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2015 | OriginalPaper | Chapter

What is in a Contour Map?

A Region-Based Logical Formalization of Contour Semantics

Authors: Torsten Hahmann, E. Lynn Usery

Published in: Spatial Information Theory

Publisher: Springer International Publishing

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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.

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Footnotes
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
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
All presented axioms, definitions and theorems are first-order sentences which are implicitly universally quantified over any variables that are not explicitly quantified.
 
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
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
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
Other conventions about label direction and positioning are also commonly used.
 
8
For brevity, this ontology excludes all definitions that are unnecessary for the characterization.
 
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.
 
Literature
1.
go back to reference Boyell, R., Ruston, H.: Hybrid techniques for real-time radar simulation. In: IEEE Fall Joint Computer Conference (IEEE 1963), pp. 445–458 (1963) Boyell, R., Ruston, H.: Hybrid techniques for real-time radar simulation. In: IEEE Fall Joint Computer Conference (IEEE 1963), pp. 445–458 (1963)
2.
go back to reference Casati, R., Varzi, A.C.: Parts and Places. MIT Press, Cambridge (1999) Casati, R., Varzi, A.C.: Parts and Places. MIT Press, Cambridge (1999)
3.
go back to reference Cohn, A.G., Renz, J.: Qualitative spatial representation and reasoning. In: van Harmelen, F., Lifschitz, V., Porter, B. (eds.) Handbook of Knowledge Representation. Elsevier, Amsterdam (2008) Cohn, A.G., Renz, J.: Qualitative spatial representation and reasoning. In: van Harmelen, F., Lifschitz, V., Porter, B. (eds.) Handbook of Knowledge Representation. Elsevier, Amsterdam (2008)
4.
go back to reference Egenhofer, M.J., Mark, D.M.: Naive geography. In: Kuhn, W., Frank, A.U. (eds.) COSIT 1995. LNCS, vol. 988, pp. 1–15. Springer, Heidelberg (1995) CrossRef Egenhofer, M.J., Mark, D.M.: Naive geography. In: Kuhn, W., Frank, A.U. (eds.) COSIT 1995. LNCS, vol. 988, pp. 1–15. Springer, Heidelberg (1995) CrossRef
5.
go back to reference Egenhofer, M.J., Sharma, J.: Topological relations between regions in \(R^{2}\) and \(Z^{2}\). In: Abel, D.J., Ooi, B.-C. (eds.) SSD 1993. LNCS, vol. 692, pp. 316–336. Springer, Heidelberg (1993) CrossRef Egenhofer, M.J., Sharma, J.: Topological relations between regions in \(R^{2}\) and \(Z^{2}\). In: Abel, D.J., Ooi, B.-C. (eds.) SSD 1993. LNCS, vol. 692, pp. 316–336. Springer, Heidelberg (1993) CrossRef
6.
go back to reference Freeman, H., Morse, S.: On searching a contour map for a given terrain elevation profile. J. Franklin Institute 284(1), 1–25 (1967) CrossRef Freeman, H., Morse, S.: On searching a contour map for a given terrain elevation profile. J. Franklin Institute 284(1), 1–25 (1967) CrossRef
7.
go back to reference Hahmann, T.: A reconciliation of logical representations of space: from multidimensional mereotopology to geometry. Ph.D. thesis, University of Toronto, Department of Computer Science (2013) Hahmann, T.: A reconciliation of logical representations of space: from multidimensional mereotopology to geometry. Ph.D. thesis, University of Toronto, Department of Computer Science (2013)
8.
go back to reference Hahmann, T., Grüninger, M.: A naïve theory of dimension for qualitative spatial relations. In: Symposium on Logical Formalizations of Commonsense Reasoning (CommonSense 2011). AAAI Press (2011) Hahmann, T., Grüninger, M.: A naïve theory of dimension for qualitative spatial relations. In: Symposium on Logical Formalizations of Commonsense Reasoning (CommonSense 2011). AAAI Press (2011)
9.
go back to reference Hahmann, T., Grüninger, M.: Region-based theories of space: mereotopology and beyond. In: Hazarika, S.M. (ed.) Qualitative Spatio-Temporal Representation and Reasoning: Trends and Future Directions, pp. 1–62. IGI, USA (2012) CrossRef Hahmann, T., Grüninger, M.: Region-based theories of space: mereotopology and beyond. In: Hazarika, S.M. (ed.) Qualitative Spatio-Temporal Representation and Reasoning: Trends and Future Directions, pp. 1–62. IGI, USA (2012) CrossRef
10.
go back to reference Hutton, C.: An account of the calculations made from the survey and measures taken at Schehallien. Philos. Trans. R. Soc. Lond. 68, 689–788 (1778) CrossRef Hutton, C.: An account of the calculations made from the survey and measures taken at Schehallien. Philos. Trans. R. Soc. Lond. 68, 689–788 (1778) CrossRef
11.
go back to reference Kweon, I., Kanade, T.: Extracting topographic terrain features from elevation maps. CVGIP: Image Underst. 59, 171–182 (1994) CrossRef Kweon, I., Kanade, T.: Extracting topographic terrain features from elevation maps. CVGIP: Image Underst. 59, 171–182 (1994) CrossRef
12.
go back to reference Masolo, C., Borgo, S., Gangemi, A., Guarino, N., Oltramari, A.: Wonderweb deliverable D18 - ontology library (final report). National Research Council - Institute of Cognitive Science and Technology, Trento, Technical report (2003) Masolo, C., Borgo, S., Gangemi, A., Guarino, N., Oltramari, A.: Wonderweb deliverable D18 - ontology library (final report). National Research Council - Institute of Cognitive Science and Technology, Trento, Technical report (2003)
13.
14.
go back to reference O’Callaghan, J., Mark, D.M.: The extraction of drainage networks from digital elevation data. Comput. Vis. Graph. Image Process. 28(3), 323–344 (1984) CrossRef O’Callaghan, J., Mark, D.M.: The extraction of drainage networks from digital elevation data. Comput. Vis. Graph. Image Process. 28(3), 323–344 (1984) CrossRef
17.
go back to reference Pike, R., Evans, I., Hengle, T.: Geomorphometry: a brief guide. In: Hengl, T., Reuter, H. (eds.) Geomorphometry: Concepts, Software Applications. Elsevier, Amsterdam (2009) Pike, R., Evans, I., Hengle, T.: Geomorphometry: a brief guide. In: Hengl, T., Reuter, H. (eds.) Geomorphometry: Concepts, Software Applications. Elsevier, Amsterdam (2009)
18.
go back to reference Rana, S. (ed.): Topological Data Structures for Surfaces. Wiley, New York (2004) Rana, S. (ed.): Topological Data Structures for Surfaces. Wiley, New York (2004)
19.
go back to reference Randell, D.A., Cui, Z., Cohn, A.G.: A spatial logic based on regions and connection. In: KR 1992: Principles of Knowledge Representation and Reasoning, pp. 165–176 (1992) Randell, D.A., Cui, Z., Cohn, A.G.: A spatial logic based on regions and connection. In: KR 1992: Principles of Knowledge Representation and Reasoning, pp. 165–176 (1992)
20.
go back to reference Rijgersberg, H., van Assem, M., Top, J.: Ontology of units of measure and related concepts. Semant. Web J. 4(1), 3–13 (2013) Rijgersberg, H., van Assem, M., Top, J.: Ontology of units of measure and related concepts. Semant. Web J. 4(1), 3–13 (2013)
21.
go back to reference Sinha, G., Kolas, D., Mark, D., Romero, B.E., Usery, L.E., Berg-Cross, G., Padmanabhan, A.: Surface network ontology design patterns for linked topographic data, May 2014 Sinha, G., Kolas, D., Mark, D., Romero, B.E., Usery, L.E., Berg-Cross, G., Padmanabhan, A.: Surface network ontology design patterns for linked topographic data, May 2014
22.
go back to reference Sinha, G., Mark, D., Kolas, D., Varanka, D., Romero, B.E., Feng, C.-C., Usery, E.L., Liebermann, J., Sorokine, A.: An ontology design pattern for surface water features. In: Duckham, M., Pebesma, E., Stewart, K., Frank, A.U. (eds.) GIScience 2014. LNCS, vol. 8728, pp. 187–203. Springer, Heidelberg (2014) Sinha, G., Mark, D., Kolas, D., Varanka, D., Romero, B.E., Feng, C.-C., Usery, E.L., Liebermann, J., Sorokine, A.: An ontology design pattern for surface water features. In: Duckham, M., Pebesma, E., Stewart, K., Frank, A.U. (eds.) GIScience 2014. LNCS, vol. 8728, pp. 187–203. Springer, Heidelberg (2014)
23.
go back to reference Stevens, S.S.: On the theory of scales of measurement. Science 103(2684), 677–680 (1946) CrossRefMATH Stevens, S.S.: On the theory of scales of measurement. Science 103(2684), 677–680 (1946) CrossRefMATH
24.
go back to reference Usery, E.L., Varanka, D., Finn, M.P.: A 125 year history of topographic mapping and GIS in the U.S. Geological Survey 1884–2009, Part 1, 1884–1980, March 2015 Usery, E.L., Varanka, D., Finn, M.P.: A 125 year history of topographic mapping and GIS in the U.S. Geological Survey 1884–2009, Part 1, 1884–1980, March 2015
Metadata
Title
What is in a Contour Map?
Authors
Torsten Hahmann
E. Lynn Usery
Copyright Year
2015
DOI
https://doi.org/10.1007/978-3-319-23374-1_18

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