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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 [email protected]
Distributed Wavelet Analysis for Sensor Networks (compass.cs.rice.edu)
2
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
Distributed Wavelet Analysis for Sensor Networks (compass.cs.rice.edu)
3
Haar Pyramid
• Simple, first transform (ICASSP ‘05) that avoids complicated neighbor designations
• Routing clusters define multiscale structure for piecewise-constant (PWC) averages and differences…
Distributed Wavelet Analysis for Sensor Networks (compass.cs.rice.edu)
4
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).
Δ1
S
W1
W2
W3
tot1
tot1
1
1
tot
tot
)( 321 tot
Δ2 Δ3
Distributed Wavelet Analysis for Sensor Networks (compass.cs.rice.edu)
5
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
Distributed Wavelet Analysis for Sensor Networks (compass.cs.rice.edu)
6
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:
“odd”
“even”
scaling
split P U
detail
split P U
detail
split P U
detail
…
Distributed Wavelet Analysis for Sensor Networks (compass.cs.rice.edu)
7
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…
Distributed Wavelet Analysis for Sensor Networks (compass.cs.rice.edu)
8
Mesh Refinement Example
Boundary sensors provide top-level scaling values to stabilize Predict step
Distributed Wavelet Analysis for Sensor Networks (compass.cs.rice.edu)
9
Computing Predict Coefficients
- predicted - updated
)( PVLet describe the neighborhood around a point VP to be predicted
')')]((),(,1[ 1, XXXVyVxP ppVj p
]))((,))((,1[ PP VyVxX
Predict coefficients at scale j are given by:
where:
x(*),y(*)
x(*),y(*)
x(*),y(*)
x(*),y(*)x(*),y(*)
x(*),y(*)
x(*),y(*)x(*),y(*)
x(*),y(*)
x(*),y(*)
x(*),y(*)
x(*),y(*)
x(*),y(*)
x(*),y(*)
x(*),y(*)
Pj,V*
Pj,V*
Pj,V*
Distributed Wavelet Analysis for Sensor Networks (compass.cs.rice.edu)
10
Updating Area Assignments
1\)(
jLevenVk
- predicted - updated
New areas are calculated by update sensors using coefficients from predict sensors as:
where describes the red neighborhood of a blue sensor.
1\)(
)(,,1,1,
jL
ppLLevenVk
kVjVjVjVj PAAA
(Aj+1*,Pj,V*(*))
(Aj+1*,Pj,V*(*))
(Aj+1*,Pj,V*(*))
(Aj+1*,Pj,V*(*))(Aj+1*,Pj,V*(*))
(Aj+1*,Pj,V*(*))
(Aj+1*,Pj,V*(*))
(Aj+1*,Pj,V*(*))
(Aj+1*,Pj,V*(*))
(Aj+1*,Pj,V*(*))
(Aj+1*,Pj,V*(*))
(Aj+1*,Pj,V*(*))
(Aj+1*,Pj,V*(*))
(Aj+1*,Pj,V*(*))
(Aj+1*,Pj,V*(*))
Aj,V*
Aj,V*
Aj,V*
Aj,V*
Aj,V*Aj,V*
Aj,V*
Aj,V*
Aj,V* Aj,V*
Distributed Wavelet Analysis for Sensor Networks (compass.cs.rice.edu)
11
Computing Update Coefficients
- predicted - updated
Update coefficients to apply to differences are calculated at the red sensors as:
ppp VjVjVj AAu ,1)(,)(, )(
Aj,V*
Aj,V*
Aj,V*
Aj,V*Aj,V*
Aj,V*
Aj,V*
Aj,V*
Aj,V*
Aj,V*
Aj,V*
Aj,V*
Aj,V*
Aj,V*
Aj,V*
Uj,n(V*)
Uj,n(V*)
Uj,n(V*)
Distributed Wavelet Analysis for Sensor Networks (compass.cs.rice.edu)
12
Calculating Wavelet Values
pVjP ,
- predicted - updated
Once predict coefficients are available, predicted sensors can calculated their scale j wavelet difference values as:
)(,1,,1, pppp VjVjVjVj sPsd
Sj+1,n(V*)(*)Sj+1,n(V*)(*)
Sj+1,n(V*)(*)
Sj+1,n(V*)(*)
Sj+1,n(V*)(*)
Sj+1,n(V*)(*)
Sj+1,n(V*)(*)
Sj+1,n(V*)(*)
Sj+1,n(V*)(*)Sj+1,n(V*)(*)
Sj+1,n(V*)(*)
Sj+1,n(V*)(*)
Sj+1,n(V*)(*)
Sj+1,n(V*)(*)
Sj+1,n(V*)(*)
Sj+1,n(V*)(*)
dj,v*
dj,v*
dj,v*
Distributed Wavelet Analysis for Sensor Networks (compass.cs.rice.edu)
13
Calculating Scaling Values
pVjU ,
- predicted - updated
Once predict coefficients are available, predicted sensors can calculated their scale j wavelet difference values as:
1\)(
)(,,1,1,
jL
ppLLevenVk
kVjVjVjVj udss
dj+1,v* uj,v*(*)
dj+1,v* uj,v*(*)
dj+1,v* uj,v*(*)
dj+1,v* uj,v*(*)
dj+1,v* uj,v*(*)
dj+1,v* uj,v*(*)
dj+1,v* uj,v*(*)
dj+1,v* uj,v*(*)
dj+1,v* uj,v*(*)
dj+1,v* uj,v*(*)
dj+1,v* uj,v*(*)
dj+1,v* uj,v*(*)
dj+1,v* uj,v*(*)
dj+1,v* uj,v*(*)
dj+1,v* uj,v*(*)
dj+1,v* uj,v*(*)
Sj,v*
Sj,v*
Sj,v*
Sj,v*
Sj,v*
Sj,v*Sj,v*
Sj,v*
Sj,v*Sj,v*
Distributed Wavelet Analysis for Sensor Networks (compass.cs.rice.edu)
14
Ideal Nonlinear Thresholding Example
50 sensors sampling a noisy quadratic bowl with a discontinuity at x=y.
Distributed Wavelet Analysis for Sensor Networks (compass.cs.rice.edu)
15
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.