J Math Imaging Vis (2012) 44:375–398 DOI 10.1007/s10851-012-0333-3 Fuzzy Connectedness Image Segmentation in Graph Cut Formulation: A Linear-Time Algorithm and a Comparative Analysis Krzysztof Chris Ciesielski · Jayaram K. Udupa · A.X. Falcão · P.A.V. Miranda Published online: 21 March 2012 © Springer Science+Business Media, LLC 2012 Abstract A deep theoretical analysis of the graph cut im- age segmentation framework presented in this paper simul- taneously translates into important contributions in several directions. The most important practical contribution of this work is a full theoretical description, and implementation, of a novel powerful segmentation algorithm, GC max . The out- put of GC max coincides with a version of a segmentation algorithm known as Iterative Relative Fuzzy Connectedness, IRFC. However, GC max is considerably faster than the clas- sic IRFC algorithm, which we prove theoretically and show experimentally. Specifically, we prove that, in the worst case scenario, the GC max algorithm runs in linear time with re- spect to the variable M =|C|+|Z|, where |C| is the image scene size and |Z| is the size of the allowable range, Z, of the associated weight/affinity function. For most implemen- K.C. Ciesielski () Department of Mathematics, West Virginia University, Morgantown, WV 26506-6310, USA e-mail: KCies@math.wvu.edu url: http://www.math.wvu.edu/~kcies K.C. Ciesielski · J.K. Udupa Department of Radiology, MIPG, University of Pennsylvania, Blockley Hall–4th Floor, 423 Guardian Drive, Philadelphia, PA, 19104-6021, USA A.X. Falcão Institute of Computing, University of Campinas, Campinas, SP, Brazil P.A.V. Miranda Department of Computer Science, IME, University of São Paulo (USP), São Paulo, SP, Brazil tations, Z is identical to the set of allowable image inten- sity values, and its size can be treated as small with respect to |C|, meaning that O(M) = O(|C|). In such a situation, GC max runs in linear time with respect to the image size |C|. We show that the output of GC max constitutes a solution of a graph cut energy minimization problem, in which the energy is defined as the ℓ ∞ norm ‖F P ‖ ∞ of the map F P that associates, with every element e from the boundary of an object P , its weight w(e). This formulation brings IRFC algorithms to the realm of the graph cut energy minimizers, with energy functions ‖F P ‖ q for q ∈[1, ∞]. Of these, the best known minimization problem is for the energy ‖F P ‖ 1 , which is solved by the classic min-cut/max-flow algorithm, referred to often as the Graph Cut algorithm. We notice that a minimization problem for ‖F P ‖ q , q ∈[1, ∞), is identical to that for ‖F P ‖ 1 , when the orig- inal weight function w is replaced by w q . Thus, any al- gorithm GC sum solving the ‖F P ‖ 1 minimization prob- lem, solves also one for ‖F P ‖ q with q ∈[1, ∞), so just two algorithms, GC sum and GC max , are enough to solve all ‖F P ‖ q -minimization problems. We also show that, for any fixed weight assignment, the solutions of the ‖F P ‖ q -minimization problems converge to a solution of the ‖F P ‖ ∞ -minimization problem (‖F P ‖ ∞ = lim q →∞ ‖F P ‖ q is not enough to deduce that). An experimental comparison of the performance of GC max and GC sum algorithms is included. This concentrates on comparing the actual (as opposed to provable worst sce- nario) algorithms’ running time, as well as the influence of the choice of the seeds on the output. Keywords Image processing · Segmentation · Graph cut · Fuzzy connectedness