Special Topics in Computer Science:

Wavelets and Filter Banks in Image Processing

Signal processing has become an essential part of contemporary scientific and technological activity. Dramatic changes have already been made in a broad range of fields such as communications, imaging, radar & sonar, and music reproduction, to name just a few. Each of these areas has developed its own algorithms, mathematics, and specialized techniques. Describing signals in terms of frequency and using the Fourier Transform for analysis has been essential in all of these fields.

Like the Fourier transform (FT), a wavelet transform (WT) represents a signal in another domain, a time-frequency domain. However, wavelet transforms are more general than the Fourier transform. Unlike the Fourier transform, wavelet transforms may describe localized signals more efficiently. For example, a wavelet transform may describe a function into different frequency components, and then study each component with a resolution matched to its scale.

Although the first wavelet transform was discovered in 1910 by Alfred Haar,  wavelet transforms have only recently been used. They are gaining widespread acceptance in many fields and already dominates some technologies such as signal compression. Knowledge of wavelet transforms is essential for any current work in processing signals.

The basis functions of the WT, scaling functions and wavelets, are often more complicated than the basis functions of the FT, sines and cosines. Unlike the FT the basis functions in the WT are localized in both the input and wavelet domain. Furthermore, like the FT, the WT is a linear operation that is invertible and can be made orthogonal. The general idea behind the wavelet transform is to represent any arbitrary function as a superposition of wavelets. Based on a mother wavelet, scaled and shifted versions of the mother wavelet can be summed to represent an arbitrary function. Like the FT, scaled and shifted versions of sine and cosine functions can be summed to represent an arbitrary function.

Lecturer

Dates

Contents

1. Introduction; Wavelet Transform (WT) and the Fourier Transform; comparison; The Fourier Transform as a decomposition;
2. Fourier Transform Theorems; The Two-Dimensional Sampling Theory;
3. Linear Systems; Low-Pass Filters (L_P) and High-Pass (H_P)Filters: Time-domain and frequency-domain.
4. The effect of the Downsampling and Upsampling in the frequency domain: Aliasing and Imaging.
5. Scaling Functions and Wavelets; The Wavelet equation; Wavelet transforms by multiresolution.
6. The Tree-Structured Filter Bank; The Pyramid algorithm; Fast Wavelet Transform (FWT); Haar Waveletes and Recursion.
7. Filter banks; Perfect reconstruction Orthogonal filter banks;
8. Halfband filters; Spectral factorization; Maxflat (Daubechies) filters.
9. Biorthoganal Wavevelets and Filter Banks; Perfect Reconstruction.
10. Image Compression; Distortion in Image Compression.

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Exam

Students will do a project that will be marked.

Literature

• A. Haar: Zur Theorie der orthogonalen Funktionsysteme. Math. Annal. 69:331-371, 1910.
• G. Strang, T. Nguyen: Wavelets and Filter Banks, Wellesley-Cambridge Press: Wellesley, MA (1996)
• I. Daubechies, Orthonormal bases of compactly supported wavelets. Commun. On Pure and Appl. Math. 41:909-996, November 1988.
• Ingrid Daubechies: Ten Lectures on Wavelets (1992)
• Stéphane Mallat: A Wavelet Tour of Signal Processing. Academic Press, 2001.
• Y. Meyer, Wavelets and operators, Cambridge Studies in Advanced Math., vol.37., Cambridge University Press, Cambridge, 1992.
• Charles K. Chui: An Introduction to Wavelets
• D. Gabor: Theory of communication. J. IEE, 93:429-457, 1946
• F. Schipp, W.R. Wade Transforms on Normed Fields. Leaflets in Mathematics, Janus Pannonius University, Pecs, 1995.
• Y. Lee, S.P. Kozaitis: Multiresolution gradient-based edge detection in noisy images using wavelet domain filters, Optical Engineering, 39(9):2405-2412, 2000.
• S.P. Kozaitis, and R. H. Cofer, Lineal feature detection using multiresolution wavelet filters, Photogrammetric Engineering &Remote Sensing, vol. 71,No. 6, 689-697, 2005.
• T. Ostrem, S. P. Kozaitis, and I. Laszlo, Multilevel fusion for enhanced feature detection, in Visual Information Processing XIV, Z. Rahman, R. A. Schowengerdt, and S. E. Reichenbach, Eds., Proc. SPIE 5817, paper 35., 2005.

Software: Matlab, only the signal processing toolbox is needed.
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