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Published in:

2015 | OriginalPaper | Chapter

What is in a Contour Map?

A Region-Based Logical Formalization of Contour Semantics

Authors: Torsten Hahmann, E. Lynn Usery

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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.
Boyell, R., Ruston, H.: Hybrid techniques for real-time radar simulation. In: IEEE Fall Joint Computer Conference (IEEE 1963), pp. 445–458 (1963)
2.
Casati, R., Varzi, A.C.: Parts and Places. MIT Press, Cambridge (1999)
3.
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.
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.
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.
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.
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.
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.
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.
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.
Kweon, I., Kanade, T.: Extracting topographic terrain features from elevation maps. CVGIP: Image Underst. 59, 171–182 (1994) CrossRef
12.
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.
Morse, S.: Concepts of use in contour map processing. Commun. ACM 12, 147–152 (1969)
14.
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
15.
Open Geospatial Consortium (OGC): OGC reference model. OGC 08–062r7, December 2011. http://​www.​opengeospatial.​org/​standards/​orm
16.
Open Geospatial Consortium (OGC): ISO 19156: geographic information - observations and measurements. OGC 10–004r3, September 2013. http://​www.​opengeospatial.​org/​standards/​om
17.
Pike, R., Evans, I., Hengle, T.: Geomorphometry: a brief guide. In: Hengl, T., Reuter, H. (eds.) Geomorphometry: Concepts, Software Applications. Elsevier, Amsterdam (2009)
18.
Rana, S. (ed.): Topological Data Structures for Surfaces. Wiley, New York (2004)
19.
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.
Rijgersberg, H., van Assem, M., Top, J.: Ontology of units of measure and related concepts. Semant. Web J. 4(1), 3–13 (2013)
21.
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.
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.
Stevens, S.S.: On the theory of scales of measurement. Science 103(2684), 677–680 (1946)
24.
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