1. Wavelets with a difference Gagan Mirchandani October 18, 2002

2. 2 1. 1.Simple intro. to wavelets and simple examples 2. 2.Some better wavelets (group theory and phase) 3. 3.Convolution & stochastic deconvolution over groups 4. 4.Application to segmentation and other things

3. 3 1. Making wavelets

4. 4 V V W 0 0 1 … V W Dilations and translations of (compact support) wavelets form the basis

5. 5 Haar scaling functions and wavelets in space V Level 0 1 1 0

6. 6 LP HP LP HP V V V W W 1 0 0 data spectrum ………… Level 1 Level 2 filter then synthesis (convolution) N N/2 N/2

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13. 13 What’s wrong with (real) wavelets? - -No spatial invariance - No convolution capability

14. 14 2. Group-based wavelets --group invariance --convolution --complex wavelet coeffs. (phase)

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16. 16 Significance of phase

17. 17 3. Convolution and stochastic deconvolution over groups

18. 18 F I L T E R x(t) y(t) k(t) x(g) k(g) y(g) standardconvolution

19. 19 + + x(g) n(g) y(g) x(g) ε(g) h(g) ¿ Stochastic deconvolution

20. 20 4. Application to segmentation and other things

21. 21 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

22. 22 Segmentation ( use of phase)

23. 23 Segmentation application

24. 24 Classification application (Brodatz texture data base ORL faces data base)

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27. 27 Last slide (Research sponsored by DEPSCoR Grant) April 2000 - April 2003 1. 1.Edge Detection - J. Ge 2. 2.Group-based convolution - M. Elfatau 3. 3.Spline-based edge detection - S. Ganapathi 4. 4.Segmentation and Classification www.uvm.edu/~mirchand