Distributed%20Wavelet%20Analysis%20for%20Sensor%20Networks:%20COMPASS%20Update

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Distributed Wavelet Analysis for Sensor Networks
COMPASS Update
Raymond Wagner Richard Baraniuk Hyeokho
Choi Shriram Sarvotham Veronique
Delouille COMPASS Project, Rice
University rwagner_at_rice.edu
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Wavelet Analysis for Sensor Networks

• GOAL replace sensor measurements with
  wavelet coefficients (enables compression,
  denoising, etc.)
• PROBLEM irregular sampling in 2-D introduces
  complications

• Wavelet filterbanks do not work for irregular
  sampling
• No clear idea of scale in the irregular 2-D
  grid
• Varying sensor density induces varying
  measurement importance
• Identifying neighbors for filtering is not
  straightforward

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Haar Pyramid

• Simple, first transform (ICASSP 05) that avoids
  complicated neighbor designations
• Routing clusters define multiscale structure for
  piecewise-constant (PWC) averages and differences

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Haar Pyramid

• Voronoi tesselation over the measurement field
  assigns support size, overcomes density
  problem.
• Using PWC approximation, 2-D problem maps to 1-D
  within a cluster.
• Slightly redundant pyramid representation (N
  differences, 1 average).

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Haar Telescope

• Update of Haar Pyramid method forming complete
  orthonormal basis (IPSN 05).
• Pairs measurements within a cluster and computes
  weighted, pairwise average/difference (PWC
  transform).
• Iterates to single average with cluster then
  iterates on set of cluster averages.

virtual telescope
two-level basis functions
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Lifting for Higher-Order Approximation

• In general, only second-generation wavelets
  constructed via lifting can cope with irregular
  sample grids.
• Lifting operates on data in the spatial domain
  via Split, Predict, and Update steps

scaling
odd
even
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Piecewise-Planar Lifting

• Piecewise-planar lifting transform can be
  constructed with planar regression Predict step.
• Delaunay triangulation of nodes (distributable)
  provides a mesh to determine neighbors.
• Pseudo-voronoi areas assigned to each node to
  begin the lifting transform, and areas updated
  after each stage.
• Odd nodes are selected in a greedy fashion,
  picking the node with smallest area such that no
  neighbors are also odd

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Mesh Refinement Example
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Computing Predict Coefficients
Predict coefficients at scale j are given
bywhere
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Updating Area Assignments
New areas are calculated by update sensors using
coefficients from predict sensors aswhere
describes the red neighborhood
of a blue sensor.
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Computing Update Coefficients
Update coefficients to apply to differences are
calculated at the red sensors as
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Calculating Wavelet Values
Once predict coefficients are available,
predicted sensors can calculated their scale j
wavelet difference values as
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Calculating Scaling Values
Once predict coefficients are available,
predicted sensors can calculated their scale j
wavelet difference values as
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Ideal Nonlinear Thresholding Example

• 50 sensors sampling a noisy quadratic bowl
  with a discontinuity at xy.

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Continuing Work

• Investigate iterative update computation
  recommended by V. Delouille.
• Develop tree overlay to describe coefficient
  descendence.
• Apply dynamic-programming based threshold
  procedure to tree.
• Devise distributed de-noising scheme based on
  Bayesean relaxation technique.