This course covers the mathematical framework for wavelets, with particular emphasis on algorithms and implementation of the algorithms. Concepts of frames, orthogonal bases, and reproducing kernel Hilbert spaces are introduced first, followed by an introduction to linear systems for continuous time and discrete time. Next, time, frequency, and scale localizing transforms are introduced, including the windowed Fourier transform and the continuous wavelet transform (CWT). Discretized CWT is studied next in the forms of the Haar and the Shannon orthogonal wavelet systems. General multi-resolution analysis is introduced, and the time domain and frequency domain properties of orthogonal wavelet systems are studied with examples of compact support wavelets. The discrete wavelet transform (DWT) is introduced and implemented. Biorthogonal wavelet systems are also described. Orthogonal wavelet packets are discussed and implemented. Wavelet regularity and the Daubechies construction is presented next. Finally the 2D DWT is discussed and implemented. Applications of wavelet analysis to de-noising and image compression are discussed together with an introduction to image coding.

Course prerequisites: 

525.627 Digital Signal Processing and the basics of linear systems.

Course instructor: 

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