Pattern Matching in Sensor Networks

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Pattern Matching in Sensor Networks

• Tom Schoellhammer

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Outline

• Localized Edge Detection in Sensor Networks
  Krishna Chintalapudi et al.
• Boundary Estimation in Sensor Networks Robert
  Nowak et al.
• My own work

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Localized Edge Detection in Sensor Fields

• Krishna Chintalapudi
• Ramesh Govindan
• USC

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Outline

• Define the notion of an edge
• Develop performance metrics for localized edge
  detection algorithms
• Edge Detection Approaches
• Statistical
• Image Processing
• Classifier Based
• Simulation Framework
• Results
• Conclusion

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Notion of an Edge

• A node knows whether it is interior or exterior
  to the phenomenon
• Edge points intersect both interior and exterior
• Basically, a step edge

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Notion of an Edge, contd.

• Ideal edge has no thickness
• Edge nodes are those nodes that are
• Interior to the phenomenon
• Lies within a tolerance radius of the ideal edge
• Tolerance radius dictates the thickness of an
  edge that users are willing to tolerate

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Metrics

• Two classes of metrics
• Robustness
• Few false negatives
• Few false positives
• Insensitive to sensor calibration error
• Insensitive to threshold settings over a broad
  range of operating conditions
• Performance
• Energy expended
• Edge thickness

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Metrics contd.

• Percent Missed Detection Errors
• Those sensors that lie within the tolerance
  radius but were not detected
• em Strue Sdet / Strue

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Metrics contd.

• Mean Thickness Ratio
• Let t(S, E) be the mean distance of all the
  sensors in the set S to the edge E
• Outliers are disregarded
• et t(Sdet, E) t(Strue, E) / t(Strue, E)

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Metrics contd.

• False Detection
• Those nodes that are declared edges but are not
• N is the total number of nodes in the field
• ef Sdet Strue / N

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Statistical Approach

• Information collected from neighborhood defined
  by probing radius R r
• A set of statistics T1, , Tn based on the
  information collected from neighbors
• A boolean decision function Phi(T1, , Tn )

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Statistical Approach ex.

• Let n be the number of 1 valued event predicates
  in a neighborhood
• Let n- be the number of 0 valued event predicates
  in a neighborhood
• T 1 - n - n- / ( n n- )
• Phi( T ) 1 if T gt gamma0, 0 otherwise

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Image Processing Approach

• A Prewitt Sobel filter is used at each node to
  calculate the gradient
• If the gradient is larger than some threshold
  then the node is an edge node

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Classifier-based Approach

• Create a partition of neighbor data
• Use a partition validity measure to decide
  whether a point is on an edge
• Linear partition
• If the line passes within the tolerance radius of
  the node then the node is an edge node

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Classifier Example
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Classifier Example
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Simulation Framework

• Two different data sets
• Linear boundary
• Randomly chosen line partitioning the sensor
  field
• Elliptical boundary
• Randomly chosen major and minor axes lengths
• Randomly chosen orientation

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Simulation Framework contd.

• Parameters
• Different node densities
• 5, 15, 30 nodes per radio range
• Sensor error model
• Event predicate is negated with probability p
• p 1, 5, 10
• Probing radius to tolerance radius ratio R/r
• 1, 1.5, 2, 2.5, 3
• Best parameters used for all simulations

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Simulation Results contd.

• Detection probability goes up for higher values
  of R/r

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Simulation Results contd.

• Unwanted detections go up with increased R/r

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Simulation Results contd.
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Simulation Results

• Classifier based approach is best

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Boundary Estimation in Sensor NetworksTheory
and Methods

• By Robert Nowak and Urbashi Mitra
• Rice University
• Slides adapted from Naim Busek

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Introduction

• Boundary Estimation
• Two or more regions of distinct behavior (e.g.
  differing mean values)
• Tradeoff between accuracy (mean squared error)
  and power usage
• MSE 1/Power

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Problem Formulation Approach

• Hierarchical structure of clusterheads
• Clusterhead election within square
• Clusterhead computes
• Letting n4J creates J1 layers in the hierarchy

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Hierarchical Organization
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Hierarchical Boundary Estimation (Features)

• High resolution at boundary Low resolution in
  homogeneous regions.
• With mild constraints on the boundary smoothness
  can derive an upper bound on MSE.
• This bound can be used to tune the tradeoff of
• MSE 1/Power
• Complexity of estimate relates directly to power
  consumption.

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Hierarchical Boundary Estimation (Partitioning)

• Sensor domain is the unit square

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Hierarchical Boundary Estimation (Partitioning)

• Sensor domain is the unit square
• Partition into n sub-squares

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Hierarchical Boundary Estimation (Partitioning)

• Sensor domain is the unit square
• Partition into n sub-squares
• Perform bottom up pruning of tree

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Hierarchical Boundary Estimation (Partitioning)

• Sensor domain is the unit square
• Partition into n sub-squares
• Perform bottom up pruning of tree

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Hierarchical Boundary Estimation (Partitioning)

• Sensor domain is the unit square
• Partition into n sub-squares
• Perform bottom up pruning of tree
• Boundary resolution is sqrt(n)

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Bounds sans Details (Accuracy-Power Trade-off)

• Total comm. cost in-network (local) cost
  out-of-network (sending to observer) cost
• Out-of-network Communication costs
• final description of the boundary O(sqrt(n))
• In-network Communication cost
• final size of tree Cost O(sqrt(n))

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Simulations

• Run for size 4k networks for k 2,,8
• 10dB signal-to-noise, MSE averaged over 50 runs

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Conclusions and Future Work

• Method for boundary estimation
• Method nearly achieves optimal trade-off
• MSE 1/Power
• Simulations agree with theory
• Future work
• Use more sophisticated boundaries
• Adding in effects of communication channel
• Tracking a time-varying boundary

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Gradient Triggered Edge Detection in Sensor
Networks
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Outline

• Problem, Terminology
• Edge Definition
• Metrics
• Detection Procedure
• Scenario Setup, Assumptions, and Results
• Additional Stuff

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The Problem

• Whats new here
• More general idea of an edge
• Not just step edges
• Data directed search
• Trigger search based on local gradient
• Search in the direction of local gradient

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Edge Point Definition

• Upper limit
• Lower limit
• Trigger gradient
• End gradient
• Horizontal extent

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Metric Rational

• Measure properties that affect the way edges are
  used
• Traversal

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Metric Mean Distance between Adjacent Edge Nodes

• Rational
• Edge should be traversable
• Compare it to radio range
• Projection onto the ideal edge imposes an
  ordering
• Calculate the mean distance between adjacent nodes

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Metric Mean Edge Thickness

• Rational
• Minimize the area that needs to be traversed
• Compare to radio range
• Calculate the mean distance from each marked
  point to the ideal edge

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Detection Procedure

• Nodes collect local data

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Detection Procedure

• Nodes collect local data
• Nodes calculate their gradient

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Detection Procedure

• Nodes collect local data
• Nodes calculate their gradient
• Node calculating a large gradient looks for the
  high and low steps

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Detection Procedure

• Nodes collect local data
• Nodes calculate their gradient
• Node calculating a large gradient looks for the
  high and low steps
• Step info flows back

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Gradient Calculation

• Gradient is estimated from a finite set of
  samples.
• Gradient estimation errors can lead to false
  positive and false negative triggers.

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Gradient Calculation
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Performance Tradeoff

• High threshold
• Small mean thickness
• Large mean distance

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Performance Tradeoff

• Low threshold
• Large mean thickness
• Small mean distance

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Limitations of the work to date

• Limitations of analysis, studies
• Havent quantified error in gradient calculation
• Haven't tried this on real data or faked-real
  data
• Limitation of approach
• Gradient estimation is poor when density is low
• Gradient estimation is poor when sampling region
  is on the order of feature size (low density)