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

1 1 1 1

1 -j -1 j

1 j -1 -j

1 -1 1 -14

4

1 1 1 1

1 -j -1 j

1 j -1 -j

1 -1 1 -1

4

4

4

4

4

4

X[n]X[n-3]

4-channel PR FB (complex)

Orthogonal,linear phaseShort delayGroup invariantShiftableFast transform, integer arithmeticPhase available

Not very regularNot translation invariant

2

WPT Extensions - 2D WPT

x1

2

0

3

x

xx

WPT phase – a gradient estimate

23x

xx01

x1

x2

x3

x 1-D DFT

1

2

0

3

V

V V

V

1V

x0

~~

~~

~

-45

Edges Phase angle

HH

-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 withglobal phase

Reconstruction withlocal (WPT) phase

1. Image segmentation

2. Texture classification

3. Motion estimation

4. Overlapped WPT

Other applications of phase

Acknowledgements

Jun Ge Xuling LuoValerie Chickanosky