Distributed Wavelet Analysis for Sensor Networks: COMPASS Update Raymond Wagner Richard Baraniuk...

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


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…


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


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


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

…


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…


8

Mesh Refinement Example

Boundary sensors provide top-level scaling values to stabilize Predict step


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*


10

Updating Area Assignments

1\)(

jLevenVk


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*


11

Computing Update Coefficients


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*)


12

Calculating Wavelet Values

pVjP ,


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*


13

Calculating Scaling Values

pVjU ,


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*


14

Ideal Nonlinear Thresholding Example

50 sensors sampling a noisy quadratic bowl with a discontinuity at x=y.


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.

Distributed Wavelet Analysis for Sensor Networks: COMPASS Update Raymond Wagner Richard Baraniuk...

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