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+ Compressed Sensing. + Mobashir Mohammad Aditya Kulkarni Tobias Bertelsen Malay Singh Hirak Sarkar Nirandika Wanigasekara Yamilet Serrano Llerena Parvathy.

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Presentation on theme: "+ Compressed Sensing. + Mobashir Mohammad Aditya Kulkarni Tobias Bertelsen Malay Singh Hirak Sarkar Nirandika Wanigasekara Yamilet Serrano Llerena Parvathy."— Presentation transcript:

1 + Compressed Sensing

2 + Mobashir Mohammad Aditya Kulkarni Tobias Bertelsen Malay Singh Hirak Sarkar Nirandika Wanigasekara Yamilet Serrano Llerena Parvathy Sudhir

3 + Introduction Mobashir Mohammad 3

4 + The Data Deluge Sensors: Better… Stronger… Faster… Challenge: Exponentially increasing amounts of data Audio, Image, Video, Weather, … Global scale acquisition 4

5 + 5

6 + Sensing by Sampling Sample N 6

7 + Sensing by Sampling (2) Sample N Compress N >> L JPEG … L L Decompress N >> L N 7

8 + Compression: Toy Example 8

9 + Discrete Cosine Transformation Transformation 9

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? Donoho

11 + Outline Compressed Sensing Constructing Φ Sparse Signal Recovery Convex Optimization Algorithm Applications Summary Future Work 11

12 + Compressed Sensing Aditya Kulkarni 12

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 rate Sampling Analog Audio Signal Compression (e.g. MP3) High-rate Low-rate Compressed Sensing 13

14 + Sparsity The concept that most signals in our natural world are sparse a.Original image c. Image reconstructed by discarding the zero coefficients 14

15 + How It Works 15

16 + Dimensionality Reduction Problem 16

17 + Sampling 17

18 + 18

19 + 19

20 + Sparsity The concept that most signals in our natural world are sparse a.Original image c. Image reconstructed by discarding the zero coefficients 20

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22 + Constructing Φ Tobias Bertelsen 22

23 + RIP - Restricted Isometry Property The distance between two points are approximately the same in the signal-space and measure-space 23

24 + RIP - Restricted Isometry Property Image: 24

25 + Randomized algorithm 25

26 + Sub-Gaussian distribution 26

27 + Johnson-Lidenstrauss Lemma 27

28 + Generalizing to RIP 28

29 + Randomized algorithm 29

30 + Sparse in another base 30

31 + 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 31

32 + Sparse Signal Recovery Malay Singh 32

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35 + 35

36 + But the problem is non-convex and very hard to solve 36

37 + We are minimizing the Euclidean distance. But the arbitrary angle of hyperplane matters 37

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39 + 39

40 + Convex Optimization Hirak Sarkar 40

41 + What it is all about … 41

42 + 42

43 + Versions of the same problem 43

44 + Formalize 44

45 + Shrinkage operator 45

46 + Algorithm 46

47 + Performance 47

48 + Single Pixel Camera Nirandika Wanigasekara 48

49 + Single Pixel Camera 49

50 + Single Pixel Camera- Architecture 50

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 51

52 + Single Pixel Camera- Architecture 52

53 + Single Pixel Camera- Photodiode 53

54 + Single Pixel Camera- Architecture 54

55 + Single Pixel Camera- measurements 55

56 + Single Pixel Camera- Architecture 56

57 + Sample image reconstructions 256*256 conventional image of black and white ‘R’ How can we improve the reconstruction further? 57

58 + Utility This device is useful when measurements are expensive Low Light Imager Conventional  Photomultiplier tube/ avalanche photodiode Single Pixel Camera  Single photomultiplier Original pixels from

59 + Utility CS Infrared Imager IR photodiode CS Hyperspectral Imager 59

60 + Compressed Sensing MRI Yamilet Serrano Llerena 60

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 61

62 + Compressed Sensing MRI Scan Process 62

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 63

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 : 64

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 65

66 + Compressed Sensing MRI Scan Process 66

67 + Image Reconstruction CS uses only a fraction of the MRI data to reconstruct image. 67

68 + Image Reconstruction 68

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. 69

70 + Summary Parvathy Sudhir Pillai 70

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 71

72 + Open Problems ‘Good’ sensing matrices Adaptive? Deterministic? Nonlinear compressed sensing Numerical algorithms Hardware design Intensity (x) 72

73 + Impact Data generation and storage Conceptual achievements Exploit minimal complexity efficiently Information theory framework Numerous application areas Legacy - Trans disciplinary research Information Software Hardware Complexity CS 73

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 74

75 + In the news 75

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 , 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, compressive-sensing-technique/ compressive-sensing-technique/ 76

77 + THANK YOU 77


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