Template for Courses s1

1. Title: Vector Space Projections

2. Credit Structure (L-T-P-Cr): 3 0 0 3

3. Course Code:

4. Program/Semester: PhD/MTech

5. Category: Science Elective

6. Prerequisites: Linear Algebra/Digital Signal Processing

7. Foundation for:

8. Abstract Content:

Problems of digital filter design to Sudoku solver can be formulated in terms of projections onto convex/non-convex sets. This course aims at introducing the concepts and method of projections onto convex sets, a Hilbert space approach to solving problems of constraint feasibility. Many convergent iterative algorithms already in existence are vector-space projection algorithms. Objective of this course is to develop formalism for a large number of similar problems. This course will help students in developing a unified perspective towards solving constraint feasibility problems.

Optional:

9. Suggested Text/s:

[1] L. Debnath and P. Mikusinski. Hilbert Spaces with Applications. Elsevier Academic Press, 2005.

[2] H. Stark and Y. Yang. Vector space projections: a numerical approach to signal and image processing, neural nets, and optics. Wiley series in telecommunications and signal processing. Wiley, 1998.

[3] D. G. Luenberger. Optimization by Vector Space Methods, John Wiley & Sons, 1968

[4] S. Boyd and L. Vandenberghe. Convex Optimization, Cambridge University Press, 2004.

10. Detailed Contents:

1. Vector Space Concepts (~ 8 lectures) a. Vector Spaces b. Normed Vector Spaces c. Inner Product Spaces d. Hilbert Spaces 2. Projections onto Convex Sets (~ 10 lectures) a. Convex Set b. Projection c. Properties of Projections d. Fundamental Theory of POCS e. POCS Algorithm and Its Numerical Aspects f. Elementary Projectors i. Linear Type Constraints ii. Error Type Constraints iii. Similarity Constraints iv. Time/Frequency Domain Constraints 3. Solutions of Linear Equations (~3 lectures) a. Convex-Set Formulation b. Convergence c. Minimum Distance Property d. Hybrid Algorithm e. Computed Tomography and Resolution Enhancement

4. Generalized Projections (~7 lectures) a. Convex versus Non-Convex Sets b. Theory of Generalized Projection c. Generalized Projection Algorithm in a Product Vector Space

5. Applications (~ 12 lectures) (a) Communications Non-uniform Sampling, Digital Filters, Estimation of Probability Density Functions, Beam Forming, Color Matching, Blind De-convolution (b) Neural Nets, Image Processing Learning in Multi-Layer Nets, Projection Method Learning Rule, Image Compression, Image Sharpening, Noise Smoothing, Image Synthesis, Restoration of Blurred and Noisy Images, High-Resolution Image Restoration (c) Introduction to Convex Optimization and applications to Signal Processing

11. Outcomes and Objectives:

Outcomes: (a) students will get introduced to Hilbert space and concepts of functional analysis (b) students will learn to include constraints in the engineering design problems (c) students will write a term paper on POCS methods for one of the application area.

12. Comments:

The course is divided into five basic modules as prescribed above. The fifth module is a project based learning module. It comprises of applications from various domains, and depending upon the student's interest a subset of applications will be selected for discussion. Students are also expected to take up a project during the course which will be based on the methods developed/learnt in the class. Beginning part of the course deals with the theory of Hilbert spaces and convex sets. Book by L. Debnath on Hilbert spaces and its application, [1], will be referred for developing concepts of Hilbert space. The rest of the course is mostly exploratory and will be taught from the book titled Vector Space Projections by Henry Stark and Youngyi Yang [2]. The book by Boyd [4] will be referred to for the part on introduction to convex optimization along with the book by Luenberger [3].