Skip to content Skip to navigation Skip to collection information

OpenStax_CNX

You are here: Home » Content » An Introduction to Compressive Sensing » Sensing matrix design

Navigation

Recently Viewed

This feature requires Javascript to be enabled.
 

Sensing matrix design

Module by: Mark A. Davenport. E-mail the author

Summary: This module provides an overview of the sensing matrix design problem in compressive sensing.

In order to make the discussion more concrete, we will restrict our attention to the standard finite-dimensional compressive sensing (CS) model. Specifically, given a signal xRNxRN, we consider measurement systems that acquire MM linear measurements. We can represent this process mathematically as

y = Φ x , y = Φ x ,
(1)

where ΦΦ is an M×NM×N matrix and yRMyRM. The matrix ΦΦ represents a dimensionality reduction, i.e., it maps RNRN, where NN is generally large, into RMRM, where MM is typically much smaller than NN. Note that in the standard CS framework we assume that the measurements are non-adaptive, meaning that the rows of ΦΦ are fixed in advance and do not depend on the previously acquired measurements. In certain settings adaptive measurement schemes can lead to significant performance gains.

Note that although the standard CS framework assumes that xx is a finite-length vector with a discrete-valued index (such as time or space), in practice we will often be interested in designing measurement systems for acquiring continuously-indexed signals such as continuous-time signals or images. For now we will simply think of xx as a finite-length window of Nyquist-rate samples, and we temporarily ignore the issue of how to directly acquire compressive measurements without first sampling at the Nyquist rate.

There are two main theoretical questions in CS. First, how should we design the sensing matrix ΦΦ to ensure that it preserves the information in the signal xx? Second, how can we recover the original signal xx from measurements yy? In the case where our data is sparse or compressible, we will see that we can design matrices ΦΦ with MNMN that ensure that we will be able to recover the original signal accurately and efficiently using a variety of practical algorithms.

We begin in this part of the course by first addressing the question of how to design the sensing matrix ΦΦ. Rather than directly proposing a design procedure, we instead consider a number of desirable properties that we might wish ΦΦ to have (including the null space property, the restricted isometry property, and bounded coherence). We then provide some important examples of matrix constructions that satisfy these properties.

Collection Navigation

Content actions

Download:

Collection as:

PDF | EPUB (?)

What is an EPUB file?

EPUB is an electronic book format that can be read on a variety of mobile devices.

Downloading to a reading device

For detailed instructions on how to download this content's EPUB to your specific device, click the "(?)" link.

| More downloads ...

Module as:

PDF | EPUB (?)

What is an EPUB file?

EPUB is an electronic book format that can be read on a variety of mobile devices.

Downloading to a reading device

For detailed instructions on how to download this content's EPUB to your specific device, click the "(?)" link.

| More downloads ...

Add:

Collection to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks

Module to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks