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What is in a Contour Map?

A Region-Based Logical Formalization of Contour Semantics

Part of the Lecture Notes in Computer Science book series (LNTCS,volume 9368)

Abstract

Contours maps (such as topographic maps) compress the information of a function over a two-dimensional area into a discrete set of closed lines that connect points of equal value (isolines), striking a fine balance between expressiveness and cognitive simplicity. They allow humans to perform many common sense reasoning tasks about the underlying function (e.g. elevation).

This paper analyses and formalizes contour semantics in a first-order logic ontology that forms the basis for enabling computational common sense reasoning about contour information. The elicited contour semantics comprises four key concepts – contour regions, contour lines, contour values, and contour sets – and their subclasses and associated relations, which are grounded in an existing qualitative spatial ontology. All concepts and relations are illustrated and motivated by physical-geographic features identifiable on topographic contour maps. The encoding of the semantics of contour concepts in first-order logic and a derived conceptual model as basis for an OWL ontology lay the foundation for fully automated, semantically-aware qualitative and quantitative reasoning about contours.

Keywords

  • Contour maps
  • Isolines
  • Knowledge representation
  • First-order logic
  • Spatial ontology
  • Region-based space
  • Naive geography
  • Physical reasoning

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Fig. 1.
Fig. 2.
Fig. 3.
Fig. 4.
Fig. 5.

Notes

  1. 1.

    Such measures form a field when the space is unbounded. We use the term field more loosely, including both bounded and unbounded variants.

  2. 2.

    OGC’s reference model and more specific standards such as GeoSPARQL and GML include coverage data types to represent fields, but offer no way of representing fields using contours.

  3. 3.

    All presented axioms, definitions and theorems are first-order sentences which are implicitly universally quantified over any variables that are not explicitly quantified.

  4. 4.

    We assume that any two measured quantities x and y with \( MQuantity (x)\) and \( MQuantity (y)\) can be directly compared using standard (in)equality so that the result is not a mere comparison of their numeric values (denoted by \(\mathrm {mValue}(x)\) and \(\mathrm {mValue}(y)\)) but takes their associated units \(\mathrm {mUnit}(x)\) and \(\mathrm {mUnit}(y)\) into account. E.g., if \(x=1\,\text {km}\) and \(y= 100\,\text {m}\), then \(x>y\) is true. All comparisons of measured quantities, even between quantities in the same unit, require a common measured qualities (\(\mathrm {mQuality}(x)=\mathrm {mQuality}(y)\)), e.g., both are elevations.

  5. 5.

    Our parent-child relations are based on spatial containment among regions and are similar to the parent-child relation in the enclosure trees from [1]. The resulting structure is closely related to the graphs known as contour trees [11] that essentially uses a dual version of our representation by representing regions as arcs and contours as nodes.

  6. 6.

    In order to capture the contour set that forms the context for the parent-child and sibling relations, we chose to model them as ternary predicates. In the derived conceptual model in Fig. 5, the parent-child and sibling relations are expressed using a new helper class each, together with new relations between the helper classes and the parents/children/siblings.

  7. 7.

    Other conventions about label direction and positioning are also commonly used.

  8. 8.

    For brevity, this ontology excludes all definitions that are unnecessary for the characterization.

  9. 9.

    The region of equal contour value is only disconnected when separated by two or more cliffs, which are points/segments of its containing contour region where it shares a portion of its boundary with one or multiple child contour regions.

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Acknowledgments

Part of this paper is based on discussions about key concepts in contour maps at the joint SOCoP (Spatial Ontology Community of Practice) and GeoVoCamp workshop in Madison, WI in June 2014, continued at a GeoVoCamp meeting at USGS in Reston, VA in December 2014. We gratefully acknowledge and thank those who contributed to these preliminary discussions: Carl Sack (at Madison), Joshua Lieberman, Dave Kolas and John (Ebo) David (at Reston), and in particular Dalia Varanka for her valuable contributions at both workshops. We further thank the workshop organizers Nancy Wiegand and Gary Berg-Cross, whose efforts to organize these events enabled those fruitful discussions. Finally, we appreciate the constructive comments of Philip Thiem and four anonymous reviewers, which helped improve the final paper.

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Hahmann, T., Usery, E.L. (2015). What is in a Contour Map?. In: Fabrikant, S., Raubal, M., Bertolotto, M., Davies, C., Freundschuh, S., Bell, S. (eds) Spatial Information Theory. COSIT 2015. Lecture Notes in Computer Science(), vol 9368. Springer, Cham. https://doi.org/10.1007/978-3-319-23374-1_18

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