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

Finite groups acting on discrete signals

Wreath products of cyclic groups

The quadtree X(1,9) with 9 levels

Group-based image processing

Multiresolution analysis, filter banks and discrete wavelets

Example

Current research directions

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

j -1 j 1 j -1 -j j -1 j 1 j -1 -j 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

WPT Extensions - 2D WPT

x x x x WPT phase – a gradient estimate

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

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

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

Original V 1 |V | Cos ( ) 1 θ ~ ~ α α

Reconstruction with global phase Reconstruction with local (WPT) phase

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

Acknowledgements Jun Ge Xuling Luo Valerie Chickanosky