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

2. 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. 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…

4. 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). Δ1Δ1 S W1W1 W2W2 W3W3 Δ2Δ2 Δ3Δ3

5. 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

6. 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 PU detail split PU detail split PU detail …

7. 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…

8. 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

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

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

11. 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: A j,V* U j,n(V*)

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

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

14. 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.

15. 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.