Compressed Sensing

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Compressed Sensing. Mobashir Mohammad Aditya Kulkarni Tobias Bertelsen Malay Singh Hirak Sarkar Nirandika Wanigasekara Yamilet Serrano Llerena Parvathy Sudhir. Compressed Sensing. Introduction. Mobashir Mohammad. The Data Deluge. Sensors: Better… Stronger… Faster… Challenge: - PowerPoint PPT Presentation

Text of Compressed Sensing

Compressed Sensing

Compressed Sensing+1Compressed SensingMobashir MohammadAditya KulkarniTobias BertelsenMalay SinghHirak SarkarNirandika WanigasekaraYamilet Serrano LlerenaParvathy Sudhir

+IntroductionMobashir Mohammad3+ The Data DelugeSensors: Better Stronger FasterChallenge:Exponentially increasing amounts of dataAudio, Image, Video, Weather, Global scale acquisition

4+4

5+The amount of data generate >> Amount that can be stored >> The tranmission rateThe idea of Use it now or loose it since 2007

5Sensing by Sampling

SampleN

6+Old fashion way of how signal processing works

6Sensing by Sampling (2)

SampleN

CompressN >> L

JPEGL

LDecompressN >> LN

7+Old fashion way of how signal processing worksAt encoder side, its very expensiveEarlier according to Moores law the clock speed doubles almost every 2 yearsHowever for the ADC the clock speed doubles almost every 6 years.Even more, the Compression is computationally very expensive

7Compression: Toy Example

8+8Discrete Cosine Transformation

Transformation9+MotivationWhy 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?

Donoho200410+OutlineCompressed SensingConstructing Sparse Signal RecoveryConvex Optimization AlgorithmApplicationsSummary Future Work11+ Compressed SensingAditya Kulkarni12+ 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 neededNyquist rateSamplingAnalogAudioSignalCompression(e.g. MP3)High-rateLow-rate

CompressedSensing13+SparsityThe concept that most signals in our natural world are sparse

Original imagec. Image reconstructed by discarding the zero coefficients14+14That is, they have concise representations when expressed properly.25,000 largest coefficientsHow It Works15

+Dimensionality Reduction Problem16+Sampling17+18+19+SparsityThe concept that most signals in our natural world are sparse

Original imagec. Image reconstructed by discarding the zero coefficients20+20That is, they have concise representations when expressed properly.25,000 largest coefficients21+Constructing Tobias Bertelsen22+ RIP - Restricted Isometry PropertyThe distance between two points are approximately the same in the signal-space and measure-space

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RIP - Restricted Isometry PropertyImage: http://www.brainshark.com/brainshark/brainshark.net/portal/title.aspx?pid=zCgzXgcEKz0z024+Randomized algorithm25+Sub-Gaussian distribution26+Johnson-Lidenstrauss Lemma27+Generalizing to RIP28+Randomized algorithm

29+Sparse in another base30+StableRobust to noise, since it satisfies RIPUniversalWorks with any orthogonal basis DemocraticAny element in has equal importanceRobust to data loss

Other MethodsRandom Fourier submatrixFast JL transform31+Sparse Signal RecoveryMalay Singh

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33+

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

36+36

We are minimizing the Euclidean distance. But the arbitrary angle of hyperplane matters37+

38+39+Convex OptimizationHirak Sarkar40+ What it is all about 41+42+Versions of the same problem43+Formalize44+Shrinkage operator

45+ Algorithm

46+Performance 47+Single Pixel CameraNirandika Wanigasekara

48+ Single Pixel Camera

49+Single Pixel Camera- Architecture

50+Single Pixel Camera- DMD ArrayDigital Micro mirror Device A type of a reflective spatial light modulatorSelectively redirects parts of the light beamConsisting 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

51+Single Pixel Camera- Architecture

52+Single Pixel Camera- Photodiode53+Single Pixel Camera- Architecture

54+Single Pixel Camera- measurements55+Single Pixel Camera- Architecture

56+Sample image reconstructions256*256 conventional image of black and white R

How can we improve the reconstruction further? 57+UtilityThis device is useful when measurements are expensive Low Light ImagerConventional Photomultiplier tube/ avalanche photodiode Single Pixel Camera Single photomultiplier

Original800160065536 pixels from 660058+UtilityCS Infrared ImagerIR photodiodeCS Hyperspectral Imager

59+Compressed Sensing MRIYamilet Serrano Llerena

60+ Compressed Sensing MRIMagnetic 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

61+Compressed Sensing MRI

Scan Process

62+Scan Process

Signal ReceivedK-space

Space where MRI data is stored

K-space trajectories:

K-space is 2D Fourier transform of the MR image63+- An MRI scan consists of series of repeated short experiments.In each experiment, different part of the Fourier-space (k-space) information of the image is collectedThe experiments finish when a full data set is gathered. - Each row of k-space contains the raw data received under a particular phase gradient, where the order in which the rows are recorded depends on the imaging sequence used; Once all of k- space has been assembled, it is Fourier transformed (2D FFT) to obtain the imageK-space is sampled using magnetic gradients

k-space is the 2D or 3D Fourier transform of the MR image

1) Cartesian Grid : - Reconstruction is simple. Apply the inverse Fast Fourier Transform - Is robust to many source of system imperfections2) Radial - It can be thought of as a private case of variable density spirals3) Spiral - It is known for its hardware efficiency, fast imaging and robustness to flow. - They are mainly used for real-time and fast imaging applications.

* Signal received : We sample the amplitude of the signal and put these numbers into a list

63In the context of CS

:Is depends on the acquisition deviceIs the Fourier BasisIs an M x N matrixIs the measured k-space data from the scannery :y = x

x :

64+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

65+Compressed Sensing MRI

Scan Process

66+Image Reconstruction

CS uses only a fraction of the MRI data to reconstruct image. 67+Because there are many possible images that may fit the acquired data, the reconstrucion constrain the final image to be commpressible or sparse, which tends to give the correct image

67Image Reconstruction

68+Benefits of CS w.r.t ResonanceAllow for faster image acquisition (essential for cardiac/pediatric imaging)Using same amount of k-space data, CS can obtain higher Resolution Images.

69+SummaryParvathy Sudhir Pillai70+ SummaryMotivationData delugeDirectly acquiring useful part of the signalKey idea: Reduce the number of samples ImplicationsDimensionality reductionLow redundancy and wastage

71+Open ProblemsGood sensing matricesAdaptive? Deterministic?Nonlinear compressed sensing Numerical algorithmsHardware design

Intensity (x)

72+ImpactData generation and storageConceptual achievementsExploit minimal complexity efficientlyInformation theory frameworkNumerous application areas

Legacy - Trans disciplinary research

InformationSoftwareHardwareComplexityCS73+Ongoing ResearchNew mathematical framework for evaluating CS schemesSpectrum sensingNot so optimalData transmission - wireless sensors (EKG) to wired base stations.90% power savings

74+But its been slow to catchon commercially, in part because of a general skepticism that sophisticated math everworks as well in practice as it does in theory.@Rice University researchers produced images with a resolution of tens of thousands of pixels using a camera whose sensor had only one pixel.in part because of a general skepticism that sophisticated math ever works as well in practice as it does in theorythat factors in the real-world performance of hardware components. bridge between these two worlds

In a series of recent papers, four members of associate professor Vladimir Stojanovics Integrated Systems Group at RLE Abari, Stojanovic, postdoc Fabian Lim and recent graduate Fred Chen applied their methodology to two applications where compressed sensing appeared to promise significant power savings. The first was

in which wireless devices would scan the airwaves to detect unused frequencies that they could use to increase their data rates. such as electrocardiogram (EKG) leads to wired base stations

At last years International Conference on Acoustics, Speech, and Signal Processing, the researchers showed that, alas, in spectrum detection, compressed sensing can provide only a relatively coarse-grained picture of spectrum allocation; even then, the power savings are fairly meager.

But in other work, they argue that encoding data from wireless sensors may be a more natural application of the technique. In a forthcoming paper in the journalIEEE Transactions on Circuits and Systems, they show that, indeed, in the case of EKG monitoring, it can provide a 90 percent reduction in the power consumed by battery-powered wireless leads. The applications the RLE researchers investigated do something similar, but rather than using mirrors to modify a signal, they use another s