{N}} (soft segmentation). For hard segmentation S ∈ { 0 for background , 1 for foreground/object to be detected } N {\displaystyle S\in \{0{\text{ for background}},1{\text{ for foreground/object to be detected}}\}^{N}} Energy function: E ( x , S , C , λ ) {\displaystyle E(x,S,C,\lambda )} where C is the color parameter and λ is the coherence parameter. E ( x , S , C , λ ) = E c o l o r + E c o h e r e n c e {\displaystyle E(x,S,C,\lambda )=E_{\rm {color}}+E_{\rm {coherence}}} Optimization: The segmentation can be estimated as a global minimum over S: arg min S E ( x , S , C , λ ) {\displaystyle {\arg \min }_{S}E(x,S,C,\lambda )} === Existing methods === Standard Graph cuts: optimize energy function over the segmentation (unknown S value). Iterated Graph cuts: First step optimizes over the color parameters using K-means. Second step performs the usual graph cuts algorithm. These 2 steps are repeated recursively until convergence Dynamic graph cuts:Allows to re-run the algorithm much faster after modifying the problem (e.g. after new seeds have been added by a user). === Energy function === Pr ( x ∣ S ) = K − E {\displaystyle \Pr(x\mid S)=K^{-E}} where the energy E {\displaystyle E} is composed of two different mod
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