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Boundary Detection
Jue Wangand
Runhe Zhang
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Outline
Boundary detection using static nodes Boundary detection using mobile nodes
Applications: Dark / Light (without Gradient) Chemical Spills (without Gradient) Temperature (with Gradient)
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Boundary detection using static nodes1. K. Chintalapudi and R. Govindan,
“Localized edge detection in sensor fields,” in Ad-hoc Networks Journal, 2003.
2. J. Cortés, S. Martinez, T. Karatas, and F. Bullo, “Coverage control for mobile sensing networks,” IEEE Transactions on Robotics and Automation, vol. 20, no. 2, 2004.
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Model
Nodes arbitrary deployed, know their location (x,y) by a localization system.
Interior of the phenomenon (I) Exterior of the phenomenon (E) Ideal Edge: set of all points (x,y), such that
every non-empty neighborhood of (x,y) intersects with both I and E
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Model
Edge Sensor In the interior of the phenomenon Lies within a pre-specified distance r of
the ideal edge (r: tolerance radius)
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Robustness and Performance
Robustness Intrinsic error Sensor calibration error Threshold settings
Performances Trade off between energy and accuracy Quality of the result: thickness of edge
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Performance
Percentage Missed Detection Errors: fraction of sensors which lie within the radius of tolerance (Strue) but were not marked as edge sensors (Sdet). em=|Strue - Sdet| / |Strue|
False Detection Errors: This represents the fraction of nodes that declared themselves to be edge sensors but should not have (Sdet-Strue) among the rest of (S-Strue) sensors.ef=|Sdet-Strue| / (N-|Strue|)
Mean Thickness ratio: Let t(S,E) be the mean distance of all the sensors in set S to the edge E.et= (t(Sdet,E)-t(Strue)) / t(Strue,E)
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Three approaches to localized edge detection Statistical approach Approach from image processing Approach from pattern recognition
Sensor gets it’s own information and probe the information within its probing radius (R)
Intuitively, larger R/r yields better performances
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The statistical approach
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The statistical approach - Example
n+ the number of 1 valued predicates
n- the number of 0 valued predicates
Choice of gamma0 depends on R/r and the performance requirements
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The statistical approach - Performance
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Image Processing Approach
Basic Idea: using high-pass filter to retain high frequencies, i.e., abrupt changes such as edges present in the image and removes all the uniformities.
Treat sensor as pixel.
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Pattern Recognition approach (classifier-based) Relies on the information provided by
sensors in the interior I being ‘significantly’ different from that by sensors in the exterior E
‘Similar’ data lie in same subnet and ‘dissimilar’ data lie in different subset.
A successful partition implies the presence of edge.
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Pattern Recognition Approach - Example Classifier: Line
L(a,b,c)=ax+by+c=0 such that all sensors with 1 are on one side of the line and those with 0 lie on the other side.
A localized edge detection scheme based on a linear classifier is: if this line passes within a distance of r from the sensor, the partition is accepted as valid and the edge is deemed as an edge sensor
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Simulation and Performance – 1
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Simulation and Performance – 2
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Simulation and Performance – 3
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Simulation and Performance - 4
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Conclusion
Three approaches for boundary detection Three measurements for performances Trade offs between energy - performances
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Boundary detection using mobile nodes1. D. Marthaler and A. L. Bertozzi, “Tracking environmental level
sets with autonomous vehicles," UCLA Computational and Applied Mathematics Reports, April 2003.
2. D. Marthaler and A. L. Bertozzi, “Collective motion algorithms for determining environmental boundaries," UCLA Computational and Applied Mathematics Reports, April 2003.
3. A. L. Bertozzi, M. Kemp, and D. Marthaler, “Determining environmental boundaries: Asynchronous communication andphysical scales," UCLA Computational and Applied Mathematics Reports, March 2004.
4. A. Savvides, J Fang, and D. Lymberopoulos, “Using mobile sensing nodes for dynamic boundary estimation," Yale University,Electrical Engineering Department Report, 2004.
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Introduction
Locating the boundary of a physical phenomenon using multiple vehicles
Platform features Ability of each agent to perform a ‘move to’ function, to
move to a specified new position on command Ability of each agent to obtain position information about
other agents A sensor for determining environmental concentration at
the agent’s location and a method for estimating the local gradient concentration
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Algorithm
Anti-collision / inflation mechanism Motion related to sensing Motion related to communication and
cooperation
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Estimation of local gradient of the environmental concentration
C(x,y): the concentration function of the environment at robot’s position V=(x,y)
P(x,y)=f(C(x,y)) that achieves a minimum at the boundary of the environmental concentration. E.g. P=-(Co-C)2, Co is boundary concentration.
results in a gradient descent toward a local minimum of the function P.
Motion related to sensing
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Motion related to sensing (cont.)
Results in a composite motion of an agent toward the boundary plus motion along level curves of the concentration function of C.
Omega determines the speed at which the agent traverses the boundary once it arrives there.
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Example
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Virtual contour
Take consideration of the location of other nodes that forms virtual contour.
Alpha, beta are some constant
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Algorithm (cont.)
The nodes exchange their location information only at ‘surface time’ intervals.
Effective of Surface time Effective of Initial position Effective of Position noise
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Effective of surfacing time (1)
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Effective of surfacing time (2)
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Effective of surfacing time (3)
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Effective of surfacing time (4)
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Effective of initialization
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Effective of position noise
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Summary and Future Work
Using PDE to solve the boundary detection problem
Multiple boundaries Moving boundaries Without knowledge of gradient
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Thank you