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Topological relations form the backbone of qualitative spatial reasoning and, as such, play a paramount role in geographic information systems. Three decades of research have provided a proliferation of sets of qualitative topological relations in both continuous and discretized spaces, but only in continuous spaces has the concept of organizing these relations into a larger framework (called a conceptual neighborhood graph) been considered. Previous work leveraged matrix differences to derive the anisotropic scaling neighborhood for these relations. In this paper, a simulation protocol is used to derive conceptual neighborhood graphs of qualitative topological relations in Z2 for the operations of translation and isotropic scaling. It is further shown that, when aggregating raster relations into their continuous counterparts and collapsing neighborhood connections within these groups, the familiar conceptual neighborhood structures for continuous regions appear.
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This content will become publicly available on September 1, 2026
Digital Relations in Z1: Discretized Time and Rasterized Lines
There is voluminous literature concerning the scope of topological relations that span various embedding spaces from R1 to R2, Z2 , S1 and S2 , and T2. In the case of the *1 spaces, those relations have been considered as conceptualizations of both spatial relations and temporal relations. Missing from that list are the set of digital relations that exist within Z1 , representing discretized time, discretized ordered line segments, or discretized linear features as embedding spaces. Discretized time plays an essential role in timeseries data, spatio-temporal information systems, and geo-foundation models where time is represented in layers of consecutive spatial rasters and/or spatial vector objects colloquially referred to as space–time cubes or spatio-temporal stacks. This paper explores the digital relations that exist in Z1 interpreted as a regular topological space under the digital Jordan curve model as well as a folded-over temporal interpretation of that space for use in spatio-temporal information systems and geo-foundation models. The digital Jordan curve model represents the maximum expressive power between discretized objects, making it the ideal paradigm for a decision support system model. It identifies 34 9-intersection relations in Z1 , 42 9-intersection + margin relations in Z1 , and 74 temporal relations in Z1 , utilizing the 9+-intersection, the commercial standard for spatial information systems for querying topological relations. This work creates opportunities for better spatio-temporal reasoning capacity within spatio-temporal stacks and a more direct interface with intuitive language concepts, instrumental for effective utilization of spatial tools. Three use cases are demonstrated in the discussion, representing each of the utilities of Z1 within the spatial data science community.
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- Award ID(s):
- 2019470
- PAR ID:
- 10655880
- Publisher / Repository:
- International Journey of Geo-Information
- Date Published:
- Journal Name:
- ISPRS International Journal of Geo-Information
- Volume:
- 14
- Issue:
- 9
- ISSN:
- 2220-9964
- Page Range / eLocation ID:
- 327
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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