### Citations

1616 |
Image Analysis and MathematiCal Morphology
- Serra
- 1982
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Citation Context ...s). Let M be a subset of a picture, and B a structuring element (i.e., a “small” subset of the grid). (MªkB) means k ≥ 0 successive erosions of M by B, with 0B = ∅. M ◦B means the opening of M by B [(=-=Serra 1982-=-)]. A distance transform (defined by the structuring element) can then be defined in terms of erosions and openings. Following (Serra 1982), the morphological skeleton S of a region M can be calculate... |

252 |
Sequential operations in digital picture processing
- Rosenfeld, Pfaltz
- 1966
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Citation Context ...or linear) set, and a skeletal curve is an abstract disambiguation of a (biomedical) skeleton (see Figure 1.1). Both concepts of skeletal curves have been “digitized”. Digital medial axes go back on (=-=Rosenfeld and Pfaltz 1966-=-), who defined them based on distance transforms. For a given connected set, they are not necessarily connected. Digital linear skeletons are defined by topology-preserving thinning [see (Rosenfeld 19... |

119 |
Toriwaki: New algorithms for euclidean distance transformations of an n-dimensional digitised picture with applications
- Saito, I
- 1994
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Citation Context ...e have: 4.1. Distance transform 37 f3(x, y) = min{(f2(x, j))2 + (y − j)2 : j = 1, ..., n} (4.32) Efficient implementations for solving this minimization problem have been discussed in various papers (=-=Saito and Toriwaki 1994-=-, Hirata 1996, Meijster et al. 2000, Toriwaki and Mori 2001, Bailey 2004). We give a geometric interpretation that illustrates the idea. For a fixed column (i.e., x constant, f2(x, j) = g(j)) and a fi... |

112 |
Connectivity in digital pictures
- Rosenfeld
- 1970
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Citation Context ... Pfaltz 1966), who defined them based on distance transforms. For a given connected set, they are not necessarily connected. Digital linear skeletons are defined by topology-preserving thinning [see (=-=Rosenfeld 1970-=-)], which is a repeated removal of “deletable elements”; today those are called simple pixels for two-dimensional (2D) pictures, or simple voxels for three-dimensional (3D) pictures. The calculation o... |

55 | A general algorithm for computing distance transforms in linear time. - Meijster, Roerdink, et al. - 2000 |

55 |
A method for obtaining skeletons using a quasi-Euclidean distance
- Montanari
- 1968
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Citation Context ...alculated with different metrics. For approximations of the Euclidean distance, some authors suggested the use of different weights for steps within a grid point neighborhood (Sanniti di Baja 1994). (=-=Montanari 1968-=-) introduced quasi-Euclidean distances, such as (d4(p, q) + 2.5. Metrics for Digital Pictures 21 Figure 2.6: Local maxima of distance maps, calculated with d4-metric (left), d8-metric (middle) and de-... |

55 | A.: A 3D 6-subiteration thinning algorithm for extracting medial lines.
- Palagyi, Kuba
- 1998
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Citation Context ...nt adjacency relations (0-, 1-, 2) define different types of directions and the number of subiterations (6, 8 or 12). Tests of simplicity are restricted to all 0-adjacent elements, as for example in (=-=Palagyi and Kuba 1998-=-, Palagyi and Kuba 1999). The location of the resulting set of skeletal curves depends on the defined sequence of subiterations. Results vary under rotation. The third strategy is applied in (Bertrand... |

51 |
The vector distance transform in two and three dimensions.
- Mullikin
- 1992
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Citation Context ... than the grid constant, and with increasing grid resolution they have almost no impact. The time complexity of the algorithm is linear O(n), where n is the total number of pixels in the picture. In (=-=Mullikin 1992-=-), the concept has been adopted for 3D space. 4.1.3 Iterative distance transform algorithms For completeness reasons, we briefly describe an iterative algorithm which uses the grassfire model and weig... |

50 | A parallel 3d 12-subiteration thinning algorithm.
- Pal´agyi, Kuba
- 1999
(Show Context)
Citation Context ...(0-, 1-, 2) define different types of directions and the number of subiterations (6, 8 or 12). Tests of simplicity are restricted to all 0-adjacent elements, as for example in (Palagyi and Kuba 1998, =-=Palagyi and Kuba 1999-=-). The location of the resulting set of skeletal curves depends on the defined sequence of subiterations. Results vary under rotation. The third strategy is applied in (Bertrand and Aktouf 1994, Saha ... |

47 |
A Thinning Algorithm for Discrete Binary Images. Comput.Graph. Image Process.
- Pavlidis
- 1980
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Citation Context ...for end elements to ensure Condition 3 of the criteria for topology preservation. Several thinning algorithms stop these iterative procedure if all remaining elements are ”multiple” [see for example (=-=Pavlidis 1980-=-, Borgefors et al. 1999)]. Multiple elements are not 72 5. Topologic skeletons uniquely defined. The identification of all multiple elements is not enough to generate digital curves or arcs. Normally,... |

40 |
Characterization of Parallel Thinning Algorithms”,
- Rosenfeld
- 1975
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Citation Context ...uivalent to the smoothed picture. It approximates the medial axis. End elements are preserved. One standard example of using directional subiterations is the four-subiteration algorithm by Rosenfeld (=-=Rosenfeld 1975-=-) (Algorithm B). A pixel p is deleted if B1: p is an object pixel. B2: p is not isolated or an end pixel, that means L8(p) > 1. B3: XH(p) = 1. B4: P (q2i+1) = 0, where i = 1, ..., 4, 1, ..., at succes... |

39 |
Continuous Skeletons from Digitized Image,”
- Montanari
- 1969
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Citation Context ...ple, for determining in 2D the labels of the d8- or d4-distance transform, the structural elements are 1 1 1 0 1 0 1 1 1 1 1 1 1 1 1 0 1 0 respectively. Grassfire Transform (Calabi and Hartnett 1968, =-=Montanari 1969-=-) proposed an alternative approach for calculating a digital version of a medial axis. The concept is called prairie fire or grassfire transform. Assume a fire front that starts at the same time at ev... |

35 |
A boolean characterization of 3-D simple points
- Bertrand
- 1996
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Citation Context ... It can be done by using algorithms derived from graph theory. The number of computations depends on the size of the components. The following characterization of 26-simple object elements is due to (=-=Malandain and Bertrand 1992-=-, Saha et al. 1991): 7. CHARACTERIZATION. An object voxel p of a 3D picture P is 26-simple in P iff (i) p is 26-adjacent to another object voxel q, and (ii) p is 6-adjacent to a non-object voxel q′, a... |

33 |
Detection of 3D Simple Points for Topology Preserving,”
- Saha, Chaudhuri
- 1994
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Citation Context ... determine simple voxels in 3D? Are these existing characterizations equivalent to one-another? – First we review some definitions. In 1994, two independent publications (Bertrand and Malandain 1994, =-=Saha and Chaudhuri 1994-=-) proposed the following: 5.2. Simple elements 61 6. CHARACTERIZATION. A voxel (p, P (p)) of a picture P is a 26-simple grid element iff it is 26-adjacent to exactly one 26-component of voxels in A26(... |

31 |
A New Shape Preserving Parallel Thinning Algorithm for 3D
- Saha, Chaudhuri, et al.
- 1997
(Show Context)
Citation Context ... 1999). The location of the resulting set of skeletal curves depends on the defined sequence of subiterations. Results vary under rotation. The third strategy is applied in (Bertrand and Aktouf 1994, =-=Saha et al. 1997-=-). Non-end elements of ideally thin curves do not satisfy pixel deletion criteria in the sense that they are non-simple (for p with exactly one 0-adjacent object voxel is χ(Kp) = 1 and Kp and Kp are c... |

28 | Hausegger K: Sequential 3D Thinning Algorithm and Its Medical Applications.
- Palágyi, Balogh, et al.
- 2001
(Show Context)
Citation Context ...sary elements or to connect elements are not required. Algorithm C generates 4-connected skeletons, algorithm D generates ”thick” skeletons and distance skeletons are not connected. The algorithm in (=-=Palagyi et al. 2001-=-, Palagyi and Kuba 1998) uses 6 subiterations (U, D, N, S, W, E) (see Figure 5.13). The algorithm follows the concept of directional thinning as algorithm B in 2D and the authors in (Palagyi and Kuba ... |

26 |
An analysis of topological properties of digitized binary pictures using local features
- Yokoi, Toriwaki, et al.
- 1975
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Citation Context ...nct 8-components of 1’s in A8(p) in case there is at least one 0 in A4(p), and the H-crossing number is always equal to the number of distinct 4-adjacent 4-components of 0’s in A8(p). 8. DEFINITION. (=-=Yokoi et al. 1975-=-) The number of distinct 4-adjacent 4-components of 1’s (0’s) is called connectivity number XY (p) (XY (p)) with: XY (p) = 4∑ i=1 ai and XY (p) = 4∑ i=1 bi where, in A8(p) and in picture P , ai = P (q... |

22 | Recognition, - Rutovitz - 1966 |

19 | 2003, Quantitative analysis of intrathoracic airway trees: methods and validation - Palagyi, Tschirren, et al. |

19 | E.: Exact Medial Axis with Euclidean Distance. - Remy, Thiel - 2005 |

13 |
A.: Topology-preserving deformations of two-valued digital pictures. Graph. Models Image Process
- Rosenfeld, Kong, et al.
- 1998
(Show Context)
Citation Context ...h are easy to compute, or easy to visualize for 2D and 3D digital pictures. 5.2.1 Equivalent characterizations of simple elements in 2D We use the grid point model and we define a simple 1 following (=-=Rosenfeld et al. 1998-=-) using a good pair α, α′: if α = 4 then α′ = 8, and if α = 8 then α′ = 4. 17. DEFINITION. A 1 of a picture P is called α-simple (α ∈ {4, 8}) if it is α-adjacent to exactly one α-component of 1’s in A... |

12 | 2001, Distance transformation and skeletonization of 3d pictures and their applications to medical images - Toriwaki, Mori |

11 |
On topology preservation
- Ma
- 1994
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Citation Context ...d the other one between the α′-connected components of non-object elements of P and the α′-connected components after deformation of P by D. We consider the following definition of a simple sequence (=-=Ma 1994-=-, Kong 1995): 19. DEFINITION. Let P = P0 be the original picture and Pi the result of deformation i. A sequence q1, q2, ...qn of distinct elements of an α-connected component in a digital picture P is... |

8 | Weighted distance transforms for volume images digitized in elongated voxel grids, - Sintorn, Borgefors - 2004 |

8 |
Uber die allgemeinen Cantorischen Kurven, Annual Meeting Deutsche Mathematiker Vereinigung,
- Urysohn
- 1923
(Show Context)
Citation Context ...ly defined) linear skeleton or a (uniquely defined) medial axis, aiming at topologic or geometric studies. Curves in general have been defined by P. Urysohn and K. Menger in the 1920s and 1930s [see (=-=Urysohn 1923-=-) and (Menger 1932)], also introducing branch or end points of curves. A curve finally defines, at a more abstract level, an undirected graph, whose nodes are identified with branch or end points, and... |

6 |
Principles and algorithms for 2-D and 3-D shrinking
- Saha, Chanda, et al.
- 1991
(Show Context)
Citation Context ...rithms derived from graph theory. The number of computations depends on the size of the components. The following characterization of 26-simple object elements is due to (Malandain and Bertrand 1992, =-=Saha et al. 1991-=-): 7. CHARACTERIZATION. An object voxel p of a 3D picture P is 26-simple in P iff (i) p is 26-adjacent to another object voxel q, and (ii) p is 6-adjacent to a non-object voxel q′, and (iii) the set o... |

6 | Well-shaped, stable and reversible skeletons from the (3,4)-distance transform - Baja, G - 1994 |

4 |
Skeleton generation from x,y boundary sequences
- Shapiro, Pisa, et al.
- 1981
(Show Context)
Citation Context ... by points on the frontier of M . (For further details, see the reference.) Middle-Line Transform There are also many proposals for geometric skeletons based on heuristics. We only give one example. (=-=Shapiro et al. 1981-=-) uses frontier segments for the generation of skeletons. The approach assumes that the frontier of an object M is very smooth. Oppositely located segments A and B of the frontier are used to locate m... |

3 | 2002, Topologies for binary or multi-level images - a review - Rosenfeld, Klette |

2 |
A 3d fully parallel 3d thinning algorithm for generating medial faces
- Ma
- 1995
(Show Context)
Citation Context ...e fire starts at every border element at the same time. Tests of simplicity for a set D in one iteration step are done by examining neighborhoods of the 0-adjacent elements of p ∈ D. For example, in (=-=Ma 1995-=-) an algorithm for 3D pictures is published where the test of simplicity for a single element depends on 30 elements, and an additional test is required to guarantee topology preservation. In 2D, sequ... |

2 |
Generating skeletons and center lines from the distance transform., Graphical Models and Image Processing
- Niblack, Gibbons, et al.
- 1992
(Show Context)
Citation Context ...elements is very noise sensitive. In general, a postprocessing step is required to connect all elements of the medial axis. In 2D, the computation of saddle points is useful to generate connectivity (=-=Niblack et al. 1992-=-). The identification of such points in 3D is expensive. Some authors [e.g., in (Toriwaki and Mori 2001)] use the Euclidean distance transform image as input for a topologic thinning algorithm. Elemen... |

2 | 2005, On the dedection of simple points in higher dimensions using cubical homology - Niethammer, Kalies, et al. |

2 |
The euclidean distance transform., Dissertation No.304
- Ragnemalm
- 1993
(Show Context)
Citation Context ...e simplicity of TPA-algorithms originally designed for grid metrics d4 and d8. Weighted distance transforms for 3D pictures based on this concept have been studied in (Borgefors 1984, Borgefors 1996, =-=Ragnemalm 1993-=-). In general, those algorithms deliver approximations of Euclidean distances. The time complexity of those algorithms is linear O(n), where n is the total number of pixels in the picture. 4.1.2 Vecto... |