Wavelets with a difference Gagan Mirchandani October 18, 2002.

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Presentation transcript:

Wavelets with a difference Gagan Mirchandani October 18, 2002

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 1. Making wavelets

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

5 Haar scaling functions and wavelets in space V Level

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

7

8

9

10

11

12

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

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

15

16 Significance of phase

17 3. Convolution and stochastic deconvolution over groups

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

x(g) n(g) y(g) x(g) ε(g) h(g) ¿ Stochastic deconvolution

20 4. Application to segmentation and other things

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 Segmentation ( use of phase)

23 Segmentation application

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

25

26

27 Last slide (Research sponsored by DEPSCoR Grant) April April Edge Detection - J. Ge 2. 2.Group-based convolution - M. Elfatau 3. 3.Spline-based edge detection - S. Ganapathi 4. 4.Segmentation and Classification