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
1 Student
review
Cost Free Online Course
Pace Self Paced
Provider edX
Language English
Hours 5-6 hours a week
Calendar 5 weeks long

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## Review for edX's Sparse Representations in Signal and Image Processing: Fundamentals 5.0 Based on 1 reviews

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5.0 2 months ago
by completed this course, spending 4 hours a week on it and found the course difficulty to be medium.
Interesting course which covers the concepts of Sparse modelling in image processing applications. Most of the course was theoretical but it did include two programming assignments based on MATLAB where we implement some of the algorithms. The requires some strong foundation in Linear algebra. Overall it is worth the time and a lot to learn.
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