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M. Weber and E. A. Lee
not handling information that is not needed. For example, the set containment relation is all the map information necessary for the FourSquare localization example
in the introduction, since the only information to be gained from checking in is
containment.
Consider the advantages of applying this maximum abstraction principle to the
spatial ontologies depicted in Fig. 3. All three ontologies, the occupancy grid, the floor
plan, and the abstract graph, represent information about the same region of space.
An IoT application could theoretically use any one of the ontologies to determine, for
example, that room 545Q is inside the DOP Center. However, it takes a certain level
of geometric understanding to extract that information from the less abstract physical
ontologies. To use the occupancy grid, our IoT application must be equipped with
an algorithm for parsing occupancy grids and determining when a collection of cells
in a grid is contained by another collection of cells. In other words, effective use of
the occupancy grid is restricted to IoT applications that are prepared in advance to
interact with robotic maps. Similar limitations hold for IoT systems using the floor
plan, or any other spatial ontology requiring geometric analysis.
But if the IoT system were designed to interact with map providers through an
abstract notion of containment, the system wouldn’t have to bother understanding the
nuances of geometry in every spatial ontology it might come across. It may instead
operate in terms of semantic localization. Perhaps the relational ontology was created
by inspecting an occupancy grid, or maybe it was a floor plan. Either way the IoT
application doesn’t have to bother knowing the specifics. As long as it can pose the
query regarding the DOP Center, room 545Q, and containment, the IoT application
can treat the source of an abstracted spatial representation as a black box.
As they get more abstract, of course, relational ontologies lose the ability to
evaluate some kinds of spatial relationships. This idea parallels the usual hierarchy
of mathematical spaces. A Euclidean space has quite a lot of mathematical structure
that may not match well with the information available sensors are able to deliver. A
Euclidean-space ontology supports reasoning about angles and orientation, concepts
that are not defined in the more abstract mathematical structures shown in Fig. 2.
The hierarchy of these mathematical spaces offers a starting point for reasoning
about combinations of maps. For example, given a Euclidean-space map of an office
space and a Set (containment) map of objects in the space, objects can be placed
approximately, with known error bounds, onto the Euclidean-space map. But much
more complicated mapping combinations will be required, since even two Euclideanspace maps may not use the same coordinate system. The concept of a spatial ontology
becomes an essential feature of location modeling.
Topological spaces can be used to construct maps that represent paths through
indoor settings. Navigation with graphs is a common concept in robotics [22], where
nodes represent waypoints in a space and edges represent paths between waypoints.
Such data structures are routinely used to construct sequences of actions to move a
robot between nodes. Additionally, Ghrist et al. [23] show that algebraic topology
can be used directly to relate the convex hull of a landmark set in a Euclidean space
to a simplex of a simplicial complex. This provides a natural abstraction mechanism
for topological maps.
M. Weber and E. A. Lee
not handling information that is not needed. For example, the set containment relation is all the map information necessary for the FourSquare localization example
in the introduction, since the only information to be gained from checking in is
containment.
Consider the advantages of applying this maximum abstraction principle to the
spatial ontologies depicted in Fig. 3. All three ontologies, the occupancy grid, the floor
plan, and the abstract graph, represent information about the same region of space.
An IoT application could theoretically use any one of the ontologies to determine, for
example, that room 545Q is inside the DOP Center. However, it takes a certain level
of geometric understanding to extract that information from the less abstract physical
ontologies. To use the occupancy grid, our IoT application must be equipped with
an algorithm for parsing occupancy grids and determining when a collection of cells
in a grid is contained by another collection of cells. In other words, effective use of
the occupancy grid is restricted to IoT applications that are prepared in advance to
interact with robotic maps. Similar limitations hold for IoT systems using the floor
plan, or any other spatial ontology requiring geometric analysis.
But if the IoT system were designed to interact with map providers through an
abstract notion of containment, the system wouldn’t have to bother understanding the
nuances of geometry in every spatial ontology it might come across. It may instead
operate in terms of semantic localization. Perhaps the relational ontology was created
by inspecting an occupancy grid, or maybe it was a floor plan. Either way the IoT
application doesn’t have to bother knowing the specifics. As long as it can pose the
query regarding the DOP Center, room 545Q, and containment, the IoT application
can treat the source of an abstracted spatial representation as a black box.
As they get more abstract, of course, relational ontologies lose the ability to
evaluate some kinds of spatial relationships. This idea parallels the usual hierarchy
of mathematical spaces. A Euclidean space has quite a lot of mathematical structure
that may not match well with the information available sensors are able to deliver. A
Euclidean-space ontology supports reasoning about angles and orientation, concepts
that are not defined in the more abstract mathematical structures shown in Fig. 2.
The hierarchy of these mathematical spaces offers a starting point for reasoning
about combinations of maps. For example, given a Euclidean-space map of an office
space and a Set (containment) map of objects in the space, objects can be placed
approximately, with known error bounds, onto the Euclidean-space map. But much
more complicated mapping combinations will be required, since even two Euclideanspace maps may not use the same coordinate system. The concept of a spatial ontology
becomes an essential feature of location modeling.
Topological spaces can be used to construct maps that represent paths through
indoor settings. Navigation with graphs is a common concept in robotics [22], where
nodes represent waypoints in a space and edges represent paths between waypoints.
Such data structures are routinely used to construct sequences of actions to move a
robot between nodes. Additionally, Ghrist et al. [23] show that algebraic topology
can be used directly to relate the convex hull of a landmark set in a Euclidean space
to a simplex of a simplicial complex. This provides a natural abstraction mechanism
for topological maps.