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!!!
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