A guide to the intrepid adventurer.

Compressive Sensing are methods that were developed in the last 15 years (post. 2000). They are really interesting methods and a subject that is not only pure mathematics because it has many applications in many fields of engineering. In this small guide I will list good resources to learn Compressive Sensing in a fun and enjoyable way.

How to use only a small part of the data and still be able to reconstruct an image. For example in a image, how to use only a small group of pixels and reconstruct the all image.

1. Video - Compressed Sensing: Overview
   https://www.youtube.com/watch?v=SbU1pahbbkc

2. Video - Compressed Sensing: Mathematical Formulation
   https://www.youtube.com/watch?v=inr-nGnVc0k

3. Video - Compressed Sensing: When It Works
   https://www.youtube.com/watch?v=hmBTACBGWJs

4. Video - Sparsity and the L1 Norm
   https://www.youtube.com/watch?v=76B5cMEZA4Y

5. Video - Underdetermined systems and compressed sensing - Matlab
   https://www.youtube.com/watch?v=otr1YwNBWfc

6. Video - Underdetermined systems and compressed sensing - Python
   https://www.youtube.com/watch?v=_-Jkq-Faa2Y

The Nyquist sampling theorem, says that we need to sample at two times the maximum frequency of the signal to have a complete reconstruction of the signal. But with Compressive Sensing we could have a much lower sample rate (number of samples by unit of time) condition to the fact that the sampling is random in time or space and that we know the sampled clock with some precision. For example, in a audio signal of two tones we can reconstruct with a sample rate that is 32x below the Nyquist sample rate and still reconstruct the signal.

1. Video - Shannon Nyquist Sampling Theorem
   https://www.youtube.com/watch?v=FcXZ28BX-xE

2. Video - Beating Nyquist with Compressed Sensing, in Python
   https://www.youtube.com/watch?v=5-LY6wBIKx8

3. Video - Beating Nyquist with Compressed Sensing in Matlab
   https://www.youtube.com/watch?v=A8W1I3mtjp8

4. Video - Beating Nyquist with Compressed Sensing in Matlab, part 2
   https://www.youtube.com/watch?v=HVR2DaaD0Xo

Imagine that you will have to place sensors in some physical way along an area (ex: the ocean or in a airfoil) to obtain data from a physical phenomenon. You couldn’t spread sensors in all the area, so you had to choose carefully a way of placing the least possible sensors in a way that maximized the reconstruction fidelity of the underling phenomenal if you had some examples of samples of a model of that phenomenon. The same can be applied to many other fields, for example to images in that you could reconstruct images, or detect images from only a small set of pixels.

1. Video - Sparse Sensor Placement Optimization for Reconstruction
   https://www.youtube.com/watch?v=BCygBhyrl8k

2. Video - Sparse Sensor Placement Optimization for Classification
   https://www.youtube.com/watch?v=zJ2z__5mepA

1. Book site
   http://databookuw.com/

2. Free book from the author of the above youtube channel
   http://databookuw.com/databook.pdf

1. Compressive Sensing – A 25 Minute Tour
   Emmanuel Candès
   https://www.raeng.org.uk/publications/other/candes-presentation-frontiers-of-engineering

2. A Systematic Review of Compressive Sensing: Concepts, Implementations and Applications
   Meenu Rani,S. B. Dhok, and R. B. Deshmukh
   https://ieeexplore.ieee.org/document/8260873

Article that has a historical vision of Compressive Sensing and info about the performance and complexity of the different algorithms used in the field.

3. Reconstruction Algorithms in Compressive Sensing: An Overview
   by André Luiz Pilastri, João Manuel R. S. Tavares
   https://sigarra.up.pt/faup/pt/pub_geral.pub_view?pi_pub_base_id=139664

4. Compressive Sensing Resources - Rice University DSP
   https://dsp.rice.edu/cs/

Simple and fast algorithm used in the optimization phase of compressed sensing (minimization with constrains).

1. Paper
   CoSaMP: Iterative signal recovery from incomplete and inaccurate samples
   https://www.sciencedirect.com/science/article/pii/S1063520308000638

2. Cosamp Github implementation
   https://github.com/avirmaux/CoSaMP

The following are some really good books in each subject, the majority which is free and the one that aren’t are low cost. This books will help you understand the underling mathematics.

1. Linear Algebra: Theory, Intuition, Code
   by Mike X Cohen

1. Algorithms for Optimization - The MIT Press
   by Mykel J. Kochenderfer

• Free book on the tab download
  https://algorithmsbook.com/optimization/

2. Convex Optimization
   by Stephen Boyd, Lieven Vandenberghe

• Free book online
  https://web.stanford.edu/~boyd/cvxbook/

1. Introduction to Probability, Statistics, and Random Processes
   by Hossein Pishro-Nik

• Free book online
  https://www.probabilitycourse.com/

1. Information Theory, Inference and Learning Algorithms
   by David J. C. MacKay

• Free book online
  https://www.inference.org.uk/itprnn/book.pdf

• Site
  https://www.inference.org.uk/itprnn/

1. Engineering Mathematics
   by Prof Anthony Croft, Dr Robert Davison, James Flint, Martin Hargreaves

• The links to all my guides are in:
  Guides on Linux - Programming - Embedded - Electronics - Aeronautics
  https://github.com/joaocarvalhoopen/Guides_Linux-Programming-Electronics-Aeronautics

Best regards,
Joao Nuno Carvalho