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Distributed Wavelet Analysis for Sensor Networks: COMPASS Update

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

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Distributed Wavelet Analysis for Sensor Networks: COMPASS Update

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  1. Distributed Wavelet Analysis for Sensor Networks: COMPASS Update Raymond Wagner Richard Baraniuk Hyeokho Choi Shriram Sarvotham Veronique Delouille COMPASS Project, Rice University rwagner@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. S W1 W2 W3 Δ1 Δ2 Δ3 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). 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. split split split U P P U P U detail detail detail 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” 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. Boundary sensors provide top-level scaling values to stabilize Predict step Mesh Refinement Example Distributed Wavelet Analysis for Sensor Networks (compass.cs.rice.edu)

  9. - predicted - updated x(*),y(*) Pj,V* x(*),y(*) x(*),y(*) x(*),y(*) Let describe the neighborhood around a point VP to be predicted x(*),y(*) x(*),y(*) Pj,V* x(*),y(*) x(*),y(*) x(*),y(*) x(*),y(*) x(*),y(*) x(*),y(*) x(*),y(*) Pj,V* x(*),y(*) x(*),y(*) Computing Predict Coefficients Predict coefficients at scale j are given by:where: Distributed Wavelet Analysis for Sensor Networks (compass.cs.rice.edu)

  10. Aj,V* Aj,V* - predicted - updated Aj,V* Aj,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+1*,Pj,V*(*)) Aj,V* (Aj+1*,Pj,V*(*)) (Aj+1*,Pj,V*(*)) Aj,V* (Aj+1*,Pj,V*(*)) (Aj+1*,Pj,V*(*)) (Aj+1*,Pj,V*(*)) Aj,V* Aj,V* Updating Area Assignments New areas are calculated by update sensors using coefficients from predict sensors as:where describes the red neighborhood of a blue sensor. Distributed Wavelet Analysis for Sensor Networks (compass.cs.rice.edu)

  11. - predicted - updated Aj,V* Uj,n(V*) Aj,V* Aj,V* Aj,V* Aj,V* Aj,V* Uj,n(V*) Aj,V* Aj,V* Aj,V* Aj,V* Aj,V* Aj,V* Aj,V* Uj,n(V*) Aj,V* Aj,V* Computing Update Coefficients Update coefficients to apply to differences are calculated at the red sensors as: Distributed Wavelet Analysis for Sensor Networks (compass.cs.rice.edu)

  12. - predicted - updated Sj+1,n(V*)(*) Sj+1,n(V*)(*) Sj+1,n(V*)(*) Sj+1,n(V*)(*) dj,v* Sj+1,n(V*)(*) dj,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* Sj+1,n(V*)(*) Sj+1,n(V*)(*) Calculating Wavelet Values Once predict coefficients are available, predicted sensors can calculated their scale j wavelet difference values as: Distributed Wavelet Analysis for Sensor Networks (compass.cs.rice.edu)

  13. Sj,v* - predicted Sj,v* - updated Sj,v* Sj,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* dj+1,v* uj,v*(*) dj+1,v* uj,v*(*) Sj,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* Calculating Scaling Values Once predict coefficients are available, predicted sensors can calculated their scale j wavelet difference values as: 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. Distributed Wavelet Analysis for Sensor Networks (compass.cs.rice.edu)

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