afosr program review:
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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. - PowerPoint PPT PresentationTRANSCRIPT
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