CS515, Fall 03: on-line syllabus

CS515, Fall 03 - syllabus







Part 1. Introduction: Representation.

linear functionals. data representation: analysis. local averaging, local differencing, sampling. time-invariance, translations and convolutions. modulations. dilation.



Part 2. Introduction: Fourier series and orthonormal systems.

Inner product spaces. The space $L_2$. orthonormality. Bessel inequality. completeness and the Riesz theorem. Parseval's identity. synthesis (reconstruction). smoothness and decay.



Part 3. Introduction: Fourier transform (L_2 theory).

Parseval's identity and reconstruction. translation, modulation, dilation and differentiation. drawbacks of fourier analysis.



Part 4. Towards MultiResolution Analysis (MRA)

The Haar system. Time-frequency localization, and Weyl-Heisenberg systems. Refinable (scaling) functions: definition and examples. Splines, and Daubechies' first scaling function. Creating scaling functions from their mask. Connections between properties of the mask and properties of the wavelets. The cascade algorithm. Convergence of the cascade algorithm.



Part 5. Construction of wavelets via MRA

The unitary extension principle. Construction of tight frames. Spline tight frames. Construction of orthonormal systems. Daubechies' orthonormal scaling functions, and wavelets.



Part 6. Introduction: Good systems

orthonormal systems. tight frames. Stable (Riesz) systems. frames. dual systems. Biorthogonal systems, and bi-frames.



Part 7. Signal Analysis

The fast wavelet/frame transform. Preprocessing. Filter banks.



Part 8. Introduction to the theory of wavelets

Approximation orders of scaling function; vanishing moments of wavelets. The transfer operator; stability; regularity of scaling functions.



Part 9. Applications

Denoising. Feature detection. Signal compression. Wavelet in 2D. Image compression.



Part `also'.

Also Midterm Exam (late in October).