Wavelets with a difference

Gagan Mirchandani

October 18, 2002

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1. Simple intro. to wavelets and simple examples

2. Some better wavelets (group theory and phase)

3. Convolution & stochastic deconvolution over groups

4. Application to segmentation and other things

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1. Making wavelets

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V

V

W

0

0

1

…V

W-1

-1

Dilations and translations of (compact support) wavelets form the basis

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Haar scaling

functions and

wavelets in space V

Level 0

1

1

00

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LP

HP

LP

HP

V

V

V

W

W

1

0

0

-1

-1data

spectrum

…………

Level 1

Level 2

filter

thensynthesis

(convolution)

NN

N/2N/2

N/2N/2

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What’s wrong with (real) wavelets?

-No spatial invariance

- No convolution capability

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2. Group-based wavelets

--group invariance--convolution

--complex wavelet coeffs. (phase)

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16Significance of phase

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3. Convolution and stochastic deconvolutionover groups

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F I L T E R

x(t) y(t)

k(t)

x(g)k(g)

y(g)

standard standard convolutionconvolution

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+ +x(g)

n(g)

y(g)

x(g)

ε(g)h(g)

¿

Stochastic deconvolution

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4. Application to segmentation and other things

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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

22Segmentation ( use of phase)

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Segmentation application

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Classification application(Brodatz texture data base

ORL faces data base)

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Last slideLast slide

(Research sponsored by DEPSCoR Grant)April 2000 - April 2003

1. Edge Detection - J. Ge

2. Group-based convolution - M. Elfatau

3. Spline-based edge detection - S. Ganapathi

4. Segmentation and Classification

www.uvm.edu/~mirchand