Experimenting with Multi-dimensional Wavelet Transformations

Tarık Arıcı and Buğra Gedik

Outline of Project Goals

Writing discrete wavelet transformation and inverse transformation wrappers (in Matlab) to handle multi-dimensional data; possible uses include: 2D Images, 3D turbulence data or multi-attribute

sensor readings Using wavelets in some example applications

Lossy compression, De-noising for images, Self-similarity analysis

Studying the phases of the wavelet filters (that delays the wavelet smoothes) and approximately computing the delay amount using DSP methods

Using this on Mammogram reconstruction Possible uses of Bayesian? (not done)

DWTR / IDWTR wrappers

Assume D dimensions Perform D sweeps, one across

each dimension, making recursive calls for each D-1 dimensional slice

Top level recursive calls go D-1 levels deep before calling the 1 dimensional wavelet transformation functions

As a result 2^D-1 detail groups and a single smooth group is constructed for each level of transformation

smoothes 7 detail groups

smoothes

3 detail groups

Example Applications: Lossy Compression

Example Applications: De-noising

Example Applications: Self-similarity Analysis

Calculate the means of the detail squares for each level and plot their log as a function of level

If the line is linear, then there is self-similarity Brownian motion is self-similar, Random data (of

course) is not

Mammogram Reconstruction

Assume all details are zero Perform inverse wavelet transformation Possible use of Bayesian Methods:

Model missing details using a Bayesian approach

Original Image

after waveletinterpolation

after fixing delay problem

DSP Perspective: Problems Related with Non-zero Phase Filtering Filtering in time domain is multiplication in frequency

domain

Phase(Y(f)) = Phase(H(f))+Phase(X(f))

h[n]X[n] y[n]

Non-zero Phase Filtering

cos(2f0t+) = cos(2f0(tf0) td = (f0)

td is constant if is a linear function of frequency

Therefore, wavelet filters should be (approximately) linear phase filters Symmetric filters have linear phase

Ex: {1, 1} (Haar), {1, 2, 1}

Least Asymmetric (LA) Wavelet Filters

Choose filter coefficients:

s.t. min |(f) – 2fv|

-L/2+1, if L =8,12,16,20

v = -L/2, if L =10, 18-L/2+2, if L =14

LA(8) and LA(12) works best.

The End!

Thanks!!!