## Ruminations on Tarjan’s union-find algorithm and connected operators

Venue: | in Mathematical Morphology: 40 Years On (Proc. of ISMM). 2005, Computational Imaging and Vision Series |

Citations: | 4 - 3 self |

### BibTeX

@INPROCEEDINGS{Géraud_ruminationson,

author = {Thierry Géraud},

title = {Ruminations on Tarjan’s union-find algorithm and connected operators},

booktitle = {in Mathematical Morphology: 40 Years On (Proc. of ISMM). 2005, Computational Imaging and Vision Series},

year = {},

pages = {105--116},

publisher = {Springer}

}

### OpenURL

### Abstract

Abstract This papers presents a comprehensive and general form of the Tarjan’s unionfind algorithm dedicated to connected operators. An interesting feature of this form is to introduce the notion of separated domains. The properties of this form and its flexibility are discussed and highlighted with examples. In particular, we give clues to handle correctly the constraint of domain-disjointness preservation and, as a consequence, we show how we can rely on “union-find ” to obtain algorithms for self-dual filters approaches and levelings with a marker function. Keywords: Union-find algorithm, reconstructions, algebraic openings and closings, domaindisjointness preservation, self-dual filters, levelings.

### Citations

254 |
Efficiency of a good but not linear set union algorithm
- Tarjan
(Show Context)
Citation Context ...uch operators is to compute from an input image the resulting partition. The Tarjan’s Union-Find Algorithm, union-find for short, computes a forest of disjoint sets while representing a set by a tree =-=[16]-=-. A connected component of points or a flat zone is thus encoded into a tree; a point becomes a node and a partition is a forest. union-find has been used to implement some connected operators; among ... |

204 | Morphological grayscale reconstruction in image analysis applications and efficient algorithms
- Vincent
- 1993
(Show Context)
Citation Context ...om our experiments, an appropriate choice for D ′ —depending on a priori knowledge about F and G—makes the union-find-based approach a serious competitor of the efficient hybrid algorithm proposed in =-=[18]-=-. Last, the case of regional extrema identification is summarized in the table below.s10 Ruminations on Tarjan’s Union-Find Algorithm and Connected Operators default init(p) border(n, p) merge(r, p) R... |

144 | Morphological Image Analysis - Soille - 2002 |

136 | The Watershed Transform: Definitions, Algorithms and Parallelization Techniques
- Roerdink, Meijster
- 1999
(Show Context)
Citation Context ...ons, we do not address any single optimization issue. Moreover, we do not enter into a comparison between union-find-based algorithms and other approaches; for those subjects, the reader can refer to =-=[13, 7]-=-. We claimed in [1] that our generic C++ image processing library, Olena [12], has been designed so that algorithms can easily be translated into programs while remaining very readable. To sustain thi... |

39 | A comparison of algorithms for connected set openings and closings
- Meijster, Wilkinson
- 2002
(Show Context)
Citation Context ...ons, we do not address any single optimization issue. Moreover, we do not enter into a comparison between union-find-based algorithms and other approaches; for those subjects, the reader can refer to =-=[13, 7]-=-. We claimed in [1] that our generic C++ image processing library, Olena [12], has been designed so that algorithms can easily be translated into programs while remaining very readable. To sustain thi... |

36 | A general approach to connected-component labeling for arbitrary image representations
- Dillencourt, Samet, et al.
- 1992
(Show Context)
Citation Context ...r a flat zone is thus encoded into a tree; a point becomes a node and a partition is a forest. union-find has been used to implement some connected operators; among them, connected component labeling =-=[2]-=-, a watershed transform [6], algebraic closing and opening [19], and component trees2 Ruminations on Tarjan’s Union-Find Algorithm and Connected Operators computation [4, 11]. A tremendous advantage o... |

21 |
From connected operators to levelings
- Meyer
- 1998
(Show Context)
Citation Context ...s. Introduction Connected operators have the important property of simplifying images while preserving contours. Several sub-classes of these operators have been formalized having stronger properties =-=[8]-=- and numerous applications have been derived from them, e.g., scale-space creation and feature analysis [17], video compression [14], or segmentation [10]. The behavior of connected operators is to me... |

18 | M.: Quasi-linear algorithm for the component tree
- Najman, Couprie
(Show Context)
Citation Context ... connected component labeling [2], a watershed transform [6], algebraic closing and opening [19], and component trees2 Ruminations on Tarjan’s Union-Find Algorithm and Connected Operators computation =-=[4, 11]-=-. A tremendous advantage of union-find lies in its simplicity. However, the descriptions of morphological operators relying on this algorithm are usually spoiled by the presence of too many implementa... |

16 |
The levelings
- Meyer
- 1998
(Show Context)
Citation Context ...sion of union-find does not compute irrelevant components or flat zones. We can now reuse the descriptions given previously of union-find-based operators to build levelings with markers as defined in =-=[9]-=-, a domain-preserving self-dual reconstruction, and partial self-dual operators defined with the infsemilattice approach in [3]. Results are summarized in the table below. D ↑◦ D ↓◦ RD sub-domain sub-... |

11 | A disjoint set algorithm for the watershed transform
- Meijster, Roerdink
- 1998
(Show Context)
Citation Context ...ed into a tree; a point becomes a node and a partition is a forest. union-find has been used to implement some connected operators; among them, connected component labeling [2], a watershed transform =-=[6]-=-, algebraic closing and opening [19], and component trees2 Ruminations on Tarjan’s Union-Find Algorithm and Connected Operators computation [4, 11]. A tremendous advantage of union-find lies in its si... |

10 | Fast morphological attribute operations using tarjan’s union-find algorithm
- Wilkinson, Roerdink
- 2000
(Show Context)
Citation Context ...ode and a partition is a forest. union-find has been used to implement some connected operators; among them, connected component labeling [2], a watershed transform [6], algebraic closing and opening =-=[19]-=-, and component trees2 Ruminations on Tarjan’s Union-Find Algorithm and Connected Operators computation [4, 11]. A tremendous advantage of union-find lies in its simplicity. However, the descriptions ... |

9 | Inf-semilattice approach to self-dual morphology
- Heijmans, Keshet
- 2001
(Show Context)
Citation Context ...jointness preservation property. Consequently we show that union-find is a simple way to get algorithms for folding induced self-dual filters [5], the inf-semilattice approach to self-dual morphology =-=[3]-=-, and levelings defined on two functions [10]. In order to keep implementation details away from algorithmic considerations, we do not address any single optimization issue. Moreover, we do not enter ... |

7 | Morphological scale-space analysis and feature extraction
- Vachier
- 2001
(Show Context)
Citation Context ...ours. Several sub-classes of these operators have been formalized having stronger properties [8] and numerous applications have been derived from them, e.g., scale-space creation and feature analysis =-=[17]-=-, video compression [14], or segmentation [10]. The behavior of connected operators is to merge most of the flat zones of an input image, thus delivering a partition which is much coarser than the inp... |

6 | Generic implementation of morphological image operators
- Darbon, Duret-Lutz
- 2002
(Show Context)
Citation Context ...any single optimization issue. Moreover, we do not enter into a comparison between union-find-based algorithms and other approaches; for those subjects, the reader can refer to [13, 7]. We claimed in =-=[1]-=- that our generic C++ image processing library, Olena [12], has been designed so that algorithms can easily be translated into programs while remaining very readable. To sustain this claim, programs g... |

5 |
Salembier’s min-tree algorithm turned into breadth first search
- Hesselink
- 2003
(Show Context)
Citation Context ... connected component labeling [2], a watershed transform [6], algebraic closing and opening [19], and component trees2 Ruminations on Tarjan’s Union-Find Algorithm and Connected Operators computation =-=[4, 11]-=-. A tremendous advantage of union-find lies in its simplicity. However, the descriptions of morphological operators relying on this algorithm are usually spoiled by the presence of too many implementa... |

2 |
image simplification filters for segmentation
- Levelings
(Show Context)
Citation Context ...ave been formalized having stronger properties [8] and numerous applications have been derived from them, e.g., scale-space creation and feature analysis [17], video compression [14], or segmentation =-=[10]-=-. The behavior of connected operators is to merge most of the flat zones of an input image, thus delivering a partition which is much coarser than the input one. In that context, a relevant approach t... |

1 |
Folding induced self-dual filters
- Mehnert, Jackway
- 2000
(Show Context)
Citation Context ...ed for the connected operators θ which verify a domain disjointness preservation property. Consequently we show that union-find is a simple way to get algorithms for folding induced self-dual filters =-=[5]-=-, the inf-semilattice approach to self-dual morphology [3], and levelings defined on two functions [10]. In order to keep implementation details away from algorithmic considerations, we do not address... |

1 |
Generic C++ image processing library, http://olena.lrde.epita.fr, free software available under
- Olena
- 2005
(Show Context)
Citation Context ...into a comparison between union-find-based algorithms and other approaches; for those subjects, the reader can refer to [13, 7]. We claimed in [1] that our generic C++ image processing library, Olena =-=[12]-=-, has been designed so that algorithms can easily be translated into programs while remaining very readable. To sustain this claim, programs given in this paper rely on our library and, thanks to it, ... |

1 |
On filters by reconstruction for size and motion simplification
- Salembier, Ruiz
- 2002
(Show Context)
Citation Context ...s of these operators have been formalized having stronger properties [8] and numerous applications have been derived from them, e.g., scale-space creation and feature analysis [17], video compression =-=[14]-=-, or segmentation [10]. The behavior of connected operators is to merge most of the flat zones of an input image, thus delivering a partition which is much coarser than the input one. In that context,... |