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AFOSR PROGRAM REVIEW:

AFOSR PROGRAM REVIEW:. INNOVATIVE SIGNAL PROCESSING FOR MILITARY DIGITAL COMMUNICATION DEPARTMENT OF MATHEMATICS AND STATISTICS UNIVERSITY OF VERMONT JUNE 6 – 8, 2002. Applications of Finite Group Theory to Signal Processing. by Richard Foote, Gagan Mirchandani and Dan Rockmore.

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AFOSR PROGRAM REVIEW:

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  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 • WPT – 4-channel PR FB • WPT extensions • WPT phase – implications • WPT applications

  11. 1 1 1 1 4 1 1 1 1 4 1 -j -1 j 1 j -1 -j 4 4 X[n] 2 X[n-3] 1 -1 1 -1 4 4 1 -1 1 -1 1 j -1 -j 4 1 -j -1 j 4 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

  12. WPT Extensions - 2D WPT

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

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

  15. H -45 -45 H Edges Phase angle ~ Relationship: Edge angle and WPT phase (V ) 1

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

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

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

  19. Other applications of phase • Image segmentation • Texture classification • Motion estimation • Overlapped WPT

  20. Acknowledgements Jun Ge Xuling Luo Valerie Chickanosky

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