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Intro

edX: Sparse Representations in Signal and Image Processing: Fundamentals

 with  Michael Elad and Yaniv Romano

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.

Syllabus

This program is composed from two separate parts:

  1. Part 1: Sparse Representations in Signal and Image Processing: Fundamentals.
  2. 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:

Overview

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

Mathematical Warm-Up

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)

Theoretical Study of P0 in the Noisy Case

Performance of Pursuit Algorithms – General

Basis-Pursuit Stability Guarantee

Thresholding Stability Guarantee: Worst-Case

OMP Stability Guarantee

Rate of Decay of the Residual in Greedy Methods

Course Summary and a Glimpse to the Future

Course Summary &
A Glimpse to the Future

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Cost Free Online Course
Provider edX
Language English
Certificates $99 Certificate Available
Hours 5-6 hours a week
Calendar 5 weeks long

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