1. AFOSR PROGRAM REVIEW: INNOVATIVE SIGNAL PROCESSING FOR MILITARY DIGITAL COMMUNICATION DEPARTMENT OF MATHEMATICS AND STATISTICS UNIVERSITY OF VERMONT JUNE 6 – 8, 2002

2. Applications of Finite Group Theory to Signal Processing by Richard Foote, Gagan Mirchandani and Dan Rockmore

3. Finite groups acting on discrete signals

4. Wreath products of cyclic groups

5. The quadtree X(1,9) with 9 levels

6. Group-based image processing

7. Multiresolution analysis, filter banks and discrete wavelets

8. Example

9. Current research directions

10. 2. APPLICATIONS WITH WPC GROUPS 1.WPT – 4-channel PR FB 2.WPT extensions 3.WPT phase – implications 4.WPT applications

11. 1 1 1 -j -1 j 1 j -1 -j 1 -1 4 4 1 1 1 -j -1 j 1 j -1 -j 1 -1 4 4 4 4 4 4 X[n] X[n-3] 4-channel PR FB (complex) Orthogonal,linear phase Short delay Group invariant Shiftable Fast transform, integer arithmetic Phase available Not very regular Not translation invariant 2

12. WPT Extensions - 2D WPT

13. x 1 2 0 3 x x x WPT phase – a gradient estimate

14. 23 x x x 0 1 x 1 x 2 x 3 x 1-D DFT 1 2 0 3 V V V V 1 V x 0 ~ ~ ~ ~ ~

16. -45 EdgesPhase angle H H -45 Relationship: Edge angle and WPT phase (V ) 1 ~

17. Spectrum: Angle -45, BW 10 Reconstruction: Angle -45, BW 10 Reconstruction: Angle -80, BW 5 Steerable filtering* with group-based filters * work with Valerie Chickanosky

18. Original V 1 |V | Cos ( ) 1 θ ~ ~ α α

21. Reconstruction with global phase Reconstruction with local (WPT) phase

22. 1. Image segmentation 2. Texture classification 3. Motion estimation 4. Overlapped WPT Other applications of phase

23. Acknowledgements Jun Ge Xuling Luo Valerie Chickanosky