ECE 710 Topic 14 - Analysis and Estimation of Signals and Images
Instructor
Professor Oleg Michailovich
Course Outline
Understanding the nature of signals of interest is an important prerequisite in applications where the signals need to be analyzed, enhanced, or recovered. In many situations, however, successfully implementing the above tasks requires finding a suitable way to model the signals. Such a modeling is nowadays regarded as a central problem in signal and image processing, whose mathematical challenge roots in the diversity and complexity of real-life data. Considering the signals of interest to be members of a properly predefined signal space is a standard way to perform the above modeling. Moreover, depending on the way the signal space is defined, one can cast a given estimation problem into either a variational (deterministic) or statistical framework. A systematic and consistent study of these estimation frameworks, understanding the relations and discrepancies between them, analyzing the criteria of optimality suggested by the frameworks, as well as learning the standard numerical methods for implementing the latter are in the focus of this course.
Syllabus
Topic 1: Sampling and Reconstruction (7 hours)
Riesz bases and frames in Hilbert spaces; Wittaker-Shannon-Nyquist sampling theorem; aliasing; general sampling theorem and approximation spaces; minimum error sampling; sampling in shift-invariant spaces; compressive sampling; non-uniform sampling.
Topic 2: Modeling and Representation of Signals and Images (6 hours)
Time-frequency representations and Heisenberg principle; global and local Fourier analysis; Gabor frames; wavelet transforms; Karhunen-Loeve basis of principal components; indepen- dent component analysis and sparse representations.
Topic 3: Basics of Numerical Optimization (5 hours)
Convex sets and functions; convex optimization problems; unconstrained minimization; gradient descent methods; conjugate gradient and Newton’s methods; constrained minimization; the Lagrangian functional; linear and quadratic optimization problems; duality.
Topic 4: Operator Equations and Regularization (7 hours)
Linear operators in Hilbert spaces and their spectral properties; invertibility and ill-posedness; Tikhonov regularization; Bayesian estimation framework; maximum-a-posteriori estimation; prior probabilities, their properties and construction.
Topic 5: Enhancement of Signals and Images (6 hours)
Linear filtering; locally adaptive filters; bi-literal filtering; total variation de-noising; wavelet de-noising; image de-blurring; super-resolution and information fusion.
Topic 6: Modern Methods of Image Processing (5 hours)
Image segmentation; active contours; image registration; image inpainting; “u + v” decomposition; pose and motion estimation; basics of visual-based control.
Course Textbooks
The students will be provided with copies of lecture notes. The recommended reading includes:
- I. Gohberg and S. Goldberg, Basic Operator Theory. Birkhauser, 1981.
- S. Mallat, A Wavelet Course in Signal Processing. Academic Press, 1998.
- B. P. Carlin and T. A. Louis, Bayes and Empirical Bayes Methods for Data Analysis. Chapman & Hall/CRC, 2000.
- S. Boyd and L. Vandenberghe, Convex Optimization. Cambridge University Press, 2004.
- Relevant journal publications.
Marking Scheme
- Home Assignments (5 assignments × 5 points): 25%
- Course project: 25%
- Final Exam: 50%