Nouveaux articles de cet auteur. Nouvelles citations des articles de cet auteur. Adresse e-mail pour l'envoi des notifications. International conference on pervasive computing, , Investigating change in space and time, , Computers, environment and urban systems 25 1 , , The events of interest involve changes to the critical points i. Four fundamental types of event appearance, disappearance, movement and switch are defined.

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Experimental investigations confirm that our algorithm is efficient, with O n overall communication complexity where n is the number of nodes in the sensor network , an even load balance and low operational latency. The accuracy of event detection is comparable to established centralized algorithms for the identification of critical points of a surface network. Our algorithm is relevant to a broad range of environmental monitoring applications of sensor networks.

Our geographic world is highly dynamic, and consequently, monitoring change over space is of considerable interest in many scientific communities. Geosensor networks have a particularly important role in environmental monitoring [ 1 ]. These wireless networks of sensor-enabled computers embedded in the geographic environment can help with capturing information about change and even responding to events. This paper is concerned specifically with qualitative spatial events connected to the critical points peaks, pits and passes of a monitored dynamic scalar field, such as a temperature, humidity, soil moisture or pollution field.

Critical points in a scalar field are points with zero slope: Critical points can be connected by critical edges e. The combination of critical points and edges forms a surface network, also known as a Morse—Smale complex [ 3 , 4 ]. These intuitive network structures capture the essential features of complex surfaces. Applications of geosensor networks to monitoring events in dynamic scalar fields must negotiate the unique resource constraints of geosensor networks. Limited resources favor algorithms that can operate in-network without centralized control by minimizing communications.

Further, energy resource constraints may prevent positioning e. In addition, the limited granularity of geosensor networks imposes restrictions on the capability to infer information about surface networks and, so, events occurring on a surface network. Based on these limitations, recent research has yielded decentralized algorithms that are capable of identifying critical points and edges in a static field monitored by a geosensor network [ 5 , 6 , 7 ].

However, to date, this work has not addressed the problem of efficiently and accurately identifying events occurring on monitored surface networks. Building on this previous research, this paper: Section 2 begins by examining in more detail the strengths and limitations of previous research on monitoring qualitative events in a dynamic field. The formal model of a geosensor network and the extended definitions of critical points are defined in Section 3 , leading to the design of a decentralized algorithm for identifying events at critical points in a dynamic scalar field Section 4.

Section 5 presents an experimental evaluation of our algorithm in terms of its the efficiency and accuracy using simulation. The results confirm the efficiency and effectiveness of the approach, discussed in Section 6. The paper concludes with a summary and suggestions for future work in Section 7. Events are defined as salient changes in state. An important question in this study is then: In short, what changes in state are salient for surface networks?

One of the most frequently-cited works relevant to this question is [ 8 ]. The approach is based on Jacobi sets [ 9 ], which delineate the paths that critical points take over time. A more empirical approach was taken by the author of [ 10 ], who identified primitive events occurring on surface networks and analyzed changes in retail activities based on these primitive events on surface networks. The author of [ 10 ] additionally defined as events the movement of critical points. Although these works do identify salient changes to surface networks, in practice, none is directly applicable to computation within a geosensor network.

The approach of [ 8 ] relies on Jacobi sets, based, in turn, on smooth, continuous functions. Such functions are known not to be suited to real-world data [ 11 , 12 ], such as the discrete observations derived from a geosensor network. The analysis of [ 10 ] relies on centralized computation and interpolation based on the geometry of the surface, which conflicts with our requirements for a decentralized and coordinate-free computing environment.

Broadening the search, an alternative to investigating events in surface networks is to look instead at events in regions and their boundaries.

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Surface networks partition a scalar field into regions. As a scalar field evolves, these regions change and evolve. This paper focuses on the evolution of catchments to monitor events occurring on surface networks, because the catchments can be used as spatial structures to collect global information about the entire system in a geosensor network. Analysis of the geometry such as the volume or centroid of regions [ 13 , 14 , 15 ] leads to identifying appearance, disappearance, merging and splitting as four primitive events that can occur in regions.

The authors of [ 16 , 17 ] also arrive at these four primitive events, using the topology of Reeb graphs to track the evolution sequences of events of burning regions. Closely related, the authors of [ 18 ] use contour trees along with geometric information about the volumes of regions to monitor essentially the same four events in turbulent vortex structures. The analysis of [ 19 , 20 ] of primitive events, involving simple, connected polygons, yields two further event types: Although the specific terms used to name these six events vary across papers, others have similarly arrived at these six events, including a range of applications in disciplines, such as meteorology [ 21 , 22 ], and tracking the evolution of social groups [ 23 , 24 ].

Thus, in the context of surface networks, this previous work suggests up to four primitive events: Other approaches are also possible. Unlike [ 19 , 20 ], the authors of [ 25 ] also consider disconnected regions i. The authors of [ 26 ] provide the pure topological events: However, in the case of scalar fields in the plane, it is not possible for a surface network to partition the space into regions with holes although this is a possibility in surface networks in other embedding spaces, such as a torus.

In summary, the key events relevant to surface networks appear to be: However, none of the approaches encountered are directly applicable to our decentralized and coordinate-free computing environment. With the exception of [ 26 ], none of the approaches described above are decentralized. However, as we have seen, the authors of [ 26 ] include several events for regions that are not directly relevant to surface networks. Further, most of the approaches above rely on geometric information about the surface and, so, are ultimately reliant on access to coordinate information about the location of critical points and their associated regions.

This paper does, however, substantially revise and extend our previous work in [ 7 ]. Our previous work defines and evaluates a decentralized and coordinate-free algorithm to identify critical points and surface networks in a static field. Based on the extended definitions of discrete surface networks, this paper not only defines basic spatial events occurring on surface networks, but also provides a decentralized algorithm to detect these events in a dynamic field.

Based on our review of the existing literature relevant to events on surface networks, we will now proceed with the design of an algorithm capable of detecting our four primitive surface network events: The algorithm is amenable to decentralized computation and capable of operating within the constraints of the limited spatial granularity of a sensor network. The formal model of a geosensor network in this paper follows the approach of [ 27 ]. The set of neighboring nodes is represented as a function nbr: Each node has a unique identity, modeled as a function id: Each node also has the ability to sense its changing environment, modeled using the sense function, sense: Note that although we allow time-varying sensed data, we assume that the structure of the communication graph and location of the nodes are static.

Using this foundational model of a geosensor network, the algorithm definitions in subsequent sections follow the decentralized algorithms design and specification style of [ 27 , 28 ]. In brief, there are four key components of decentralized algorithms: Restrictions concern the assumptions made about the environments in which an algorithm will operate. For example, there are no restrictions in terms of the structure of the communication networks in our algorithm.

We do not require spatial information, such as coordinate information. However, we assume our sensor networks are static and the communications are reliable see Algorithm 1, Line 1. System events define the external stimuli that nodes can respond to, such as receiving a message from another node or sensing a change to a monitored environmental variable see Algorithm 1, Line 4 or 7.

When a system event occurs, a node will react by initiating an atomic, terminating sequence of operations, called an action see Algorithm 1, Line 6. System states allow nodes to respond to the same events with different actions based on the effects of previous system events and actions see Algorithm 1, Line 3 or 6. This section explains the definitions of critical points for the finite spatial granularity of the discrete point data that are generated by a geosensor network.

These definitions will be used in the following Section 4 for the algorithm explanation. The ascent descent vector of a node is defined as the unique directed edge from that node to its one-hop neighbor with the highest lowest sensed value of all neighbors.

For example, a weak peak in Figure 1 has communication links with four neighboring nodes. Identification of a strong and a weak peak, ascent vectors and an ascent bridge Contour lines describe a scalar field showing the difference in elevation between consecutive contour lines. The sensed values can be estimated using a contour map. In addition, a weak peak can be connected to a strong peak via an ascent bridge.

An ascent bridge is an edge from a weak peak v to a neighboring node whose ascent vector points away from v or symmetrically for a weak pit, strong pit and descent bridge. These basic structures are illustrated in Figure 1. Using this information, it is then possible to design decentralized algorithms to identify for each node the strong peak and pit associated with that node i.

Examples of pass-edges are shown in Figure 2.

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Two thick black lines are pass-edges for which associated peaks and pits are different i. Representative pass Contour lines describe a scalar field showing the difference in elevation between consecutive contour lines. Multiple pass-edges were grouped together as a pass-cluster. Moving to a dynamic scenario, however, it becomes highly inefficient to continually monitor events occurring on a group of pass-edges. Therefore, in this paper, we add a further definition of the representative pass amongst a pass-cluster.

For example, let R be the set of pass edges that connect two specified pairs of peaks and pits in the surface network. Focusing on a representative pass, rather than a potentially large set of pass-edges, it becomes easier and more efficient to monitor events occurring on passes. It is possible that two representative passes occur as one-hop neighbors, akin to a monkey saddle in a continuous surface.

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In practice, such monkey saddles do occur in our sensor networks, but due to network granularity effects, rather than being a true reflection of the topography of the underlying surface. In other words, monkey saddles typically occur as a result of adverse network connectivity leading to certain spatially nearby nodes not being one-hop network neighbors.

Thus, in our algorithms, we also include procedures for coordination amongst neighboring representative passes to account for such granularity effects. In addition to a monkey saddle, it is important to note that degenerate critical points can occur due to the same values. Such a plateau is mainly generated by the discrete quantization while extracting surface networks. The authors in [ 4 , 29 ] deal with degenerate critical points by using perturbation. However, real data from geosensor networks are unlikely to contain identical sensed values.

This paper assumes that there are no plateaus between one-hop neighbor nodes. As argued in Section 2 , there are four primitive events occurring on surface networks: Previous approaches to monitoring such events e. In keeping with the resource constraints imposed by sensor networks, in this paper, we develop a decentralized algorithm that can monitor surface events without coordinate information.

For the ease of explanation, we present first the monitoring of events on peaks and pits and then the monitoring of events on passes. This section examines the design of a decentralized algorithm for monitoring events occurring on peaks and pits. The following subsection addresses the problem of monitoring events on passes.

## Decentralized spatial computing foundations of geosensor networks - Laurentian University

The network is initialized by decentrally identifying strong peaks and pits. In brief, each node broadcasts its sensed value. Nodes can then locally determine their ascent and descent vectors and whether they are a peak or pit. Flooding of a single initialization message from each identified peak and pit can then be used to enable every node in the network to be informed of its unique strong pit and peak, as well as discern apart weak and strong peaks and build gradient ascent and descent bridges.

The result of initialization is to partition the nodes into regions. Nodes in each region are associated with a unique pair of peak and pit identifiers. This figure is adapted from [ 30 ]. As the dynamic field evolves, our algorithm operates by inspecting these catchment areas for changes that indicate events occurring on the surface network. If one catchment area is divided into two between consecutive time steps, this indicates that a new peak has appeared on the surface network. Conversely, if two catchment areas are merged into one between consecutive time step, this indicates that one of the peaks has disappeared from the surface network.

Based on catchment areas, it is now possible to specify a decentralized spatial algorithm to monitor all of the events occurring on peaks and pits.

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For the ease of explanation, this part of the algorithm is split into four components Algorithms 1—4 based on the types of events occurring. In the sequel, we also only discuss the case for peaks; identification of events for pits occurs in a symmetric fashion. Algorithm 1 responds to changes in the dynamic field.

Each node monitors locally any changes in its sensed value. When a change is detected, a node broadcasts an update message upd8 to its neighbors Algorithm 1, Line 6. If a node needs to update its peak identifier pkid following a change in its ascent vector, it must then initiate a cascade of notifications about this change to its neighbors Algorithm 1, Lines 12— Detecting changes in gradient vectors in this way provides the basis for all higher level monitoring of the events occurring on critical points Algorithm 1, Lines 19— A node that transitions from state peak a strong peak to state idle a non-peak indicates that a peak has moved.

Such transitions are detected in Algorithm 2. It is similarly straightforward to detect peak disappearance Algorithm 3. If this wipk message reaches a node that has a different peak identifier, that node can then infer that the peak represented by the node that initiated the wipk message has disappeared. Algorithm 3 can be summarized as follows:. Interestingly, when a node that was previously a peak receives a rwpk message, it may have already changed its peak identifier: In this case, this message simply confirms that a peak disappeared.

Lastly, Algorithm 4 presents a mechanism to monitor the appearance of peaks. As is common in decentralized algorithm design, we make no assumptions in our algorithm about network synchronization such as message ordering or bounded communication delays. Combined with the lack of centralized control inherent in decentralized algorithms, this lack of coordination makes it more challenging to monitor a peak appearance, when compared to events such as peak movement or disappearance.

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