## Imprecision in Finite Resolution Spatial Data (1997)

Venue: | GeoInformatica |

Citations: | 43 - 7 self |

### BibTeX

@ARTICLE{Worboys97imprecisionin,

author = {Michael Worboys},

title = {Imprecision in Finite Resolution Spatial Data},

journal = {GeoInformatica},

year = {1997},

volume = {2},

pages = {257--279}

}

### Years of Citing Articles

### OpenURL

### Abstract

An important component of spatial data quality is the imprecision resulting from the resolution at which data are represented. Current research on topics such as spatial data integration and generalisation needs to be well-founded on a theory of multi-resolution. This paper provides a formal framework for treating the notion of resolution and multi-resolution in geographic spaces. It goes further to develop an approach to reasoning with imprecision about spatial entities and relationships resulting from finite resolution representations. The approach is similar to aspects of rough and fuzzy set theories. The paper concludes by providing the beginnings of a geometry of vague spatial entities and relationships. Keywords: uncertainty, vagueness, rough set, fuzzy set, resolution, spatial reasoning, data quality 1. Introduction The notion of spatial resolution is fundamental to many aspects of the representation of spatial data, and a proper formulation of a multi-resolution data model is...

### Citations

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Citation Context ...atial uncertainty and imprecision A currently fashionable approach to locational uncertainty in the GIS literature is the application of fuzzy reasoning, originated in a more general setting by Zadeh =-=[27]-=-, [28]. By far the most common application of fuzzy reasoning techniques is to reasoning about and representing the locations of boundaries. For example Leung [15] constructs a model in which a bounda... |

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Citation Context ...s `degree of imprecision' of spatial entities and relationships resulting from a collection of finite resolutions. Extensions are under investigation that apply the Dempster-Shafer theory of evidence =-=[21]-=- to spatial reasoning with variable precision and belief levels. Applications investigated at this stage have been limited to reasoning, under conditions of imprecise observation, about the presence, ... |

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Citation Context ...e. Of course, any resolution space will be closed with respect to the lattice operations defined above. We can note that in general a resolution lattice is not modular (and therefore not distributive =-=[6]). A modular lattice-=- L has the property that for all x, y, z �� S, if zsx then x �� (y �� z) = (x �� y) �� z To see that a resolution lattice is in general not modular, let S = {a, b, c, d}, and R 1 =... |

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Citation Context ...uncertainty and imprecision A currently fashionable approach to locational uncertainty in the GIS literature is the application of fuzzy reasoning, originated in a more general setting by Zadeh [27], =-=[28]-=-. By far the most common application of fuzzy reasoning techniques is to reasoning about and representing the locations of boundaries. For example Leung [15] constructs a model in which a boundary is ... |

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Citation Context ...calculus of individuals [1], [2] has been set in a many-sorted first-order logic [5], the focus of the theory being a reflexive, symmetric, connection relation between spatial regions. Cohn and Gotts =-=[3]-=- provide some ideas for the vague treatment of connection. A unified treatment of mereology and topology is undertaken in [23]. To illustrate our approach, we discuss the vague extension to the connec... |

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Citation Context ...hole relations Two fundamental spatial relationships are part of (a mereological relationship) and connection (a topological relationship). Mereology has been discussed by several authors (e.g. [22], =-=[23]-=-). Clarke's calculus of individuals [1], [2] has been set in a many-sorted first-order logic [5], the focus of the theory being a reflexive, symmetric, connection relation between spatial regions. Coh... |

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Citation Context ...onships are part of (a mereological relationship) and connection (a topological relationship). Mereology has been discussed by several authors (e.g. [22], [23]). Clarke's calculus of individuals [1], =-=[2]-=- has been set in a many-sorted first-order logic [5], the focus of the theory being a reflexive, symmetric, connection relation between spatial regions. Cohn and Gotts [3] provide some ideas for the v... |

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Citation Context ...n order to be able to use it effectively. Information on quality arises both in the modelling, representation and storage of data in databases, and also in their analysis, reasoning and visualization =-=[17]-=-. 2.1 Components of spatial data quality Data quality is multi-dimensional, with components ranging from subjective aspects such as fitness-for-purpose, to objective measurables like deviation from ob... |

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Citation Context ...d connection (a topological relationship). Mereology has been discussed by several authors (e.g. [22], [23]). Clarke's calculus of individuals [1], [2] has been set in a many-sorted first-order logic =-=[5]-=-, the focus of the theory being a reflexive, symmetric, connection relation between spatial regions. Cohn and Gotts [3] provide some ideas for the vague treatment of connection. A unified treatment of... |

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Citation Context ...of multiple resolutions. This section discusses some of the properties of the spatial relationships between different objects. There is little in the literature on this topic, although a recent paper =-=[8]-=- contains some similar underlying ideas to those set out below, but in a different formal framework. As usual, let S be a set, and R be a resolution space on S. We assume throughout this section that ... |

36 |
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Citation Context ...by the theory of rough sets. As rough sets have not been considered much in the GIS literature, we take some space to introduce the basic ideas. 5 The starting point of the theory of rough sets (e.g. =-=[18]-=-, [19], [20]) is that entities can only be perceived by making observations about them, and that the observations provide information at differing degrees of precision and accuracy. Any particular obs... |

31 |
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Citation Context ... seen as based on some resolution structure. Even in the simplest case of a physical resolution dependent upon pixel size, problems arise from the uncertainty imposed by the lack of precision. Fisher =-=[10]-=- writes that `the pixel, the elementary unit of analysis in remote sensing and the usual vehicle for integrating data between GIS and remote sensing, is a delusion which may become a snare for the unw... |

27 |
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Citation Context ...al spatial data model is seen as based upon some resolution 4 structure. The well-known triangulated irregular networks (TIN) representation, and the `realms' representations of G��ting and Schnei=-=der [13]-=-, both provide example of representations with respect to particular resolution structures. The hierarchical terrain models of De Floriani and her colleagues [11] give examples of representations with... |

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Citation Context ...arity or resolution at which the observation is made, or the information is represented. Error as a component of data quality has been quite widely treated in the GIS literature. Hunter and Goodchild =-=[14]-=- survey means of communicating and reasoning with spatial data error, which include ignoring it, epsilon bands, misclassification matrices, map reliability diagrams, fuzzy logic, probability surfaces,... |

21 |
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Citation Context ...es change continuously from values associated with one region to that of its adjacent region. Leung's methods have something in common with the approach taken later in this paper. Wang and Brent Hall =-=[25]-=- describe Leung's partitioning of a region into its core (`the area whose characteristics are most compatible with the linguistic proposition characterising the region') and its boundary (`the area wh... |

15 |
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Citation Context ...lon bands, misclassification matrices, map reliability diagrams, fuzzy logic, probability surfaces, and variability diagrams. They provide a case study of some possible treatments of error. Couclelis =-=[4]-=- notes that vagueness can result from the inherent nature of the object (`the South of England' is inherently more vague than `the county of Surrey', although both regions have some inherent vagueness... |

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Citation Context ...ble with the linguistic proposition characterising the region'). The fuzzy approach to geographic regional boundaries (and thus to the regions themselves) has been researched by several authors (e.g. =-=[7]-=-, [15], [16], [25]). The essence of the approach is that uncertainty of membership of a location in a region is indicated by a real number between 0 and 1, where a membership value of 0 indicates that... |

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Citation Context ...n geographic data quality are generated at all stages of the data life-cycle, from capture, through input, manipulation and analysis, to the presentation of results. It is now widely recognised (e.g. =-=[12]-=-) that data quality is an important component of the description of the data, that each data entity should carry information describing its quality, that each operation on the data should have associa... |

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Citation Context ...colleagues [11] give examples of representations with respect to multi-resolution structures. Other work that takes a formal approach to spatial and temporal resolution has been undertaken by Euzenat =-=[9]-=-. 2.2 Formal treatments of spatial uncertainty and imprecision A currently fashionable approach to locational uncertainty in the GIS literature is the application of fuzzy reasoning, originated in a m... |

11 |
The nature of boundaries on 'area-class' maps, Cartographica
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Citation Context ...e linguistic proposition characterising the region'). The fuzzy approach to geographic regional boundaries (and thus to the regions themselves) has been researched by several authors (e.g. [7], [15], =-=[16]-=-, [25]). The essence of the approach is that uncertainty of membership of a location in a region is indicated by a real number between 0 and 1, where a membership value of 0 indicates that the locatio... |

10 |
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Citation Context ...resentations of G��ting and Schneider [13], both provide example of representations with respect to particular resolution structures. The hierarchical terrain models of De Floriani and her colleag=-=ues [11]-=- give examples of representations with respect to multi-resolution structures. Other work that takes a formal approach to spatial and temporal resolution has been undertaken by Euzenat [9]. 2.2 Formal... |

10 |
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Citation Context ...d in a more general setting by Zadeh [27], [28]. By far the most common application of fuzzy reasoning techniques is to reasoning about and representing the locations of boundaries. For example Leung =-=[15]-=- constructs a model in which a boundary is represented as a zone in which attribute values change continuously from values associated with one region to that of its adjacent region. Leung's methods ha... |

6 |
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Citation Context ...use particular difficulties is provided by highly dynamic situations, such as in transportation networks or battlefield scenarios, where observations are subject to noise, conflict and incompleteness =-=[26]-=-. The arguments contained in this paper focus on the contribution that imprecision makes to uncertainty. Given the nature of digital computation, all data, spatial or otherwise, can be represented at ... |

4 |
and soft sets
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Citation Context ...y of rough sets. As rough sets have not been considered much in the GIS literature, we take some space to introduce the basic ideas. 5 The starting point of the theory of rough sets (e.g. [18], [19], =-=[20]-=-) is that entities can only be perceived by making observations about them, and that the observations provide information at differing degrees of precision and accuracy. Any particular observation is ... |

3 |
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Citation Context ... theory of rough sets. As rough sets have not been considered much in the GIS literature, we take some space to introduce the basic ideas. 5 The starting point of the theory of rough sets (e.g. [18], =-=[19]-=-, [20]) is that entities can only be perceived by making observations about them, and that the observations provide information at differing degrees of precision and accuracy. Any particular observati... |

3 |
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Citation Context ...d region of the Euclidean plane). A resolution R of S is a finite partition of S. Alternatively, a resolution is defined by the equivalence relation r that gives rise to the finite partition R. As in =-=[24] we -=-call an element x �� R a resel. We note that a resolution is any partition of the underlying set into a finite number of subsets. The partition may arise from a pixellation of the space and be a r... |