Download Experimenting with Multi-dimensional Wavelet Transformations

Experimenting with Multidimensional Wavelet
Transformations
Tarık Arıcı and Buğra Gedik
Outline of Project Goals
 Writing discrete wavelet transformation and inverse
transformation wrappers (in Matlab) to handle multidimensional 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, Selfsimilarity 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 smoothes
calls for each D-1 dimensional slice
 Top level recursive calls go D-1
levels deep before calling the 1
dimensional wavelet transformation
functions
7 detail groups
 As a result 2^D-1 detail groups and
a single smooth group is
constructed for each level of
transformation
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
Original Image
after wavelet
interpolation
after fixing
delay problem
 Assume all details are zero
 Perform inverse wavelet transformation
 Possible use of Bayesian Methods:
 Model missing details using a Bayesian approach
DSP Perspective: Problems Related
with Non-zero Phase Filtering
 Filtering in time domain is multiplication in frequency
domain
X[n]
h[n]
y[n]
 Phase(Y(f)) = Phase(H(f))+Phase(X(f))
Non-zero Phase Filtering
 cos(2pf0t+f) = cos(2pf0(t+f/(2pf0))
= f/(2pf0)
td is constant if f is a linear function of
frequency
 td

 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(f) – 2pfv|
v=
-L/2+1, if L =8,12,16,20
-L/2, if L =10, 18
-L/2+2, if L =14
 LA(8) and LA(12) works best.
The End!
 Thanks!!!