## Shape Matching: Similarity Measures and Algorithms (2001)

Citations: | 91 - 1 self |

### BibTeX

@INPROCEEDINGS{Veltkamp01shapematching:,

author = {Remco C. Veltkamp},

title = {Shape Matching: Similarity Measures and Algorithms},

booktitle = {},

year = {2001},

pages = {188--197},

publisher = {}

}

### Years of Citing Articles

### OpenURL

### Abstract

Shape matching is an important ingredient in shape retrieval, recognition and classification, alignment and registration, and approximation and simplification. This paper treats various aspects that are needed to solve shape matching problems: choosing the precise problem, selecting the properties of the similarity measure that are needed for the problem, choosing the specific similarity measure, and constructing the algorithm to compute the similarity. The focus is on methods that lie close to the field of computational geometry.

### Citations

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Citation Context ...ays wanted. Indeed, human perception does not always find that shape � is equally similar to �, as� is to �. In particular, a variant � of prototype � is often found more similar to � th=-=an vice versa [53]. A more frequently encountered form-=-ulation of the triangle inequality is the following: � �� � � ��� � � �� � . Similarity measures for partial matching, giving a small distance � �� � if a part of �... |

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Citation Context ...ments and other global object features such as area, circularity, eccentricity, compactness, major axis orientation, Euler number, concavity tree, shape numbers, can all be used for shape description =-=[9]-=-, [42]. 1.1.2 Modal matching Rather than working with the area of a 2D object, the boundary can be used instead. Samples of the boundary can be described with Fourier descriptors, the coefficients of ... |

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Citation Context ... problems. First some related work is mentioned in the next subsection. 1.1 Related work Matching has been approached in a number of ways, including tree pruning [55], the generalized Hough transform =-=[8]-=- or pose clustering [51], geometric hashing [59], the alignment method [27], statistics [40], deformable templates [50], relaxation labeling [44], Fourier descriptors [35], wavelet transform [31], cur... |

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Citation Context ...rff distance is not a metric since it fails the triangle inequality. Deciding whether there is a translation plus scaling that brings the partial Hausdorff distance under a given threshold is done in =-=[30]-=- by means of a transformation spacessubdivision scheme. The running time depends on the depth of subdivision of transformation space. For pattern matching, the Hausdorff metric is often too sensitive ... |

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Citation Context ...entioned along with the description of the measure, when the algorithm is specific for that measure. This section mentions a few algorithms that are more general. 5.1 Voting schemes Geometric hashing =-=[33, 59]-=- is a method that determines if there is a transformed subset of the query point set that matches a subset of a target point set. The method first constructs a single hash table for all target point s... |

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Citation Context ...les [5]. The name Earth Mover’s Distance is coined by Jorge Stolfi. The transport distance has been used in heat transform problems [43], object contour matching [19], and color-based image retrieva=-=l [46]. Let ��� be the flow from location �� to ��, � the matrix of elements �-=-����, and��� the ‘ground distance’ between �� and ��. Then the transport distance is ÈÑ ÈÒ Ñ�Ò� �� �� ������ ÈÑ ÈÒ �� �� ��� under... |

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Citation Context ...nstruct a shape of fewer elements (points, segments, triangles, etc.), that is still similar to the original. There are many heuristics for approximating polygonal curves [45] and polyhedral surfaces =-=[26]-=-. Optimal methods construct an approximation with the fewest elements given a maximal dissimilarity, or with the smallest dissimilarity given the maximal number of elements. (Checking the former dissi... |

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Citation Context ...in a number of ways, including tree pruning [55], the generalized Hough transform [8] or pose clustering [51], geometric hashing [59], the alignment method [27], statistics [40], deformable templates =-=[50]-=-, relaxation labeling [44], Fourier descriptors [35], wavelet transform [31], curvature scale space [36], and neural networks [21]. The following subsections treat a few methods in more detail. They a... |

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Citation Context ...tching has been approached in a number of ways, including tree pruning [55], the generalized Hough transform [8] or pose clustering [51], geometric hashing [59], the alignment method [27], statistics =-=[40]-=-, deformable templates [50], relaxation labeling [44], Fourier descriptors [35], wavelet transform [31], curvature scale space [36], and neural networks [21]. The following subsections treat a few met... |

170 |
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Citation Context ...1 Related work Matching has been approached in a number of ways, including tree pruning [55], the generalized Hough transform [8] or pose clustering [51], geometric hashing [59], the alignment method =-=[27]-=-, statistics [40], deformable templates [50], relaxation labeling [44], Fourier descriptors [35], wavelet transform [31], curvature scale space [36], and neural networks [21]. The following subsection... |

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Citation Context ... in the next subsection. 1.1 Related work Matching has been approached in a number of ways, including tree pruning [55], the generalized Hough transform [8] or pose clustering [51], geometric hashing =-=[59]-=-, the alignment method [27], statistics [40], deformable templates [50], relaxation labeling [44], Fourier descriptors [35], wavelet transform [31], curvature scale space [36], and neural networks [21... |

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Citation Context ...same parameter Ø � � ℄, such that « � ¬ � ,and« � ¬ � .The Fréchet distance is the minimum over all monotone increasing parameterizations « Ø and ¬ Ø of the maximal distance ��=-=� � « Ø �� ¬ Ø , Ø � � ℄. [4] cons-=-iders the computation of the Fréchet distance for the special case of polylines. Deciding whether the Fréchet distance is smaller than a given constant, can be done in time Ç ÑÒ . Based on this r... |

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Citation Context ... [51], geometric hashing [59], the alignment method [27], statistics [40], deformable templates [50], relaxation labeling [44], Fourier descriptors [35], wavelet transform [31], curvature scale space =-=[36]-=-, and neural networks [21]. The following subsections treat a few methods in more detail. They are based on shape representations that depend on the global shape. Therefore, they are not robust agains... |

123 |
High-level Vision
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Citation Context ... tally a vote in the corresponding entry of the table. Again the winner entry determines the matching transformation. The complexity of matching a single query set is Ç ÆÑ Ò . In the alignment met=-=hod [27, 54]-=-, for each triplet of points from the query set, and each triplet from the target set, we compute the transformation between them. With each such transformation, all the other points from the target s... |

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Citation Context ... and other global object features such as area, circularity, eccentricity, compactness, major axis orientation, Euler number, concavity tree, shape numbers, can all be used for shape description [9], =-=[42]-=-. 1.1.2 Modal matching Rather than working with the area of a 2D object, the boundary can be used instead. Samples of the boundary can be described with Fourier descriptors, the coefficients of the di... |

89 |
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Citation Context ...izes the Hausdorff distance � � �� � can be determined in time Ç ÑÒ ÐÓ� ÑÒ when the underlying metric is Ä or Ä [14]. For other ÄÔ metrics, Ô � � ���� it can be com=-=puted in time Ç ÑÒ Ñ Ò « ÑÒ ÐÓ� Ñ Ò [29]. (-=-« Ò is the inverse Ackermann function, a very slowly increasing function.) This is done using the upper envelopes of Voronoi surfaces. Computing the optimal rigid motion Ö (translation plus rotatio... |

79 |
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Citation Context ...elated work is mentioned in the next subsection. 1.1 Related work Matching has been approached in a number of ways, including tree pruning [55], the generalized Hough transform [8] or pose clustering =-=[51]-=-, geometric hashing [59], the alignment method [27], statistics [40], deformable templates [50], relaxation labeling [44], Fourier descriptors [35], wavelet transform [31], curvature scale space [36],... |

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Citation Context ...ce (Ä ). The Hausdorff distance À �� � is the maximum of �� �� � and �� ��� : À �� � � Ñ�Ü�� � �� � � � � ��� �. For finite point set=-=s, it can be computed using Voronoi diagrams in time Ç Ñ Ò ÐÓ� Ñ Ò [2]. Given two finit point sets �-=-�� and �, the translation � £ that minimizes the Hausdorff distance � � �� � can be determined in time Ç ÑÒ ÐÓ� ÑÒ when the underlying metric is Ä or Ä [14]. For other ÄÔ m... |

72 | Geometric Pattern Matching under Euclidean Motion
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Citation Context ...be done in Ç Ñ Ò � ÐÓ� ÑÒ time [28]. This is done using dynamic Voronoi diagrams. Given a real value ¯, deciding if there is a rigid motion such that À Ö � �� � ¯ can be done in=-= time Ç Ñ Ò Ñ Ò ÐÓ� ÑÒ [13]-=-. Given the high complexities of these problems, it makes sense to look at approximations. Computing an approximate optimal Hausdorff distance under translation and rigid motion is discussed in [1]. T... |

70 | Convexity rule for shape decomposition based on discrete contour evolution
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Citation Context ...late. Matching of two objects can be done by matching points of annihilation in the ×� � plane [36]. Another way of reducing curvature changes is based on the turning angle function (see section =-=4.5) [34]-=-. Matching with the curvature scale space is robust against slight affine distortion, as has been experimentally determined [37]. Be careful, however, to use this property for fish recognition, see se... |

62 |
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Citation Context ...f piles of dirt from the huge cattle stables of king Augeas by Hercules [5]. The name Earth Mover’s Distance is coined by Jorge Stolfi. The transport distance has been used in heat transform problem=-=s [43], object contour matching [19], and colo-=-r-based image retrieval [46]. Let ��� be the flow from location �� to ��, � the matrix of elements ���, and��� the ‘ground distance’ between �� and ��. Then the... |

59 |
Point pattern matching by relaxation
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Citation Context ...ding tree pruning [55], the generalized Hough transform [8] or pose clustering [51], geometric hashing [59], the alignment method [27], statistics [40], deformable templates [50], relaxation labeling =-=[44]-=-, Fourier descriptors [35], wavelet transform [31], curvature scale space [36], and neural networks [21]. The following subsections treat a few methods in more detail. They are based on shape represen... |

56 | Techniques for assessing polygonal approximation of curves
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Citation Context ...mation and simplification: construct a shape of fewer elements (points, segments, triangles, etc.), that is still similar to the original. There are many heuristics for approximating polygonal curves =-=[45]-=- and polyhedral surfaces [26]. Optimal methods construct an approximation with the fewest elements given a maximal dissimilarity, or with the smallest dissimilarity given the maximal number of element... |

48 |
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Citation Context ...s segments, or points with normal vectors [10], and for other transformations than affine transformations. A comparison between geometric hashing, pose clustering, and the alignment method is made in =-=[60]-=-. Other voting schemes exist, for example taking a probabilistic approach [39]. 5.2 Subdivision Schemes As mentioned above, deciding whether there is a translation plus scaling that brings the partial... |

46 | Efficient image retrieval through vantage objects
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Citation Context ...ssimilarity measure on � and �: � �� � �sÊ �¢� ×s¢� × �Ô �× ¡ �Ô , see figure 5. ¢� × ¢� × Figure 5: Rectangles enclosed by ¢� × , ¢� × , and dotted =-=lines are used for evaluation of dissimilarity. In [58]-=-, for the purpose of retrieving hieroglyphic shapes, polyline curves do not have the same length, so that partial matching can be performed. In that case the starting point of the shorter one is moved... |

45 | On dynamic Voronoi diagrams and the minimum Hausdorff distance for point sets under euclidean motion in the plane
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Citation Context ...reasing function.) This is done using the upper envelopes of Voronoi surfaces. Computing the optimal rigid motion Ö (translation plus rotation), minimizing À Ö � �� can be done in Ç Ñ Ò ��=-=� ÐÓ� ÑÒ time [28]. This is done using dy-=-namic Voronoi diagrams. Given a real value ¯, deciding if there is a rigid motion such that À Ö � �� � ¯ can be done in time Ç Ñ Ò Ñ Ò ÐÓ� ÑÒ [13]. Given the high complexities o... |

41 | Deformable Prototypes for Encoding Shape Categories in Image Databases
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Citation Context ...the value � of � for which Ñ �� ��� is minimal. If the value of � for which Ñ ����� is minimal is equal to � , then point � of one shape and point � of the other shap=-=e match. See for example [22] and [49]-=- for variations on this basic technique of modal matching. 1.1.3 Curvature scale space Another approach is the use of a scale space representation of the curvature of the contour of 2D objects. Let th... |

40 | Matching shapes with a reference point
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Citation Context ...Ò [13]. Given the high complexities of these problems, it makes sense to look at approximations. Computing an approximate optimal Hausdorff distance under translation and rigid motion is discussed in=-= [1]-=-. The Hausdorff distance is very sensitive to noise: a single outlier can determine the distance value. For finite point sets, a similar measure that is not as sensitive is the partial Hausdorff dista... |

38 |
Parameterized Point Pattern Matching and Its Application to Recognition of Object Families
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Citation Context ...t and efficient solutions to geometric problems. First some related work is mentioned in the next subsection. 1.1 Related work Matching has been approached in a number of ways, including tree pruning =-=[55]-=-, the generalized Hough transform [8] or pose clustering [51], geometric hashing [59], the alignment method [27], statistics [40], deformable templates [50], relaxation labeling [44], Fourier descript... |

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Citation Context ... metric. Translating convex polygons so that their centroids coincide also gives an approximate solution for the symmetric difference, which is at most 11/3 of the optimal solution under translations =-=[3]. This also holds for -=-a set of transformations � other than translations, if the following holds: the centroid of �, � , is equivariant under the transformations, i.e. � � � � � for all � in � ,and� i... |

34 | Improvements on bottleneck matching and related problems using geometry
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Citation Context ...proximation � � to the real bottleneck distance � . An approximate matching between � and � with � � the furthest matched pair, such that � � � � � ¯ � , can be computed wit=-=h a less complex algorithm [17]. The decision -=-problem for translations, deciding whether there exists a translation � such that � � �� � � ¯ can also be solved, but takes considerably more time [17]. Because of the high degree in t... |

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Citation Context ...r a given threshold is done in [30] by a progressive subdivision of transformation space. The subdivision of transformation space is generalized to a general ‘geometric branch and bound’ framework=-= in [25]. -=-Here the optimal transformation is approximated to any desired accuracy. The matching can be done with respect to any transformation group �, for example similarity (translation, rotation, and scali... |

29 |
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Citation Context ...ng distance Æ�Å �� � is the minimum over all correspondences � of È × ����� � ����� , with � ����� the difference between the tangent angles at �� a=-=nd ��. It can be computed using dynamic programming [16]. This measure is not a metric, since -=-it does not obey the triangle inequality. The relaxed nonlinear elastic matching distance Æ�ÅÖ is a variation of Æ�Å, where the stretch × ����� of ���� �� ��� is ... |

28 |
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Citation Context ...e integral can be replaced by a summation. The infinite sequence of moments, Ô� Õ � � ����, uniquely determines the shape, and vice versa. Variations such as Zernike moments are descri=-=bed in [32] and [12]-=-. Based on such moments, a number of functions, moment invariants, can be defined that are invariant under certain transformations such as translation, scaling, and rotation. Using only a limited numb... |

28 | Partial matching of planar polylines under similarity transformations
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Citation Context ...spect to the turning function, which can be done in Ç ÑÒ ÐÓ� ÑÒ time [6]. Partial matching under scaling, in addition to translation and rotation, is more involved. It can be done in time Ç =-=Ñ Ò , see [15]. ×s4-=-.6 Fréchet Distance The Hausdorff distance is often not appropriate to measure the dissimilarity between curves. For all points on �, the distance to the closest point on � may be small, but if w... |

23 | Efficient pose clustering using a randomized algorithm
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Citation Context ...than affine transformations. A comparison between geometric hashing, pose clustering, and the alignment method is made in [60]. Other voting schemes exist, for example taking a probabilistic approach =-=[39]-=-. 5.2 Subdivision Schemes As mentioned above, deciding whether there is a translation plus scaling that brings the partial Hausdorff distance under a given threshold is done in [30] by a progressive s... |

20 | Relaxing the triangle inequality in pattern matching
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Citation Context ...e distance from the man to the horse is large, so � Ñ�Ò� �ÒØ�ÙÖ � �ÒØ�ÙÖ��ÓÖ×� � � Ñ�Ò� �ÓÖ×� does not hold. It therefore makes sense to formulate a=-=n even weaker form, the relaxed triangle inequality [18]: � �� � � ���-=-�� � � �� � , for some constant � . Figure 2: Under partial matching, the triangle inequality does not hold. Continuity properties It is often desirable that a similarity function has some... |

19 |
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Citation Context ...mpler sorting algorithm leads to an asymptotic running time of Ç ÑÒ ÐÓ� ÑÒ . Still, the parametric search is not easy to implement. A simpler algorithm, which runs in time Ç ÑÒ Ñ Ò ÐÓ�=-=�� ÑÒ is given in [20]. A variat-=-ion of the Fréchet distance is obtained by dropping the monotonicity condition of the parameterization. The resulting Fréchet distance � �� � is a semimetric: zero distance need not mean tha... |

18 |
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(Show Context)
Citation Context ...ther than working with the area of a 2D object, the boundary can be used instead. Samples of the boundary can be described with Fourier descriptors, the coefficients of the discrete Fourier transform =-=[56]-=-. Another form of shape decomposition is the decomposition into an ordered set of eigenvectors, also called principal components. Again, the noisy high order components can be discarded, using only th... |

17 | Tekalp, Shape similarity matching for query-by-example. Pattern Recognition
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(Show Context)
Citation Context ...etermine the value � of � for which Ñ �� ��� is minimal. If the value of � for which Ñ ����� is minimal is equal to � , then point � of one shape and point � of the o=-=ther shape match. See for example [22]-=- and [49] for variations on this basic technique of modal matching. 1.1.3 Curvature scale space Another approach is the use of a scale space representation of the curvature of the contour of 2D object... |

15 |
Shape Recognition Using Metrics on the Space of Shapes
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(Show Context)
Citation Context ...cattle stables of king Augeas by Hercules [5]. The name Earth Mover’s Distance is coined by Jorge Stolfi. The transport distance has been used in heat transform problems [43], object contour matchin=-=g [19], and color-based image retrieval [46]. Let �-=-���� be the flow from location �� to ��, � the matrix of elements ���, and��� the ‘ground distance’ between �� and ��. Then the transport distance is ÈÑ ÈÒ Ñ... |

14 |
Improved algorithms for robust point pattern matching and applications to image registration
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(Show Context)
Citation Context ... recursively, the algorithm is also expected to work well in applications in which the minima lie in small clusters. A speed-up can be achieved by combining the progressive subdivision with alignment =-=[38]-=-. 6 Concluding remarks We have discussed a number of shape similarity properties. More possibly useful properties are formulated in [57]. It is a challenging research task to construct similarity meas... |

11 |
Approximate decision algorithms for approximate congruence
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(Show Context)
Citation Context ...putational complexity, it is interesting to look at approximations with a factor ¯: � � �� � � ¯ � � � £ �Ì ,where� £ is the optimal translation. Finding such a translation =-=can be done in Ç Ò �� time [48]-=-. Variations on the bottleneck distance are the minimum weight distance, the most uniform distance, and the minimum deviation distance. 4.4 Hausdorff Distance In many applications, for example stereo ... |

9 |
Testing the congruence of d-dimensional point sets
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(Show Context)
Citation Context ...ith the best known time complexity run in Ç Ò ÐÓ� Ò time if � is translations, scaling, or homotheties (translation plus scaling), and Ç Ò ��� � ÐÓ� Ò time for rotations, isome=-=tries, and similarities [11]. 4.2 ÄÔ Distance, Minkowski Dista-=-nce Many similarity measures on shapes are based on the ÄÔ distance between two points. For two points Ü� Ý in Ê � ,theÄÔdistance is defined as ÄÔ Ü� Ý � È� �� �Ü�sÝ�... |

9 |
Matching and learning structural and spatial representation with neural networks
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- 1995
(Show Context)
Citation Context ...59], the alignment method [27], statistics [40], deformable templates [50], relaxation labeling [44], Fourier descriptors [35], wavelet transform [31], curvature scale space [36], and neural networks =-=[21]-=-. The following subsections treat a few methods in more detail. They are based on shape representations that depend on the global shape. Therefore, they are not robust against occlusion, and do not al... |

8 | Partial surface matching by using directed footprints
- Barequet, Sharir
- 1996
(Show Context)
Citation Context ...the target. The complexity of matching a single query set is Ç ÆÑ � Ò . Variations of these methods also work for geometric features other than points, such as segments, or points with normal ve=-=ctors [10]-=-, and for other transformations than affine transformations. A comparison between geometric hashing, pose clustering, and the alignment method is made in [60]. Other voting schemes exist, for example ... |

7 |
An error metric for binary images, in: Robust Computer Vision: Quality of Vision Algorithms
- Baddeley
- 1992
(Show Context)
Citation Context ...ion space. For pattern matching, the Hausdorff metric is often too sensitive to noise. For finite point sets, the partial Hausdorff distance is not that sensitive, but it is no metric. Alternatively, =-=[7] observes that the Hausdorff distance of �� � � �, -=-� having a finite number of elements, can be written as À �� � � ×ÙÔÜ � �� Ü� �s� Ü� � �. The supremum is then replaced by an average: ¡ Ô �� � � ���... |