1. Experimenting with Multi-dimensional Wavelet Transformations Tarık Arıcı and Buğra Gedik

2. 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)

3. 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 7 detail groups smoothes smoothes 3 detail groups

4. Example Applications: Lossy Compression

5. Example Applications: De-noising

6. 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

7. 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 wavelet interpolation after fixing delay problem

8. 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]

9. Non-zero Phase Filtering • cos(2pf0t+f) = cos(2pf0(t+f/(2pf0)) • td = f/(2pf0) • td is constant if f 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}

10. Least Asymmetric (LA) Wavelet Filters • Choose filter coefficients: s.t. min |f(f) – 2pfv| -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.

11. The End! • Thanks!!!