Abstract
Techniques based on minimal graph cuts have become a standard tool for solving combinatorial optimization problems arising in image processing and computer vision applications. These techniques can be used to minimize objective functions written as the sum of a set of unary and pairwise terms, provided that the objective function is submodular. This can be interpreted as minimizing the \(l_1\)-norm of the vector containing all pairwise and unary terms. By raising each term to a power p, the same technique can also be used to minimize the \(l_p\)-norm of the vector. Unfortunately, the submodularity of an \(l_1\)-norm objective function does not guarantee the submodularity of the corresponding \(l_p\)-norm objective function. The contribution of this paper is to provide useful conditions under which an \(l_p\)-norm objective function is submodular for all \(p\ge 1\), thereby identifying a large class of \(l_p\)-norm objective functions that can be minimized via minimal graph cuts.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
As a counterexample, consider the two-label pairwise term \(\phi \) given by \(\phi (0,0)=3\), \(\phi (1,1)=0\), and \(\phi (0,1)=\phi (1,0)=2\). It is easily verified that \(\phi \) is submodular, while \(\phi ^2\) is not.
References
Allène, C., Audibert, J.Y., Couprie, M., Keriven, R.: Some links between extremum spanning forests, watersheds and min-cuts. Image Vis. Comput. 28(10), 1460–1471 (2010)
Boykov, Y., Funka-Lea, G.: Graph cuts and efficient ND image segmentation. Int. J. Comput. Vis. 70(2), 109–131 (2006)
Boykov, Y., Veksler, O., Zabih, R.: Fast approximate energy minimization via graph cuts. IEEE Trans. Pattern Anal. Mach. Intell. 23(11), 1222–1239 (2001)
Couprie, C., Grady, L., Najman, L., Talbot, H.: Power watershed: a unifying graph-based optimization framework. IEEE Trans. Pattern Anal. Mach. Intell. 33(7), 1384–1399 (2011)
Kolmogorov, V., Rother, C.: Minimizing nonsubmodular functions with graph cuts-a review. IEEE Trans. Pattern Anal. Mach. Intell. 29(7), 1274–1279 (2007)
Kolmogorov, V., Zabin, R.: What energy functions can be minimized via graph cuts? IEEE Trans. Pattern Anal. Mach. Intell. 26(2), 147–159 (2004)
Levi, Z., Zorin, D.: Strict minimizers for geometric optimization. ACM Trans. Graph. (TOG) 33(6), 185 (2014)
Miranda, P.A., Falcão, A.X.: Links between image segmentation based on optimum-path forest and minimum cut in graph. J. Math. Imaging Vis. 35(2), 128–142 (2009)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer Nature Switzerland AG
About this paper
Cite this paper
Malmberg, F., Strand, R. (2018). When Can \(l_p\)-norm Objective Functions Be Minimized via Graph Cuts?. In: Barneva, R., Brimkov, V., Tavares, J. (eds) Combinatorial Image Analysis. IWCIA 2018. Lecture Notes in Computer Science(), vol 11255. Springer, Cham. https://doi.org/10.1007/978-3-030-05288-1_9
Download citation
DOI: https://doi.org/10.1007/978-3-030-05288-1_9
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-05287-4
Online ISBN: 978-3-030-05288-1
eBook Packages: Computer ScienceComputer Science (R0)