Wavelet Signal Processing

The main objective of this course is to conduct a study on wavelet theory from both theoretical and application perspective (such as signal compression, denoising, communication systems, and recognition) in a coherence manner. The course first builds up background on LTI operators and Fourier analysis (including Shannon’s sampling theory) and its short comings. Windowed Fourier Transform (WFT) and Continuous Wavelet Transform (CWT) are introduced along with analysis of admissibility condition for mother wavelet. To derive and understand discrete version of WT, i.e., DWT, frame theory is studied deeply along with discussion on translation-invariant dyadic wavelet transform. One of most important aspect in wavelet theory, i.e., regularity analysis and vanishing moments of wavelets along with detection of singularity and zeros-crossing using WT is discussed. DWT is introduced from Multiresolution Analysis (MRA) point of view. Mallat’s algorithm (which relates MRA and WT with QMF banks) is studied. Different factors affecting wavelet design are studied and spline wavelets (e.g., quadratic spline, cubicspline) are designed. Finally, other wavelet bases such as wavelet packets and local cosine bases are introduced for different applications. In addition to this, approximation theory is presented to study the error incurred while using the orthogonal basis. To improve the approximation of complex signals such as music recordings, we study general nonorthogonal signal decompositions such as basis pursuit and matching pursuit. Another topic of interest in signal processing is signal denoising which estimates signals buried in noise. This problem is studied with help Bayes and Minimax estimators and finally Donoho’s denoising algorithm is studied. At last different applications of WT, i.e., image/video compression, restoration, denoising, etc. are studied.