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## Chamfer Distances with Integer Neighborhoods

Citations: | 3 - 0 self |

### Citations

403 |
Distance transformations in digital images.
- Borgefors
- 1986
(Show Context)
Citation Context ...e integer value at (i, j) by N(i, j) and we call r the scaling factor. The function N defined on Mp is called an integer neighborhood. The method was pioneered by Borgefors from 1984 onwards, in [1], =-=[2]-=- and [3]. She determined the optimal w for p = 1 and p = 2 with respect to the maximum absolute error: � � sup �w ((M, j)) − 0≤j≤M � M 2 + j2 � � � for some large M ∈ Z>0. She also determined the opti... |

244 |
Distance transformations in arbitrary dimensions,”
- Borgefors
- 1984
(Show Context)
Citation Context ...te the integer value at (i, j) by N(i, j) and we call r the scaling factor. The function N defined on Mp is called an integer neighborhood. The method was pioneered by Borgefors from 1984 onwards, in =-=[1]-=-, [2] and [3]. She determined the optimal w for p = 1 and p = 2 with respect to the maximum absolute error: � � sup �w ((M, j)) − 0≤j≤M � M 2 + j2 � � � for some large M ∈ Z>0. She also determined the... |

155 |
Distance functions on digital pictures.
- Rosenfeld, Pfaltz
- 1968
(Show Context)
Citation Context ... in the (B)-case, the scaling factor is set automatically to r = n. No previous neighborhoods have been suggested under condition (D), other than the classical city block distance transformation (cf. =-=[7]-=-), which corresponds to 1N D 1 (listed in Table 4). 8 Conclusion The present paper shows that very good approximations of the Euclidean distance can be obtained by a uniform choice of the distance fun... |

54 |
A method for obtaining skeletons using a quasi-Euclidean distance
- Montanari
- 1968
(Show Context)
Citation Context ...ror are given in the general case by wp(i, j) = (1 − ep) � i 2 + j 2 under the condition (B) by w B ⎧ ⎨ |i| if 1 ≤ |i| ≤ p, j = 0 p (i, j) = |j| ⎩ � B 1 − ep if 1 ≤ |j| ≤ p, i = 0 � � ∗ (i, j) ∈ Mp , =-=(6)-=- � � i 2 + j 2 for all other vectors in M ∗ p and under the condition (D) by the Euclidean distance w D p (i, j) = � i2 + j2 � � ∗ (i, j) ∈ Mp . (8) These neighborhoods can not be written in terms of ... |

35 |
Local Distances for Distance Transformations in Two or Three Dimension
- Verwer
- 1991
(Show Context)
Citation Context ...ined the optimal w for p = 1, p = 2 and p = 3 under the restriction that (B) : w(i, 0) = |i| for all i . Borgefors presented some good integer neighborhoods for p = 1, p = 2 and p = 3.sIn 1991 Verwer =-=[11]-=- computed the optimal w for all p with respect to the maximum relative error � � � � w(i, j) � � e := lim sup �� − 1� . (1) (i,j) � i2 + j2 � Besides he gave several integer neighborhoods for p = 1 an... |

17 |
Les distances de chanfrein en analyse d’images : fondements et applications
- Thiel
- 1994
(Show Context)
Citation Context ...ger neighborhood on M1 with scaling factor r. Suppose that n0 ≤ n1 ≤ 2n0. Then the maximum relative error of N is given by � e = max 1 − 1 r min � n0, 1 2 n1 � √ 2 , 1 � n r 2 0 + (n1 − n0) 2 � − 1 . =-=(9)-=- Note that n1 n0 should approximate √ 2 so that every reasonable neighborhood will satisfy the condition n0 ≤ n1 ≤ 2n0. Similarly, every good approximation to the Euclidean distance will satisfy the f... |

12 |
Distance transforms: metrics, algorithms and applications
- Verwer
- 1991
(Show Context)
Citation Context ...i, j) ≥ � i 2 + j 2 for all (i, j) . Distance functions that satisfy (D) are used in applications where it is vital that distances are not underestimated, such as collision avoidance in robotics (cf. =-=[10]-=-). In the present paper we give a formula for the maximum relative error of an integer neighborhood under some mild restrictions. Moreover we introduce a general method to generate good integer neighb... |

11 |
Discrete distance operator on rectangular grids. Pattern Recogn Lett 16:911–23
- Coquin, Bolon
- 1995
(Show Context)
Citation Context ...s he gave several integer neighborhoods for p = 1 and p = 2. In his 1994 PhD thesis [9] Thiel presented numerous examples of integer neighborhoods for p = 2, p = 3 and p = 6. In 1995 Coquin and Bolon =-=[4]-=- extended the theory to pixels on a rectangular lattice instead of a square one. All their integer neighborhoods refer to the square case, with p = 1, p = 2 and p = 3. In unpublished work [5] Hajdu, H... |

9 |
Distances defined by neighborhood sequences", R e c o w
- Yamashita, Ibaraki
- 1986
(Show Context)
Citation Context ..., w = N r is extended to a distance function w on the whole of Z 2 , by taking w(v) as the minimal length over all possible paths from the origin to v composed of steps from Mp. Yamashita and Ibaraki =-=[12]-=- proved that the induced distance is indeed a metric. (The proof is also given by Verwer in [10].)s35 32 29 27 26 25 26 27 29 32 32 28 25 22 21 20 21 22 25 28 29 25 21 18 16 15 16 18 21 25 27 22 18 14... |

4 |
Another comment on a note on ’distance transformation in digital images
- Borgefors
- 1991
(Show Context)
Citation Context ...r value at (i, j) by N(i, j) and we call r the scaling factor. The function N defined on Mp is called an integer neighborhood. The method was pioneered by Borgefors from 1984 onwards, in [1], [2] and =-=[3]-=-. She determined the optimal w for p = 1 and p = 2 with respect to the maximum absolute error: � � sup �w ((M, j)) − 0≤j≤M � M 2 + j2 � � � for some large M ∈ Z>0. She also determined the optimal w fo... |

2 | Tijdeman: Approximation of the Euclidean distance by chamfer distances
- Hajdu, Hajdu, et al.
- 2007
(Show Context)
Citation Context ...nd Bolon [4] extended the theory to pixels on a rectangular lattice instead of a square one. All their integer neighborhoods refer to the square case, with p = 1, p = 2 and p = 3. In unpublished work =-=[5]-=- Hajdu, Hajdu and Tijdeman have determined the optimal values of w for all p, with respect to the maximum relative error, under the restriction (B) and also under the condition (D) : w(i, j) ≥ � i 2 +... |

1 |
be downloadable from: www.math.leidenuniv.nl/~scholtus/chamfer.htm
- Will
(Show Context)
Citation Context ...y theoretical value. 4 Computing the Maximum Relative Error We now give upper bounds for the maximum relative error of good approximating integer neighborhoods. We omit the proofs, which are given in =-=[8]-=-. For p = 1, the integer neighborhood N has two distinct values, and a scaling factor r. N(i, j) = � n0 if |i| + |j| = 1 n1 if |i| = |j| = 1 (7)sTheorem 1. [8] Let N be an integer neighborhood on M1 w... |