A Mathematical Model for Simplifying Representations of Objects in a Geographic Information System

The study of operations on representations of objects is well documented in the realm of spatial engineering. However, the mathematical structure and formal proof of these operational phenomena are not thoroughly explored. Other works have often focused on query-based models that seek to order classes and instances of objects in the form of semantic hierarchies or graphs. In some models, nodes of graphs represent objects and are connected by edges that represent different types of coarsening operators. This work, however, studies how the coarsening operator "simplification" can manipulate partitions of finite sets, independent from objects and their attributes. Partitions that are "simplified first have a collection of elements filtered (removed), and then the remaining partition is amalgamated (some sub-collections are unified). Simplification has many interesting mathematical properties. A finite composition of simplifications can also be accomplished with some single simplification. Also, if one partition is a simplification of the other, the simplified partition is defined to be less than the other partition according to the simp relation. This relation is shown to be a partial-order relation based on simplification. Collections of partitions can not only be proven to have a partial- order structure, but also have a lattice structure and are complete. In regard to a geographic information system (GIs), partitions related to subsets of attribute domains for objects are called views. Objects belong to different views based whether or not their attribute values lie in the underlying view domain. Given a particular view, objects with their attribute n-tuple codings contained in the view are part of the actualization set on views, and objects are labeled according to the particular subset of the view in which their coding lies. Though the scope of the work does not mainly focus on queries related directly to geographic objects, it provides verification for the existence of particular views in a system with this underlying structure. Given a finite attribute domain, one can say with mathematical certainty that different views of objects are partially ordered by simplification, and every collection of views has a greatest lower bound and least upper bound, which provides the validity for exploring queries in this regard.

Public Commons for Geospatial Data: A Conceptual Model

A wide variety of spatial data collection efforts are ongoing throughout local, state and federal agencies, private firms and non-profit organizations. Each effort is established for a different purpose but organizations and individuals often collect and maintain the same or similar information. The United States federal government has undertaken many initiatives such as the National Spatial Data Infrastructure, the National Map and Geospatial One-Stop to reduce duplicative spatial data collection and promote the coordinated use, sharing, and dissemination of spatial data nationwide. A key premise in most of these initiatives is that no national government will be able to gather and maintain more than a small percentage of the geographic data that users want and desire. Thus, national initiatives depend typically on the cooperation of those already gathering spatial data and those using GIs to meet specific needs to help construct and maintain these spatial data infrastructures and geo-libraries for their nations (Onsrud 2001). Some of the impediments to widespread spatial data sharing are well known from directly asking GIs data producers why they are not currently involved in creating datasets that are of common or compatible formats, documenting their datasets in a standardized metadata format or making their datasets more readily available to others through Data Clearinghouses or geo-libraries. The research described in this thesis addresses the impediments to wide-scale spatial data sharing faced by GIs data producers and explores a new conceptual data-sharing approach, the Public Commons for Geospatial Data, that supports user-friendly metadata creation, open access licenses, archival services and documentation of parent lineage of the contributors and value- adders of digital spatial data sets.