+

Compressed Sensing

+

Compressed Sensing

Mobashir Mohammad

Aditya Kulkarni

Tobias Bertelsen

Malay Singh

Hirak Sarkar

Nirandika Wanigasekara

Yamilet Serrano Llerena

Parvathy Sudhir

3

+ Introduction

Mobashir Mohammad

4+The Data Deluge

Sensors: Better… Stronger… Faster…

Challenge: Exponentially increasing amounts of data

Audio, Image, Video, Weather, … Global scale acquisition

5+

6+Sensing by Sampling

Sample

N

7+Sensing by Sampling (2)

Sample

N CompressN >> L

JPEG…

L

L DecompressN >> L

N

8+Compression: Toy Example

9+Discrete Cosine Transformation

Transformation

10+Motivation

Why go to so much effort to acquire all the data when most of the what we get will be thrown away?

Cant we just directly measure the part that wont end up being thrown away?

Donoho2004

11

+

Outline

• Compressed Sensing• Constructing Φ• Sparse Signal Recovery• Convex Optimization

Algorithm• Applications• Summary • Future Work

12

+ Compressed Sensing

Aditya Kulkarni

13+What is compressed sensing?

A paradigm shift that allows for the saving of time and space during the process of signal acquisition, while still allowing near perfect signal recovery when the signal is needed

Nyquist rateSampling

AnalogAudioSignal

Compression(e.g. MP3)

High-rate Low-rate

CompressedSensing

14+Sparsity

The concept that most signals in our natural world are sparse

a. Original imagec. Image reconstructed by discarding the zero coefficients

15+How It Works

16+

𝒚=𝚽 𝒙

Dimensionality Reduction Problem

I. Measure

II. Construct sensing

matrix

III. Reconstruct

17+Sampling

¿

𝑁×𝑁

measurements

sparse signal

nonzeroentries

𝑦 𝑥Φ=I

18+

¿

𝑀×𝑁

measurements

sparse signal

nonzeroentries

𝑦 𝑥Φ

𝐾 <𝑀≪𝑁

19+

¿

𝑁×𝑁

𝑁×1

nonzeroentries

𝑥 𝛼Ψ

nonzeroentries

𝑁×1

20+Sparsity

The concept that most signals in our natural world are sparse

a. Original imagec. Image reconstructed by discarding the zero coefficients

21+

¿

𝑀×𝑁

𝑦 𝛼Φ Ψ

𝑁×𝑁 𝑁×1𝑀×1

22

+ Constructing Φ

Tobias Bertelsen

23+RIP - Restricted Isometry Property

The distance between two points are approximately the same in the signal-space and measure-space

A matrix satisfies the RIP of order K if there exists a such that:

holds for all -sparse vectors and

Or equally

holds for all 2K-sparse vectors

24+RIP - Restricted Isometry Property RIP ensures that measurement error does not blow up

Image: http://www.brainshark.com/brainshark/brainshark.net/portal/title.aspx?pid=zCgzXgcEKz0z0

25+Randomized algorithm

1. Pick a sufficiently high

2. Fill randomly according to some random distribution

Which distribution?

How to pick ?

What is the probability of satisfying RIP?

26+Sub-Gaussian distribution

Defined by Tails decay at least as fast as the Gaussian E.g.: The Gaussian distribution, any bounded distribution

Satisfies the concentration of measure property (not RIP):

For any vector and a matrix with sub-Gaussian entries, there exists a such that

holds with exponentially high probability where is a constant only dependent on

27+Johnson-Lidenstrauss Lemma

Generalization to a discrete set of vectors

For any vector the magnitude are preserved with:

For all P vectors the magnitudes are preserved with:

To account for this must grow with

28+Generalizing to RIP

RIP:

We want to approximate all -sparse vectors with unit vectors

The space of all -sparse vectors is made up of -dimensional subspaces – one for each position of non-zero entries in

We sample points on the unit-sphere of each subspace

29+Randomized algorithm

Use sub-Gaussian distribution

Pick

Exponentially high probability of RIP

Formal proofs and specific formulas for constants exists

30+Sparse in another base

We assumed the signal itself was sparse

What if the signal is sparse in another base, i.e. is sparse.

must have the RIP

As long as is an orthogonal basis, the random construction works.

31+Characteristics of Random

Stable Robust to noise, since it satisfies RIP

Universal Works with any orthogonal basis

Democratic Any element in has equal importance Robust to data loss

Other Methods Random Fourier submatrix Fast JL transform

32

+ Sparse Signal Recovery

Malay Singh

33+The Hyperplane of

34+ Norms for N dimensional vector x

‖𝑥‖𝑝={ (∑𝑗=1𝑁

|𝑥 𝑗|𝑝)1𝑝 if𝑝>0

|𝑠𝑢𝑝𝑝 (𝑥)| if𝑝=0max

𝑗=1,2 ,… ,𝑁|𝑥 𝑗| if𝑝=∞

Unit Sphere of quasinorm

Unit Sphere of norm

Unit Sphere of norm

Unit Sphere of norm

35+ Balls in higher dimensions

36+How about minimization

But the problem is non-convex and very hard to solve

37+We do the minimization

We are minimizing the Euclidean distance. But the arbitrary angle of hyperplane matters

38+What if we convexify the to

39+Issues with minimization

is non-convex and minimization is potentially very difficult to solve.

We convexify the problem by replacing by . This leads us to Minimization.

Minimizing results in small values in some dimensions but not necessarily zero. provides a better result because in its

solution most of the dimensions are zero.

40

+ Convex Optimization

Hirak Sarkar

41+What it is all about …

Find a sparse representation

Here and Moreover

Two ways to solve

(P1) where is a measure of sparseness

(P2)

42+How to chose and

Take the simplest convex function

A simple

Final unconstrained version

43+Versions of the same problem

44+Formalize

Nature of Convex Differentiable

Basic Intuition Take an arbitrary Calculate Use the shrinkage operator Make corrections and iterate

45+

Shrinkage operator We define the shrinkage operator as follows

46+ Algorithm

Input: Matrix ignal measurement parameter sequence

Output: Signal estimate

Initialization:

47+Performance

For closed and convex function any the algorithm converges within finite steps

For and a moderate number of iterations needed is less than 5

48

+ Single Pixel Camera

Nirandika Wanigasekara

49+Single Pixel Camera

What is a single pixel camera An optical computer sequentially measures the Directly acquires random linear measurements without first

collecting the pixel values

50+Single Pixel Camera- Architecture

51+Single Pixel Camera- DMD Array

Digital Micro mirror Device

A type of a reflective spatial light modulator

Selectively redirects parts of the light beam

Consisting of an array of N tiny mirrors

Each mirror can be positioned in one of two states(+/-10 degrees)

Orients the light towards or away from the second lens

52+Single Pixel Camera- Architecture

53+Single Pixel Camera- Photodiode

Find the focal point of the second lens

Place a photodiode at this point

Measure the output voltage of the photodiode

The voltage equals , which is the inner product between and the desired image .

54+Single Pixel Camera- Architecture

55+Single Pixel Camera- measurements

A random number generator (RNG) sets the mirror orientations in a pseudorandom 1/0 pattern

Repeats the above process for times

Obtains the measurement vector and

Now we can construct the system in the

𝑦 𝑗 𝑥Φ j

¿

56+Single Pixel Camera- Architecture

+ 57

Sample image reconstructions

256*256 conventional image of black and white ‘R’

Image reconstructed from

How can we improve the reconstruction further?

58+Utility This device is useful when measurements are

expensive

Low Light Imager Conventional Photomultiplier tube/ avalanche photodiode Single Pixel Camera Single photomultiplier

Original 800 1600

65536 pixels from 6600

59+Utility CS Infrared Imager

IR photodiode

CS Hyperspectral Imager

60

+ Compressed Sensing MRI

Yamilet Serrano Llerena

61+Compressed Sensing MRI

Magnetic Resonance Imaging (MRI)Essential medical imaging tool with slow data acquisition process.

Applying Compressed Sensing (CS) to MRI offers that:• We can send much less

information reducing the scanned time

• We are still able to reconstruct the image in based on they are compressible

62+Compressed Sensing MRI

Scan Process

63+Scan Process

Signal Received K-space

Space where MRI data is stored

K-space trajectories:

K-space is 2D Fourier transform of the MR image

64+In the context of CS

Φ :• Is depends on the acquisition device• Is the Fourier Basis• Is an M x N matrix

• Is the measured k-space data from the scanner

y :

y = Φ x

x :

65+Recall ...

The heart of CS is the assumption that x has a sparse representation.

Medical Images are naturally compressible by sparse coding in an appropriate transform domain (e.g. Wavelet Transform)

Not significant

66+Compressed Sensing MRI

Scan Process

67+Image Reconstruction

CS uses only a fraction of the MRI data to reconstruct image.

68+Image Reconstruction

69+Benefits of CS w.r.t Resonance

Allow for faster image acquisition (essential for cardiac/pediatric imaging)

Using same amount of k-space data, CS can obtain higher Resolution Images.

70

+ Summary

Parvathy Sudhir Pillai

71+Summary

Motivation Data deluge Directly acquiring useful part of the signal

Key idea: Reduce the number of samples

Implications

Dimensionality reduction

Low redundancy and wastage

72+Open Problems

‘Good’ sensing matrices Adaptive? Deterministic?

Nonlinear compressed sensing

Numerical algorithms

Hardware design

Intensity (x)

Phase ()

Coefficients ()

73+Impact

Data generation and storage

Conceptual achievements Exploit minimal complexity efficiently Information theory framework

Numerous application areas

Legacy - Trans disciplinary research Information

SoftwareHardware

Complexity

CS

74+Ongoing Research

New mathematical framework for evaluating CS schemes Spectrum sensing

Not so optimal

Data transmission - wireless sensors (EKG) to wired base stations. 90% power savings

75+In the news

76+References

Emmanuel Candes, Compressive Sensing - A 25 Minute Tour, First EU-US Frontiers of Engineering Symposium, Cambridge, September 2010

David Schneider, Camera Chip Makes Already-Compressed Images, IEEE Spectrum, Feb 2013

T.Strohmer. Measure what should be measured: Progress and Challenges in Compressive Sensing. IEEE Signal Processing Letters, vol.19(12): pp.887-893, 2012.

Larry Hardesty, Toward practical compressed sensing, MIT news, Feb 2013

Tao Hu and Mitya Chklovvskii, Reconstruction of Sparse Circuits Using Multi-neuronal Excitation (RESCUME), Advances in Neural Information Processing Systems, 2009

http://inviewcorp.com/technology/compressive-sensing/

http://ge.geglobalresearch.com/blog/the-beauty-of-compressive-sensing/

http://www.worldindustrialreporter.com/bell-labs-create-lensless-camera-through-compressive-sensing-technique/

http://www.lablanche-and-co.com/

77+

THANK YOU