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 smoothes 7 detail groups 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(2  f 0 t+  ) = cos(2  f 0 (t  f 0 )  t d =  (  f 0 ) t d 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}

10. Least Asymmetric (LA) Wavelet Filters Choose filter coefficients: s.t. min |  (f) – 2  fv| -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!!!