This course introduces the fundamentals of the field of sparse representations, starting with its theoretical concepts, and systematically presenting its key achievements. We will touch on theory and numerical algorithms.
Modeling data is the way we – scientists – believe that information should be explained and handled. Indeed, models play a central role in practically every task in signal and image processing. Sparse representation theory puts forward an emerging, highly effective, and universal such model. Its core idea is the description of the data as a linear combination of few building blocks – atoms – taken from a pre-defined dictionary of such fundamental elements.
A series of theoretical problems arise in deploying this seemingly simple model to data sources, leading to fascinating new results in linear algebra, approximation theory, optimization, and machine learning. In this course you will learn of these achievements, which serve as the foundations for a revolution that took place in signal and image processing in recent years.
This program is composed from two separate parts:
Part 1: Sparse Representations in Signal and Image Processing: Fundamentals.
Part 2: Sparse Representations in Image Processing: From Theory to Practice.
While we recommend taking both courses, each of them can be taken independently of the other. The duration of each course is five weeks, and each part includes: (i) knowledge-check questions and discussions, (ii) series of quizzes, and (iii) Matlab programming projects. Each course will be graded separately, using the average grades of the questions/discussions [K] quizzes [Q], and projects [P], by Final-Grade = 0.1K + 0.5Q + 0.4P.
The following table includes more details of the topics we will cover in the first course:
What This Field is All About? Take 1: A New Transform
What is this field all about? Take 2: Modeling Data
A Closer Look at the SparseLand Model
Who Works on This and Who Are We?
Several examples: Applications Leveraging this Model
This Course: Scope and Style
Underdetermined Linear Systems & Regularization
The Temptation of Convexity
A Closer Look at L1 Minimization
Conversion of (P1) to Linear Programming
Seeking Sparse Solutions
Promoting Sparse Solutions
The L0 Norm and the (P0) Problem
A Signal Processing Perspective
Theoretical Analysis of the Two-Ortho Case
The Two-Ortho Case
An Uncertainty Principle
From Uncertainty to Uniqueness
Theoretical Analysis of the General Case
Introducing the Spark
Uniqueness for the General Case via the Spark
Uniqueness via the Mutual-Coherence
Spark-Coherence Relation: A Proof
Uniqueness via the Babel-Function
Upper-Bounding the Spark
Demo - Upper Bounding the Spark
Constructing Grassmanian Matrices
Demo - Constructing Grassmanian Matrices
Greedy Pursuit Algorithms - The Practice
Defining Our Objective and Directions
Greedy Algorithms - The Orthogonal Matching Pursuit
Variations over the Orthonormal Matching Pursuit
The Thresholding Algorithm
A Test Case: Demonstrating and Testing Greedy Algorithms
Relaxation Pursuit Algorithms
Relaxation of the L0 Norm – The Core Idea
A Test Case: Demonstrating and Testing Relaxation Algorithms
Guarantees of Pursuit Algorithms
Our Goal: Theoretical Justification for the Proposed Algorithms
Equivalence: Analyzing the OMP Algorithm
Equivalence: Analyzing the THR Algorithm
Equivalence: Analyzing the Basis-Pursuit Algorithm – Part 1
Equivalence: Analyzing the Basis-Pursuit Algorithm – Part 2
From Exact to Approximate Sparse Solutions
General Motivation: Why Approximate?
Pursuit Algorithms: OMP and BP Extensions
IRLS Solution of the Basis Pursuit
IRLS Solution of the Basis Pursuit: A Demo
The Unitary Case – A source of Inspiration – Part 1
The Unitary Case – A source of Inspiration – Part 2
ADMM Solution of the Basis Pursuit
Analyzing the Approximate Pursuit Problem
Uniqueness vs. Stability – Gaining Intuition
The Restricted Isometry Property (RIP)
Key Properties of the Restricted Isometry Property (RIP)
MOOCs stand for Massive Open Online Courses. These arefree online courses from universities around the world (eg. StanfordHarvardMIT) offered to anyone with an internet connection.
How do I register?
To register for a course, click on "Go to Class" button on the course page. This will take you to the providers website where you can register for the course.
How do these MOOCs or free online courses work?
MOOCs are designed for an online audience, teaching primarily through short (5-20 min.) pre recorded video lectures, that you watch on weekly schedule when convenient for you. They also have student discussion forums, homework/assignments, and online quizzes or exams.