## What metrics can be approximated by geo-cuts, or global optimization of length/area and flux (2005)

### Cached

### Download Links

- [www.adastral.ucl.ac.uk]
- [research.microsoft.com]
- [www.csd.uwo.ca]
- DBLP

### Other Repositories/Bibliography

Venue: | In ICCV |

Citations: | 50 - 10 self |

### BibTeX

@INPROCEEDINGS{Kolmogorov05whatmetrics,

author = {Vladimir Kolmogorov},

title = {What metrics can be approximated by geo-cuts, or global optimization of length/area and flux},

booktitle = {In ICCV},

year = {2005},

pages = {564--571}

}

### Years of Citing Articles

### OpenURL

### Abstract

In [3] we showed that graph cuts can find hypersurfaces of globally minimal length (or area) under any Riemannian metric. Here we show that graph cuts on directed regular grids can approximate a significantly more general class of continuous non-symmetric metrics. Using submodularity condition [1, 11], we obtain a tight characterization of graph-representable metrics. Such “submodular” metrics have an elegant geometric interpretation via hypersurface functionals combining length/area and flux. Practically speaking, we extend “geo-cuts ” algorithm [3] to a wider class of geometrically motivated hypersurface functionals and show how to globally optimize any combination of length/area and flux of a given vector field. The concept of flux was recently introduced into computer vision by [13] but it was mainly studied within variational framework so far. We are first to show that flux can be integrated into graph cuts as well. Combining geometric concepts of flux and length/area within the global optimization framework of graph cuts allows principled discrete segmentation models and advances the state of the art for the graph cuts methods in vision. In particular, we address the “shrinking ” problem of graph cuts, improve segmentation of long thin objects, and introduce useful shape constraints. 1.

### Citations

860 | An experimental comparison of mincut/max-flow algorithms for energy minimization in vision - Boykov, Kolmogorov - 2001 |

742 | What energy functions can be minimized via graph cuts
- Kolmogorov, Zabin
- 2004
(Show Context)
Citation Context ...der any Riemannian metric. Here we show that graph cuts on directed regular grids can approximate a significantly more general class of continuous non-symmetric metrics. Using submodularity condition =-=[1, 11]-=-, we obtain a tight characterization of graph-representable metrics. Such “submodular” metrics have an elegant geometric interpretation via hypersurface functionals combining length/area and flux. Pra... |

695 | Interactive graph cuts for optimal boundary and region segmentation
- Boykov, Jolly
- 2001
(Show Context)
Citation Context ...-dimensional settings. Discrete graph-cuts methods are easy to implement and their discrete framework is sufficiently flexible to include various forms of regional, boundary, or geometric constraints =-=[2, 3]-=-. Non-parametric implicit representation of hypersurfaces via graph cuts poses no restrictions on topological properties of segments. Yuri Boykov University of Western Ontario London, ON, Canada yuri@... |

188 | Computing geodesics and minimal surfaces via graph cuts
- Boykov, Kolmogorov
(Show Context)
Citation Context ...October 2005 vol.., p.1 What Metrics Can Be Approximated by Geo-Cuts, or Global Optimization of Length/Area and Flux Vladimir Kolmogorov Microsoft Research Cambridge, UK vnk@microsoft.com Abstract In =-=[3]-=- we showed that graph cuts can find hypersurfaces of globally minimal length (or area) under any Riemannian metric. Here we show that graph cuts on directed regular grids can approximate a significant... |

114 | Pseudo-Boolean optimization
- Boros, Hammer
- 2002
(Show Context)
Citation Context ...der any Riemannian metric. Here we show that graph cuts on directed regular grids can approximate a significantly more general class of continuous non-symmetric metrics. Using submodularity condition =-=[1, 11]-=-, we obtain a tight characterization of graph-representable metrics. Such “submodular” metrics have an elegant geometric interpretation via hypersurface functionals combining length/area and flux. Pra... |

107 | Flux Maximizing Geometric Flows
- Vasilevsky, Siddiqi
- 2001
(Show Context)
Citation Context ...tivated hypersurface functionals and show how to globally optimize any combination of length/area and flux of a given vector field. The concept of flux was recently introduced into computer vision by =-=[13]-=- but it was mainly studied within variational framework so far. We are first to show that flux can be integrated into graph cuts as well. Combining geometric concepts of flux and length/area within th... |

75 | Globally optimal regions and boundaries as minimum ratio weight cycles
- Jermyn, Ishikawa
(Show Context)
Citation Context ...a unit of time. Surface’s orientation determines the sign of flux. In fact, flux optimization has been previously considered in image analysis by a number of researchers. Image segmentation method in =-=[8]-=- uses minimum ratio cycle algorithm to find a contour with the largest ratio of (image gradient) flux and length. Global solution in [8] is limited to 2D contours. In a more general context, flux-base... |

50 | Regularized laplacian zero crossings as optimal edge integrators
- Kimmel, Bruckstein
(Show Context)
Citation Context ...ding gradient flow equation was first derived. Practical effectiveness of flux within variational framework (level-sets) was demonstrated in medical image segmentation [13, 6] and in edge integration =-=[9]-=-. In particular, [13] showed that flux optimization helps to segment narrow elongated structures like vessels. To the best of our knowledge, we are the first to demonstrate that flux can be integrated... |

28 | Flux invariants for shape
- Dimitrov, Damon, et al.
(Show Context)
Citation Context ...he vessel. Presence of length/area in the functional helps to add regularity. We also discuss a novel way of using flux for a shape constraint. Models using distance maps for shape priors are popular =-=[12, 7, 5]-=-. Our shape constraint for N-D image segmentation uses flux of distance map gradients. Two examples of such vector fields are shown in Figure 7. The gradients are for a signed distance map from a give... |

25 | Statistical shape knowledge in variational motion segmentation
- Cremers, Schnörr
- 2003
(Show Context)
Citation Context ...he vessel. Presence of length/area in the functional helps to add regularity. We also discuss a novel way of using flux for a shape constraint. Models using distance maps for shape priors are popular =-=[12, 7, 5]-=-. Our shape constraint for N-D image segmentation uses flux of distance map gradients. Two examples of such vector fields are shown in Figure 7. The gradients are for a signed distance map from a give... |

23 | S.J.: A discrete global minimization algorithm for continuous variational problems. harvard computer science technical report
- Kirsanov, Gortler
- 2004
(Show Context)
Citation Context ...tation of the material may assume that the reader is familiar with the main ideas on how regional and boundary cues and hard constraints can be integrated within graph cuts [2]. A very recent work in =-=[10]-=- offers somewhat complementary approach where continuous hypersurfaces are explicitly represented by facets obtained by slicing space with a large number of lines/planes. Formed cells are used as irre... |

20 |
Statistical models for medical image analysis
- Leventon
- 2000
(Show Context)
Citation Context ...he vessel. Presence of length/area in the functional helps to add regularity. We also discuss a novel way of using flux for a shape constraint. Models using distance maps for shape priors are popular =-=[12, 7, 5]-=-. Our shape constraint for N-D image segmentation uses flux of distance map gradients. Two examples of such vector fields are shown in Figure 7. The gradients are for a signed distance map from a give... |

19 | A multi-scale geometric flow for segmenting vasculature in mri
- Descoteaux, Collins, et al.
- 2004
(Show Context)
Citation Context ...osed in [13] where the corresponding gradient flow equation was first derived. Practical effectiveness of flux within variational framework (level-sets) was demonstrated in medical image segmentation =-=[13, 6]-=- and in edge integration [9]. In particular, [13] showed that flux optimization helps to segment narrow elongated structures like vessels. To the best of our knowledge, we are the first to demonstrate... |