## Globally optimal regions and boundaries as minimum ratio weight cycles (2001)

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Venue: | IEEE Transactions on Pattern Analysis and Machine Intelligence |

Citations: | 74 - 2 self |

### BibTeX

@ARTICLE{Jermyn01globallyoptimal,

author = {Ian H. Jermyn and Hiroshi Ishikawa},

title = {Globally optimal regions and boundaries as minimum ratio weight cycles},

journal = {IEEE Transactions on Pattern Analysis and Machine Intelligence},

year = {2001},

volume = {23},

pages = {1075--1088}

}

### Years of Citing Articles

### OpenURL

### Abstract

Abstract. We describe a new form of energy functional for the modelling and identification of regions in images. The energy is defined on the space of boundaries in the image domain, and can incorporate very general combinations of modelling information both from the boundary (intensity gradients,...), and from the interior of the region (texture, homogeneity,. We describe two polynomial-time digraph algorithms for finding the global minima of this energy. One of the algorithms is completely general, minimizing the functional for any choice of modelling information. It runs in a few seconds on a 256 × 256 image. The other algorithm applies to a subclass of functionals, but has the advantage of being extremely parallelizable. Neither algorithm requires initialization. 1.

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Citation Context ...r, because it uses an energy optimization criterion explicitly, is the work on active contours and deformable models. The seminal works4 Ian H. Jermyn and Hiroshi Ishikawa in this area is Kass et al. =-=[16]-=- and Blake and Zisserman [3], and much subsequent work follows this both in the form of the energy functionals used, and in algorithmic techniques. Typical energies include data terms that favour cont... |

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Citation Context ...hm works however, it is possible to restrict attention to those curves passing through a fixed point, which is not possible with the minimum ratio weight cycle algorithm described here. Shi and Malik =-=[28]-=- use a generalized eigenvalue method to find normalized cuts of an image graph, and use this to partition the image by iterating the algorithm. The algorithm is a sophisticated and general clustering ... |

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Citation Context ...ed label-correcting algorithm instead removes vertices u from a list and updates the vertices v for which 〈u, v〉 ∈ E. If v is not in the list, it is added. Both these algorithms are pseudo-polynomial =-=[1]-=-. There is an O(mn) implementation of the generic label-correcting algorithm that uses a queue as the list structure, adding updated vertices to the back. The dequeue implementation of the modified la... |

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Citation Context ...optimization criterion explicitly, is the work on active contours and deformable models. The seminal works4 Ian H. Jermyn and Hiroshi Ishikawa in this area is Kass et al. [16] and Blake and Zisserman =-=[3]-=-, and much subsequent work follows this both in the form of the energy functionals used, and in algorithmic techniques. Typical energies include data terms that favour contours passing through points ... |

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Citation Context ...of the algorithms find only local minima, or approximations to global minima over this limited set of contours. Globally minimum closed contours are not found. The interesting paper of Zhu and Yuille =-=[32]-=- uses both region and boundary information within an energy optimization model for image segmentation. The work brings active contour and region growing techniques together within a Bayesian framework... |

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Citation Context ...ntrolled feature of the energy itself as it is for linear energies. In [12], a different algorithm was proposed for the optimization of this form of energy. This ‘minimum mean weight cycle’ algorithm =-=[15]-=- was not general, optimizing only a small subset of the possible instances of the energy, and in addition, had no sensible continuum interpretation. This introduced an undesirable dependence on the di... |

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Citation Context ...cribe some specific models of regions and boundaries, and demonstrate the results of applying these models to images. 2. Related work Early work on contour-based grouping includes Parent and Zucker’s =-=[25]-=- work using relaxation methods, Sha’ashua and Ullman’s [27] work on saliency networks, and Guy and Medioni’s [10] work using voting schemes. Elder and Zucker [8] developed a method for finding closed ... |

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Citation Context ...ges. 2. Related work Early work on contour-based grouping includes Parent and Zucker’s [25] work using relaxation methods, Sha’ashua and Ullman’s [27] work on saliency networks, and Guy and Medioni’s =-=[10]-=- work using voting schemes. Elder and Zucker [8] developed a method for finding closed contours using chains of tangent vectors, but they drastically prune the search space to render tractable the exp... |

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Citation Context ...onnection between minimal energy curves (“elastica”) and stochastic processes [24], and discussed their application to computer vision. Williams, Thornber and Mahamud [21, 30] and Williams and Jacobs =-=[31]-=- find closed curves in edge maps using stochastic completion fields closely related to elastica. There is a close connection between their work and the minimum mean weight energy discussed in section ... |

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Citation Context ... much less than Karp’s algorithm. Further details of the time and space bounds are to be found in appendix A. There are also close to linear time negative cycle detection algorithms for planar graphs =-=[17]-=-. These would allow improved performance in the case of single images, but we prefer to describe the general case here since it generalises to higher dimensions, the application of which to stereo and... |

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Citation Context ...aphs. Amini et al. [2] use dynamic programming as part of a gradient descent procedure. Montanari [23] uses dynamic programming to find the minimum energy path between given end-points. Geiger et al. =-=[9]-=- use initialization with a series of points, and a choice of window around those points, to delineate the space of contours considered. In all cases initialization and restricted regions of the image ... |

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Citation Context ...emonstrate the results of applying these models to images. 2. Related work Early work on contour-based grouping includes Parent and Zucker’s [25] work using relaxation methods, Sha’ashua and Ullman’s =-=[27]-=- work on saliency networks, and Guy and Medioni’s [10] work using voting schemes. Elder and Zucker [8] developed a method for finding closed contours using chains of tangent vectors, but they drastica... |

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Citation Context ...s are almost never solved globally. Motivated by the above considerations, and by psychological work, which since the Gestalt movement has emphasized the importance of contour closure in human vision =-=[13, 14, 18, 6, 7]-=-, we propose an energy functional on boundaries in images that takes the form of a ratio of two integrals around the boundary. Our approach thus falls into the second category discussed above, that of... |

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Citation Context ...e set themselves. This work is based on edge maps rather than intensity gradients as such. Mumford first pointed out the connection between minimal energy curves (“elastica”) and stochastic processes =-=[24]-=-, and discussed their application to computer vision. Williams, Thornber and Mahamud [21, 30] and Williams and Jacobs [31] find closed curves in edge maps using stochastic completion fields closely re... |

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Citation Context ...lobal solutions, an approach that can be interpreted in terms of shortest paths in directed acyclic graphs. Amini et al. [2] use dynamic programming as part of a gradient descent procedure. Montanari =-=[23]-=- uses dynamic programming to find the minimum energy path between given end-points. Geiger et al. [9] use initialization with a series of points, and a choice of window around those points, to delinea... |

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Citation Context ...mic programming techniques to minimize contour energies in an effort to reach global solutions, an approach that can be interpreted in terms of shortest paths in directed acyclic graphs. Amini et al. =-=[2]-=- use dynamic programming as part of a gradient descent procedure. Montanari [23] uses dynamic programming to find the minimum energy path between given end-points. Geiger et al. [9] use initialization... |

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Citation Context ...of the use of gradient descent however, the algorithm can find only local minima of the energy functional used, even in the case where only a single region is sought. The paper by Cox, Rao, and Zhong =-=[4]-=- is particularly related to our work. They use a ratio energy also, of a generalised area to a generalised length, and use a “pinned ratio” algorithm to optimise it. This algorithm finds the optimal c... |

52 | The Effect of Contour Closure on the Rapid Discrimination of Two-Dimensional Shapes
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Citation Context ...s are almost never solved globally. Motivated by the above considerations, and by psychological work, which since the Gestalt movement has emphasized the importance of contour closure in human vision =-=[13, 14, 18, 6, 7]-=-, we propose an energy functional on boundaries in images that takes the form of a ratio of two integrals around the boundary. Our approach thus falls into the second category discussed above, that of... |

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Citation Context ...ormalized cuts of an image graph, and use this to partition the image by iterating the algorithm. The algorithm is a sophisticated and general clustering method, and as such, although Leung and Malik =-=[20]-=- extend the work interestingly by incorporating weak contour continuity information into the region-based model, does not seem amenable to the natural inclusion of boundary information. The big advant... |

33 | Globally optimal regions and boundaries
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Citation Context ...invariance can be broken, for example by the addition of an area term, but this becomes a modelling decision, rather than an uncontrolled feature of the energy itself as it is for linear energies. In =-=[12]-=-, a different algorithm was proposed for the optimization of this form of energy. This ‘minimum mean weight cycle’ algorithm [15] was not general, optimizing only a small subset of the possible instan... |

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Citation Context ...a region can always be modelled by its characteristic function, which is a field on the image), they do correspond conceptually to distinctions in the human visual system. It seems likely for example =-=[26, 29]-=- that human perception of motion is based both on generic low-level computations, as well as on the identification and tracking of specific objects. They also lead to different visions of how to proce... |

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Citation Context ...-dependent (a directed graph in which opposing edges have weights of opposite sign). Although there 1 There is also a linear programming approach to the problem, described by Dantzig, Blatner and Rao =-=[5]-=-.sGlobally Optimal Regions and Boundaries 11 is no way to conclude that there is a negative cycle in this more general case, a problem still arises. Minimizing cycles may self-intersect or even self-o... |

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Citation Context ...o boundary length. In this paper, we introduce a different polynomial-time algorithm, the ‘minimum ratio weight cycle’ algorithm, that globally optimizes any instance of the energy. The new algorithm =-=[19, 22]-=- outperforms the old one not only in its much broader range of applicability, but even in its time and space requirements in the instances that the old algorithm could solve. We describe the previous ... |

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Citation Context .... Mumford first pointed out the connection between minimal energy curves (“elastica”) and stochastic processes [24], and discussed their application to computer vision. Williams, Thornber and Mahamud =-=[21, 30]-=- and Williams and Jacobs [31] find closed curves in edge maps using stochastic completion fields closely related to elastica. There is a close connection between their work and the minimum mean weight... |

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Citation Context ...o boundary length. In this paper, we introduce a different polynomial-time algorithm, the ‘minimum ratio weight cycle’ algorithm, that globally optimizes any instance of the energy. The new algorithm =-=[19, 22]-=- outperforms the old one not only in its much broader range of applicability, but even in its time and space requirements in the instances that the old algorithm could solve. We describe the previous ... |

12 | Region extraction from multiple images
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Citation Context ...oved performance in the case of single images, but we prefer to describe the general case here since it generalises to higher dimensions, the application of which to stereo and motion is described in =-=[11]-=-. Note that the way the algorithm works shows what was claimed in section 1, that the energy in equation 1 is equivalent (as far as global minima go) to a linear energy of the form N[∂R] − βD[∂R], but... |

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Citation Context .... Mumford first pointed out the connection between minimal energy curves (“elastica”) and stochastic processes [24], and discussed their application to computer vision. Williams, Thornber and Mahamud =-=[21, 30]-=- and Williams and Jacobs [31] find closed curves in edge maps using stochastic completion fields closely related to elastica. There is a close connection between their work and the minimum mean weight... |

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Citation Context ...a region can always be modelled by its characteristic function, which is a field on the image), they do correspond conceptually to distinctions in the human visual system. It seems likely for example =-=[26, 29]-=- that human perception of motion is based both on generic low-level computations, as well as on the identification and tracking of specific objects. They also lead to different visions of how to proce... |

4 |
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Citation Context ...s are almost never solved globally. Motivated by the above considerations, and by psychological work, which since the Gestalt movement has emphasized the importance of contour closure in human vision =-=[13, 14, 18, 6, 7]-=-, we propose an energy functional on boundaries in images that takes the form of a ratio of two integrals around the boundary. Our approach thus falls into the second category discussed above, that of... |