#### Date of Award

Spring 5-14-2016

#### Level of Access Assigned by Author

Campus-Only Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

#### Department

Spatial Information Science and Engineering

#### Advisor

Max Egenhofer

#### Second Committee Member

M. Kate Beard-Tisdale

#### Third Committee Member

Torsten Hahmann

#### Additional Committee Members

Andre Khalil

Reinhard Moratz

#### Abstract

The world of spatial information has been painstakingly studied over the past forty years and, for the attainment of compact and meaningful systems, split into a triumvirate of domains for qualitative spatial reasoning: *topology*, *direction*, and *distance*. As advances in computation have led to quicker computation, the need for smaller, cognitively held systems is reduced, opening the door for larger reasoning systems that violate Occam’s Razor.

This dissertation attempts to bring separated concepts of qualitative spatial reasoning together by using the concept of *spatial partitions*. Partitions are mutually exclusive sets that exhaustively cover an embedding space. Qualitative direction relations over regions are currently constructed from these partitions and are based on intersection models. In this dissertation, topological relations are used to refine the qualitative direction relations over regions currently in the literature and graph theoretic definitions and theorems are presented to apply these concepts to arbitrary spatial partitions of co-dimension 0 to their embedding space.

This combination is called *topological augmentation*. Rather than computing binary set intersections between a figure object and ground tiling structure, topological augmentation expands this approach by computing a topological set intersection resulting in a topological relation from the region-region relations, providing additional insights into the *qualitative extent* of an object. Such an approach allows for a reasoning system with over 3,000 relations, relations that can be applied to particular semantic definitions based upon their context.

With the creation of such a verbose system, the interplay between direction and topology is explored and refined in a novel way by integrating the two separate forms of information before the reasoning task is attempted. While previous research has attempted combinations of direction and topology, the more verbose system presented in this work provides a transfer between direction and topology that is reduced in cardinality, a large benefit in the world of volunteered geographic information. A similar effect is also shown in the computation of converse relations and compositions using the verbose system as a guideline, moving the standard of composition in areal direction relations from the domain of *weak composition* to that of *strong composition*.

The key benefit of this approach is in the aspect of computing the composition of direction relation matrices. While a decade of research has attempted to solidify this result, numerous attempts have tried and failed to produce a systematic and verifiable composition for a given pair of objects, known as the *strong composition*, instead being successful only in the computation of the *weak composition*, that which can occur for at least one element of the set represented by the combination of objects. Through the use of topological augmentation, the strong composition of direction relation matrices is attained. Patterns in the composition are identified based on the coincidence of tile boundaries with object boundaries, a key insight that can produce crisper compositions at a minimal cost in complexity. This particular cost of complexity (from binary to topological) also serves the purpose of integrating direction relations into commercial GISs based on topological means through reverse engineering, given that the interface for computing topological relations and querying over them already exists.

Finally, composition is analyzed relative to the properties of a relation algebra, a fundamental question in the realm of artificial intelligence. It is demonstrated that the composition and converse properties of the direction relation matrix (with or without topological augmentation) do not produce a proper involution systematically, and thus the spatial representation cannot form a relation algebra. Many relation pairs as a composition are involutable, but not all. These properties are based on symmetry within the space relative to minimum bounding rectangle relations. The nature of involution in this case motivates the choices of particular forms of a relation between pairs of objects, one of the first scenarios in spatial literature where the choice of the order of the relation matters for a purpose other than a cosmetic one.

Future work is proposed to integrate direction relations into commercial GISs and to further explore the remaining properties of relation algebra relative to partition-based direction relations. Since the current gold standard is not involutable, a research agenda is proposed to determine whether or not a worthwhile direction-based information system can be developed that is endowed with universal involution, thus mitigating the effects of choices of relational order.

#### Recommended Citation

Dube, Matthew Paul, "Algebraic Refinements of Direction Relations through Topological Augmentation" (2016). *Electronic Theses and Dissertations*. 2696.

https://digitalcommons.library.umaine.edu/etd/2696