Compressed Sensing Compressed Sensing n Mobashir Mohammad n
+ Compressed Sensing
+ Compressed Sensing n Mobashir Mohammad n Aditya Kulkarni n Tobias Bertelsen n Malay Singh n Hirak Sarkar n Nirandika Wanigasekara n Yamilet Serrano Llerena n Parvathy Sudhir
3 + Introduction Mobashir Mohammad
+ 4 The Data Deluge n Sensors: Better… Stronger… Faster… n Challenge: n Exponentially increasing amounts of data n Audio, Image, Video, Weather, … n Global scale acquisition
+ 6 Sensing by Sampling Sample N
+ 7 Sensing by Sampling (2) N >> L Sample N Compress L JPEG … N >> L L Decompress N
+ 8 Compression: Toy Example
+ 9 Discrete Cosine Transformation
+ 10 Motivation n Why go to so much effort to acquire all the data when most of the what we get will be thrown away? n Cant we just directly measure the part that wont end up being thrown away? Donoho 2004
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? n 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 Analog Audio Signal Nyquist rate Sampling High-rate Compressed Compression (e. g. MP 3) Sensing Low-rate
+ 14 Sparsity n The concept that most signals in our natural world are sparse a. Original image c. Image reconstructed by discarding the zero coefficients
+ 15 How It Works
+ Dimensionality Reduction Problem 16
+ 17 Sampling
+ 18
+ 19
+ 20 Sparsity n The concept that most signals in our natural world are sparse a. Original image c. Image reconstructed by discarding the zero coefficients
+ 21
22 + Constructing Φ Tobias Bertelsen
+ 23 RIP - Restricted Isometry Property n The distance between two points are approximately the same in the signal-space and measure-space
+ 24 RIP - Restricted Isometry Property n Image: http: //www. brainshark. com/brainshark. net/portal/title. aspx? pid=z. Cgz. Xgc. EKz 0 z 0
+ 25 Randomized algorithm n
+ 26 Sub-Gaussian distribution n
+ 27 Johnson-Lidenstrauss Lemma n
+ 28 Generalizing to RIP n
+ 29 Randomized algorithm n
+ 30 Sparse in another base n
+ 31 n Stable n n Universal n n Works with any orthogonal basis Democratic n n n Robust to noise, since it satisfies RIP Any element in has equal importance Robust to data loss Other Methods n n Random Fourier submatrix Fast JL transform
32 + Sparse Signal Recovery Malay Singh
+ 33
+ 34
+ 35
+ 36 But the problem is non-convex and very hard to solve
+ 37 We are minimizing the Euclidean distance. But the arbitrary angle of hyperplane matters
+ 38
+ 39 n
40 + Convex Optimization Hirak Sarkar
+ 41 What it is all about … n
+ 42 n
+ 43 Versions of the same problem n
+ 44 Formalize n
+ 45 Shrinkage operator n
+ 46 Algorithm n
+ 47 Performance n
48 + Single Pixel Camera Nirandika Wanigasekara
+ 49 Single Pixel Camera n
+ 50 Single Pixel Camera- Architecture
+ 51 Single Pixel Camera- DMD Array n Digital Micro mirror Device n A type of a reflective spatial light modulator n Selectively redirects parts of the light beam n Consisting of an array of N tiny mirrors n Each mirror can be positioned in one of two states(+/-10 degrees) n Orients the light towards or away from the second lens
+ 52 Single Pixel Camera- Architecture
+ 53 Single Pixel Camera- Photodiode n
+ 54 Single Pixel Camera- Architecture
+ 55 Single Pixel Camera- measurements n
+ 56 Single Pixel Camera- Architecture
+ 57 Sample image reconstructions n 256*256 conventional image of black and white ‘R’ n How can we improve the reconstruction further?
+ 58 Utility n This device is useful when measurements are expensive n Low Light Imager n Conventional Photomultiplier tube/ avalanche photodiode n Single Pixel Camera Single photomultiplier Original 800 1600 65536 pixels from 6600
+ 59 Utility n CS Infrared Imager n n 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 2 D Fourier transform of the MR image
+ 64 In the context of CS Φ : • • • y = Φ x Is depends on the acquisition device Is the Fourier Basis Is an M x N matrix y : • Is the measured k-space data from the scanner x :
+ 65 Recall. . . n 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 n Allow for faster image acquisition (essential for cardiac/pediatric imaging) n Using same amount of k-space data, CS can obtain higher Resolution Images.
70 + Summary Parvathy Sudhir Pillai
+ 71 Summary n Motivation n Data deluge n Directly acquiring useful part of the signal n Key idea: Reduce the number of samples n Implications n Dimensionality reduction n Low redundancy and wastage
+ 72 Open Problems n ‘Good’ sensing matrices n Adaptive? Deterministic? n Nonlinear compressed sensing n Numerical algorithms n Hardware design Intensity (x)
+ 73 Impact n Data generation and storage n Conceptual achievements n Exploit minimal complexity efficiently n Information theory framework n Numerous application areas n Legacy - Trans disciplinary research Information Hardware C S Software Complexity
+ 74 Ongoing Research n New mathematical framework for evaluating CS schemes n Spectrum sensing n n Not so optimal Data transmission - wireless sensors (EKG) to wired base stations. n 90% power savings
+ 75 In the news
+ 76 References n Emmanuel Candes, Compressive Sensing - A 25 Minute Tour, First EU-US Frontiers of Engineering Symposium, Cambridge, September 2010 n David Schneider, Camera Chip Makes Already-Compressed Images, IEEE Spectrum, Feb 2013 n T. Strohmer. Measure what should be measured: Progress and Challenges in Compressive Sensing. IEEE Signal Processing Letters, vol. 19(12): pp. 887 -893, 2012. n Larry Hardesty, Toward practical compressed sensing, MIT news, Feb 2013 n Tao Hu and Mitya Chklovvskii, Reconstruction of Sparse Circuits Using Multi-neuronal Excitation (RESCUME), Advances in Neural Information Processing Systems, 2009 n http: //inviewcorp. com/technology/compressive-sensing/ n http: //ge. geglobalresearch. com/blog/the-beauty-of-compressive-sensing/ n http: //www. worldindustrialreporter. com/bell-labs-create-lensless-camera-throughcompressive-sensing-technique/ n http: //www. lablanche-and-co. com/
+ 77 THANK YOU
- Slides: 77
