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Sampl 2015 intro

  1. 1. 1/20 SAMPL 2015 Workshop “Xampling The Future” Monday, June 22nd, 2015
  2. 2. 2 Workshop Goal Expose SAMPL team activity in the area of sub-Nyquist sampling and super-resolution with applications to MRI Ultrasound Body sensor networks Cognitive radio Radar Microscopy and optical imaging Quantum systems Defect detection in microchips Initiate industrial and academic collaborations Recruit talented researchers to join our activities
  3. 3. 3 SAMPL Group Research Group: M.Sc. David Cohen Gal Mazor Kfir Aberman Tanya Chernyakova Amir Kiperwas Ph.D. Post Doc Deborah Cohen Lior Weizman Shahar Tsiper Regev Cohen Oren Solomon
  4. 4. 4 SAMPL Staff Yair Keller Lab Engineer Eli Shoshan Systems Advisor Yoram Or-Chen Advisor Idan Shmuel RF engineer Robert Ifraimov Hardware Engineer Alon Eilam System Engineer Aviad Arobas Computer Administrator Shahar Tsiper RF activity support David Cohen Radar activity support Oren Solomon Bioimaging activity support Anat Zaslavsky Personal Assistant
  5. 5. 5 Undergraduate Projects 20 projects each semester All projects are part of research activities Students participate in conferences and demos worldwide Many of the projects are performed in collaboration with industry Optimized Micro- Beamforming for Medical US Silent MRI with Steady Gradients Increased Sampling Capacity in Optical Communication Joint Spectrum Blind Reconstruction Synthetic Aperture Radar Simulator
  6. 6. 6 Optics Team Prof. Moti Segev Prof. Oren Cohen Ph.D. Maor Mutzafi Ph.D. Dikla Oren Ph.D. Yoav Shechtman Ph.D. Pavel Sidorenko
  7. 7. 7 Close Collaborators (Partial) Technion Labs: SIPL, HSDSL, VLSI Intersection of information theory and sampling theory: Prof. Andrea Goldsmith (Stanford), Alon Kipnis, Yuxin Chen ADC design aspects and sub-Nyquist sampling: Prof. Boris Murmann (Stanford), Doug Adams, Niki Hammler, John Spaulding Ultrasound: Prof. Dan Adam (Technion), Avinoam Bar-Zion, Zvi Friedman and Arcady Kempinski (GE), Jeremy Bercoff (Supersonic US), Dr. Shai Tejman-Yarden (Sheba Medical Center) MRI: Dr. Dafna Ben-Bashat (Ichilov), Dr. Assaf Tal (Weizmann) Phase Retrieval and Optics: Prof. John Miao (UCLA), Prof. Emmanuel Candes (Stanford), Prof. Babak Hassibi (Caltech), Kishore Jaganathan, Prof. Shahar Mandelson (Technion), Prof. Amir Beck (Technion) Radar: Prof. Alex Haimovich (NJIT), Marco Rossi, Prof. Arye Nehorai (WUSTL), Zhao Tan
  8. 8. 8 Industry and Medical Partners Industry Partners: National Instruments General Electric Keysight Technologies Texas Instruments Rafael Medical Partners: Rambam Hospital Sheba Medical Center Tel Aviv Sourasky Medical Center Funding: Intel University Industry Research Corporation Magnet Metro 450 ICORE center ERC Consolidator Grant ISF - Israel Science Foundation BSF Program for Transformative Science
  9. 9. 9 SAMPL Lab Lab inauguration: Spring 2013 Main areas: graduate students + discussion room, communication, medical imaging, computer space
  10. 10. 10 SAMPL Lab About 20 student projects each semester One of the biggest labs in the department Many awards: Best demo award at ICASSP2014 for radar Best demo award at ICASSP2014 for cognitive radio Herschel Rich innovation award for ultrasound Kasher Prize Magnet award, and many more …
  11. 11. 11 SAMPL Lab Undergraduate students participate in research papers: Many students participate in writing conference papers and many attend conferences worldwide to present their work Several projects have led to full journal papers Many undergraduate students in our lab continued to higher degrees and received the Meyer award Tens of live demos of our technology are presented in major conferences worldwide
  12. 12. 12 SAMPL Vision Tight connection between deep theory and engineering Impact basic science as well as technology and society Teaching and mentoring of students All undergraduate students in the lab are involved in state-of- the-art research and demo development Train students to do research and write scientific papers early on in their academic studies Only lab worldwide that develops wideband sub-Nyquist receivers Continue to develop new theory combined with technology design through industry involvement while creating human capital and training the future generation of researchers
  13. 13. 13 Agenda General overview of sub-Nyquist sampling Part I: Medical Imaging (MRI, Ultrasound, ECG) Part II: Communications and Radar Lunch Break Lab Awards Part III: Optics and superresolution Throughout the day there will be posters outside which expand on the ideas presented Sessions will include live demos of our prototypes
  14. 14. 14 Thanks Research students and lab team Eli Shoshan Suzie Eid Ina Rivkin Yoram Or-Chen Anat Zaslavsky
  15. 15. 15 Xampling: Sub Nyquist Sampling Cognitive radioRadar Ultrasound Pulses DOA Estimation 15
  16. 16. 16 Very high sampling rates: hardware excessive solutions High DSP rates Digital worldAnalog world Signal processing Image denoising Analysis… Processing Music Radar Speech Image… ADCs, the front end of every digital application, remain a major bottleneck Sampling: “Analog Girl in a Digital World…” Judy Gorman 99 Sampling rate must be at least twice the highest frequency Sampling Analog-to-Digital (ADC) H. Nyquist C. Shannon
  17. 17. 17 Today’s Paradigm The Separation Theorem: Circuit design experts design samplers at Nyquist rate or higher DSP/machine learning experts process the data Typical first step: Throw away (or combine in a “smart” way e.g. dimensionality reduction) much of the data … Logic: Exploit structure prevalent in most applications to reduce DSP processing rates However, the analog step is one of the costly steps Can we use the structure to reduce sampling rate + first DSP rate (data transfer, bus …) as well? ADC first DSP steps, bus, data transfer DSP high rate, generic low rate exploits structure x(t) c[n]
  18. 18. 18 Proposed Paradigm The Separation Theorem: Circuit design experts design samplers at Nyquist rate or higher DSP/machine learning experts process the data Typical first step: Throw away (or combine in a “smart” way e.g. dimensionality reduction) much of the data … Can we use the structure to reduce sampling rate + first DSP rate (data transfer, bus …) as well? ADC first DSP steps DSP high bandwidth exploits structure low rate exploits structure x(t) c[n]Pre-Processing
  19. 19. 19 Reduce storage/reduce sampling rates Reduce processing rates Reduce power consumption and energy Increase resolution Increase speed such as frame rate in imaging Increase the number of signals that can be concurrently processed Enable technologies that are currently infeasible (such as fast 3D imaging, wideband sensing, rural and portable imaging and more) We exploit structure in the analog domain in order to reduce sampling and processing rates and enable new technologies Xampling: Low-Rate Sampling Sample only the info that is needed Xampling = Compression+Sampling
  20. 20. 20 Group Objectives We examine the basic pillars of sampling and information theory: Theory: Developing the fundamental limits of rate reduction based on signal structure and the information theory of sub-sampled and structured channels Hardware: Developing the theory and hardware of mixed analog-digital hardware prototypes that exploit signal structure at sub-Nyquist rates Applications: Demonstrating the broad benefit of low rate sampling in applications ranging from wireless communication to medical imaging Goal of the presentation: Survey the main principles in exploiting analog structure Provide a variety of different applications and benefits Technical details can be found in the references and in lectures throughout the day
  21. 21. 21 Motivation Xampling: Compression + sampling of analog signals Sub-Nyquist sampling without structure Applications of sub-Nyquist sampling: Sub-Nyquist spectrum sensing Sub-Nyquist radar Sub-Nyquist ultrasound Nonlinear compressed sensing and optics Talk Outline
  22. 22. 22 Part 1: Motivation
  23. 23. 23 Structured Analog Models Can be viewed as bandlimited (subspace) But sampling at rate is a waste of resources For wideband applications Nyquist sampling may be infeasible Multiband communication: Question: How do we treat structured analog models efficiently? Unknown carriers – non-subspace
  24. 24. 24 Cognitive Radio Cognitive radio mobiles utilize unused spectrum ``holes’’ Spectral map is unknown a-priori, leading to a multiband model Federal Communications Commission (FCC) frequency allocation Licensed spectrum highly underused: E.g. TV white space, guard bands and more Shared Spectrum Company (SSC) – 16-18 Nov 2005
  25. 25. 25 Sometimes reconstructing the covariance rather than the signal itself is enough: Support detection Statistical analysis Parameter estimation (e.g. DOA) Assumption: Wide-sense stationary ergodic signal If all we want to estimate is the covariance then we can substantially reduce the sampling rate even without structure! Power Spectrum Reconstruction What is the minimal sampling rate to estimate the signal covariance? Cognitive Radios Financial time Series analysis Deborah Cohen
  26. 26. 26 Structured Analog Models Digital matched filter or super-resolution ideas (MUSIC etc.) (Quazi,Brukstein, Shan,Kailath,Pallas,Jouradin,Schmidt,Saarnisaari,Roy,Kumaresan,Tufts …) But requires sampling at the Nyquist rate of The pulse shape is known – No need to waste sampling resources! Medium identification: Unknown delays – non-subspace Channel Question (same): How do we treat structured analog models efficiently? Similar problem arises in radar, UWB communications, timing recovery problems …
  27. 27. 27 Ultrasound Relatively simple, radiation free imaging Tx pulse Ultrasonic probe Rx signal Unknowns Echoes result from scattering in the tissue The image is formed by identifying the scatterers Cardiac sonography Obstetric sonography
  28. 28. 28 To increase SNR and resolution an antenna array is used SNR and resolution are improved through beamforming by introducing appropriate time shifts to the received signals Requires high sampling rates and large data processing rates One image trace requires 128 samplers @ 20M, beamforming to 150 points, a total of 6.3x106 sums/frame Processing Rates Scan Plane Xdcr Focusing the received beam by applying nonlinear delays  2 2 1 1 1 ( ; ) 4( ) sin 4( ) 2 M m m m m t t t t c t c M                  
  29. 29. 29 Can we reduce analog sampling rates? Can we perform nonlinear beamforming on the sub-Nyquist samples without interpolating back to the high Nyquist-rate grid digitally? Challenges Compressed Beamforming Goal: reduce ultrasound machine size at same resolution Enable 3D imaging Increase frame rate Enable remote wireless ultrasound Re Tanya Chernyakova
  30. 30. 30 Subwavelength Imaging + Phase Retrieval Diffraction limit: The resolution of any optical imaging system is limited by half the wavelength This results in image smearing Furthermore, optical devices only measure magnitude, not phase 100 nm 474 476 478 480 482 484 486 462 464 466 468 470 472 474 476 Collaboration with the groups of Moti Segev and Oren Cohen Sketch of an optical microscope: the physics of EM waves acts as an ideal low-pass filter Nano-holes as seen in electronic microscope Blurred image seen in optical microscope λ=514nm
  31. 31. 31 Imaging via “Sparse” Modeling Model FT intensity Frequency [1/]Frequency[1/] -5 0 5 -6 -4 -2 0 2 4 6 Diffraction-limited (low frequency) intensity measurements Model Fourier transform Circles are 100 nm diameter Wavelength 532 nm SEM image Sparse recovery 474 476 478 480 482 484 486 462 464 466 468 470 472 474 476 Blurred image Szameit et al., Nature Materials, ‘12 Recovery of sub-wavelength images from highly truncated Fourier power spectrum Sparse phase retrieval methods
  32. 32. 32 Part 2: Xampling Theory Cognitive radioRadar Ultrasound Pulses DOA Estimation
  33. 33. 33 Theory: Union of Subspaces Model: Mixed estimation detection problem Lu and Do 08, Mishali and Eldar 09
  34. 34. 34 Union of Subspaces Model: Standard approach: Look at sum of all subspaces Signal bandlimited to High rate Lu and Do 08, Mishali and Eldar 09
  35. 35. 35 Union of Subspaces Model: Allows to keep low dimension in the problem model Low dimension translates to low sampling rate Theorem Lu and Do 08, Mishali and Eldar 09 Multiband Sampling: 2NB Pulse streams: 2L
  36. 36. 36 Xampling Xampling: Compression + Sampling Prior to analog sampling reduce bandwidth by projecting data onto low dimensional analog space Creates aliasing of the data Sample the data at low rate using standard ADCs in such a way that in the digital domain we get a compressed sensing problem Typically set up problem in frequency: low rate processing, robustness Results in low rate, low bandwidth, simple hardware and low computational cost x(t) Acquisition Compressed sensing and processing recovery Analog preprocessing Low rate (bandwidth) Mishali and Eldar, 10
  37. 37. 37 ~~ ~~ Rate should be 2L if we have L pulses Naïve attempt: direct sampling at low rate Most samples do not contain information!! Multiband problem: Rate should be 2NB Most bands do not have energy – which band should be sampled? Low Rate Acquisition: Difficulty Low rate ADC Analog preprocessing
  38. 38. 38 Alias all energy to baseband before sampling (analog projection) Can sample at low rate Resolve ambiguity in the digital domain ~~ ~~ Smear pulse before sampling (analog projection – bandwidth reduction) Each sample contains energy Resolve ambiguity in the digital domain Intuitive Solution: Pre-Processing Low rate ADC
  39. 39. 39 Subspace techniques developed in the context of array processing (such as MUSIC, ESPRIT etc.) Compressed sensing Connections between CS and subspace methods: Malioutov, Cetin, and Willsky, Davies and Eldar, Lee and Bresler, Kim, Lee and Ye, Fannjiang, Austin, Moses, Ash and Ertin For nonlinear sampling: Quadratic compressed sensing (Shechtman et. al 11, Eldar and Mendelson 12, Ohlsson et. al 12, Janganathan 12) More generally, nonlinear compressed sensing (Beck and Eldar 12, Bahman et. al 11) Digital Recovery
  40. 40. 40 Compressed Sensing Candes, Romberg, Tau 06, Donoho 06
  41. 41. 41 Compressed Sensing
  42. 42. 42 Optimal Xampling Hardware Sampling Reconstruction AWGN We derive two lower bounds on the performance of UoS estimation: Fundamental limit – regardless of sampling technique or rate Lower bound for a given sampling rate Allows to determine optimal sampling method Can compare practical algorithms to bound (det. by ) Sampling with sinusoids is optimal under a wide set of inputs! Ben-Haim, Michaeli, and Eldar 11Aliasing The minimal MSE is obtained with where are the eigenfunctions of Theorem (Generalized KLT)
  43. 43. 43 Xampling Hardware sums of exponentials The filter H(f) allows for additional freedom in shaping the tones and reduces the bandwidth The channels can be collapsed to a single channel
  44. 44. 44 Shannon Meets Nyquist: Capacity Perspective We can formulate our problem in a communication setting and design sub-Nyquist sampling methods that optimize the channel capacity What is the capacity-achieving sub-Nyquist sampler and optimal input? It turns out that our Xampling methods are optimal in terms of capacity as well under a wide range of settings! Chen, Eldar and Goldsmith 13 )(th ][ny ( )n t )(tx EncoderMessage signal structure
  45. 45. 45 Channel-Blind (Universal) Sampling For each channel state, there is a sampled capacity loss with respect to the known channel case Robustness measure: Minimax Sampled Capacity Loss over all realizations Chen, Goldsmith and Eldar 13 Model: Multiband channel The sampler is designed independent of instantaneous channel realization! maximize capacity Encoder x(t)
  46. 46. 46 Minimax Capacity Loss Xampling system (with random modulator) achieves Minimax Capacity Loss!  Complementary viewpoint on optimality of Xampling hardware  Proves optimality of random sampling )(th )(t LPF 1[ ]y n [ ]iy n [ ]my n  q1(t)  qi (t)  qm (t)r(t) y1(t) yi (t) ym (t) )(tx LPF LPF random modulation coefficients α: undersampling factor β: band sparsity ratio Optimal channel-blind sampling strategy from information-theoretic perspective achieved with sub-Gaussian distribution binary entropy function
  47. 47. 47 Part 3: Xampling Without Structure
  48. 48. 48 Until now we exploited analog structure to reduce sampling rates Two examples of reducing rate without structure: Covariance estimation Joint work With Prof. Geert Leus and Deborah Cohen Accounting for quantization effects Joint work with Prof. Andrea Goldsmith and Alon Kipnis Reducing Rate Without Structure In both cases optimal performance can be achieved at rates lower than Nyquist since we are not interested in recovering the full analog signal!
  49. 49. 49 Until now we ignored quantization Quantization introduces inevitable distortion to the signal Since the recovered signal will be distorted due to quantization do we still need to sample at the Nyquist rate? Reducing Rate with Quantization 01001001001 010010… quantizer Source Coding [Shannon]Sampling Theory ˆ[ ]y n[ ]y n 2log (#levels) bit/sec sR f Goal: Unify sampling and rate distortion theory ( )x t Kipnis, Goldsmith and Eldar 15
  50. 50. 50 Standard source coding: For a given discrete-time process y[n] and a given bit rate R what is the minimal achievable distortion Our question: For a given continuous-time process x(t) and a given bit rate R what is the minimal distortion What sampling rate is needed to achieve the optimal distortion? Unification of Rate-Distortion and Sampling Theory )(th( )x t [ ]y n ( )n t ENC DEC R f s ˆ( )x t 2 ˆ( ) inf [ ] [ ]D R y n y n  2 ˆinf ( , ) inf ( ) ( )sf sD f R x t x t  [ ]y n ENC DEC R ˆ[ ]y n
  51. 51. 51 Quantizing the Samples: Source Coding Perspective Preserve signal components above “noise floor” q , dictated by R Distortion corresponds to mmse error + signal components below noise floor Theorem (Kipnis, Goldsmith, Weissman, Eldar 2013) 2 2 1 ( , ) log ( ) / 2 fs fss X Y R f S f df       2 2 ( , ) ( ) min{ ( ), } fs fss sX Y X Y D f mmse f S f df     
  52. 52. 52 Can we achieve D(R) by sampling below fNyq? Yes! For any non-flat PSD of the input Optimal Sampling Rate ( , ) ( ) for ( )! s s DR D R f D R f f R   Shannon [1948]: “we are not interested in exact transmission when we have a continuous source, but only in transmission to within a given tolerance” No optimality loss when sampling at sub-Nyquist (without input structure)!
  53. 53. 53 Part 4: Applications “In theory, theory and practice are the same. In practice, they are not.” Albert Einstein
  54. 54. 54 The Modulated Wideband Converter ~~ ~~ Time Frequency Mishali and Eldar, 11 B B
  55. 55. 55 Single Channel Realization ~~ Time Frequency Mishali and Eldar, 11 2𝑛𝑁𝑇𝑝 𝑁 𝑇𝑝 𝑥(𝑡) 𝑝(𝑡) 1 2𝑇𝑝 2 2𝑇𝑝 Bandwidth NB ~~ 𝐻(𝑓) 𝑦 𝑛 𝑇𝑝 – periodic 𝑝(𝑡) gives the desired aliasing effect 𝐻(𝑓)
  56. 56. 56 Nyquist rate: 6 GHz Xampling rate: 360 MHz (6% of Nyquist rate) Wideband receiver mode: 49 dB dynamic range, SNDR > 30 dB ADC mode: 1.2v peak-to-peak full-scale, 42 dB SNDR = 6.7 ENOB Parameters: Performance: Cognitive Radio MWC analog front-end Mishali, Eldar, Dounaevsky, and Shoshan, 2010 Cohen et. al. 2014 6% of Nyquist rate!
  57. 57. 57 Cognitive Radio Demo at ICASSP 2014 Lustig et al., 2008
  58. 58. 58 Further Details Later Today Live Demo Robustness to noise DoA estimation Distributed collaborative detection Shahar Tsiper Deborah Cohen
  59. 59. 59 Streams of Pulses Xampling requires using a small set of Fourier coefficients Pulses can be entirely recovered from only 2L Fourier coefficients Efficient hardware: Gedalyahu, Tur, Eldar 10, Tur, Freidman, Eldar 10 Theorem (Tur, Eldar and Friedman 11) Sum-of-Sincs filter with compact support 𝑥 𝑡 𝑠∗ −𝑡 𝐹𝐹𝑇 𝑐 𝑘
  60. 60. 60 Selecting The Active Frequencies For good resolution and CS properties we need wide frequency aperture To avoid ambiguities we need at least two close frequencies Can randomly place frequencies over wide aperture Our choice: Use a small set of bandpass filters spread randomly over a wide frequency range BPF4 BPF3 BPF2 Analog signal Band-pass Filter 1 Low rate ADC Baseband down-convertor Multichannel filter: Wide aperture Close frequencies
  61. 61. 61 Xampling in Radar Distance to target Target velocity Demand for high range resolution radar requires high bandwidth signals on the order of 100s Mhz to several Ghz Classic matched filtering requires sampling and processing at the Nyquist rate Long time-on-target needed for good Doppler resolution When using multiple antennas need to space them at the spatial Nyquist rate – many antennas! Targets k Tx/Rx sensors Plane wave Bar-Ilan and Eldar 14, Itzhak et. al. 14, Rossi, Haimovich and Eldar 14
  62. 62. 62 Later Today Omer Bar-Ilan: sub-Nyquist Pulse-Doppler radar – detect targets at high resolution from sub-Nyquist samples Deborah Cohen: Reduced time-on-target – target detection with a small number of pulses Alex Dikopoltsev: Cognitive radar – transmitting only where the bandwidth is free David Cohen: sub-Nyquist MIMO radar – detecting targets using a small number of antennas, each sampled at a sub-Nyquist rate In addition: Deborah Cohen: Resolve range ambiguity in Doppler radar Kfir Aberman: Applications to SAR Deborah CohenOmer Bar-Ilan
  63. 63. 63 Xampling of Radar Pulses 1/30 of the Nyquist Rate analog filter banks ADCs splitters low pass filter Itzhak et. al. 2012 in collaboration with NI Robert Ifraimov Idan Shmuel
  64. 64. 64 Radar Demo System at ICASSP 2014
  65. 65. 65 -80 -60 -40 -20 0 20 40 60 80 0 20 40 60 80 100 120 140 160 Standard Imaging We obtain a 32-fold rate reduction Enable handheld wireless devices for rural medicine, emergency imaging in the field/ambulance Enable 3D imaging High frame rate for cardiac imaging 3328 real-valued samples, per line 360 complex-valued samples, per line -80 -60 -40 -20 0 20 40 60 80 0 20 40 60 80 100 120 140 160 -80 -60 -40 -20 0 20 40 60 80 0 20 40 60 80 100 120 140 160 100 complex-valued samples, per line ~1/10 of Nyquist per element ~1/32 of Nyquist per element Sub-Nyquist Ultrasound Imaging Chernyakova and Eldar 14 Tanya Chernyakova Alon Eilam
  66. 66. 66 Wireless Ultrasound Imaging A wireless probe performs Xampling and transmits the low rate data to a server for processing Frequency Domain Beamforming and image reconstruction is performed by the server The image is sent for display on a monitor Xampler Alon Eilam
  67. 67. 67 Phase Retrieval: Nonlinear Compressed Sensing Arises in many fields: microscopy, crystallography, astronomy, optical imaging, and more Given an optical image illuminated by coherent light, in the far field we obtain the Fourier transform of the image Optical measurement devices measure the photon flux, which is proportional to the magnitude squared of the field Fourier + Absolute value 2 [ ] [ ]y k X k[ ]x n Can we extend compressed sensing results to the nonlinear case? Crystallography
  68. 68. 68 Phase Is Important! Fourier Transform Magnitude Fourier Transform Magnitude Inverse Fourier Transform Phase Phase Inverse Fourier Transform
  69. 69. 69 Phase Retrieval Applications Ultra-short optical pulse measurement Coherent Diffractive Imaging Crystallography [1] R. Trebino et al., JOSA A 10, 5 1101-1111 (1993) [2] MM Seibert et al. Nature 470, 78-81 (2011) [3] D Shechtman et al. PRL 53, 20, 1951-1952 (1984) [1] [2] [3]
  70. 70. 70 Phase Retrieval Difficult to analyze theoretically when recovery is possible No uniqueness in 1D problems (Hofstetter 64) Uniqueness in 2D if oversampled by factor 2 (Hayes 82) No guarantee on stability No known algorithms to achieve unique solution Analysis of Random Measurements: 𝑦𝑖 = 𝑎𝑖, 𝑥 2 + 𝑤𝑖 noise 𝑥 ∈ 𝑅 𝑁 4𝑁 − 2 measurements needed for uniqueness (Balan, Casazza, Edidin o6, Bandira et. al 13) random vector How to solve objective function? Stable Phase Retrieval (Eldar and Mendelson 14): 𝑁log(𝑁) measurements needed for stability 𝑘log(𝑁/𝑘) measurements needed for stability with sparse input Solving 𝑖=1 𝑀 𝑦𝑖 − 𝑎𝑖, 𝑥 2 𝑝 1 < 𝑝 ≤ 2 provides stable solution
  71. 71. 71 Nonlinear Sparse Recovery General theory and algorithms for nonlinear sparse recovery Derive conditions for optimal solution Use them to generate algorithms Necessary Conditions: L-stationarity Iterative Hard Thresholding CW-minima Greedy Sparse Simplex (OMP) Beck and Eldar, 13 min 𝑓 𝑥 s.t. 𝑥 0 ≤ 𝑘 Generalization of compressed sensing algorithms to the nonlinear setting!
  72. 72. 72 GESPAR: GrEedy Sparse PhAse Retrieval Generalization of matching pursuit to phase retrieval Local search method with update of support For given support solution found via Damped Gauss Newton Efficient and more accurate than current techniques 1. For a given support: minimizing objective over support by linearizing the function around current support and solve for 𝑦 𝑘 𝑧 𝑘 = 𝑧 𝑘−1 + 𝑡 𝑘(𝑦 𝑘 − 𝑧 𝑘−1) 2. Find support by finding best swap: swap index with small value 𝑥𝑖 with index with large value 𝛻𝑓(𝑥𝑗) Shechtman, Beck and Eldar, 13 determined by backtracking
  73. 73. 73 Introducing Redundancy Oblique illumination Candes, Eldar, Strohmer , Voroninski 12 Multiple masks (phase diversity) Candes, Li and Soltanolkotabi 13 Short-time Fourier transform (STFT) Jaganathan, Eldar and Hassibi 15 FROG/XFROG Ptychogrpahy – scanning CDI: CDI with several overlapping illumination patterns All of these techniques lead to redundant magnitude measurements which enhance recovery STFT Recovery Results (Jaganathan, Eldar and Hassibi 15): does not vanish anywhere then almost all signals can be recovered uniquely  If / 2 andL W N x n 
  74. 74. 74 Sparsity Based Subwavelength CDI Circles are 100 nm diameter Wavelength 532 nm SEM image Sparse recovery 474 476 478 480 482 484 486 462 464 466 468 470 472 474 476 Blurred image Diffraction-limited (low frequency) intensity measurements Model Fourier transformModel FT intensity Frequency [1/] Frequency[1/] -5 0 5 -6 -4 -2 0 2 4 6 Szameit et al., Nature Materials, 12
  75. 75. 75 Sparsity Based Ankylography Concept: A short x-ray pulse is scattered from a 3D molecule combined of known elements. The 3D scattered diffraction pattern is then sampled in a single shot Recover a 3D molecule using 2D sample Short pulse X-ray K.S. Raines et al. Nature 463, 214 ,(2010). Mutzafi et. al., (2013).
  76. 76. 76 Compressed sampling and processing of many analog signals Wideband sub-Nyquist samplers in hardware Significant rate reduction in both analog and digital while retaining sufficient quality Merging information theory and sampling theory Extensions to nonlinear measurements Many applications and many research opportunities: extensions to other analog and digital problems, robustness, hardware, many open theoretical questions in nonlinear domain … Exploiting structure can lead to a new sampling paradigm which combines analog + digital and to superresolution imaging Conclusions
  77. 77. 77 Xampling Website webee.technion.ac.il/people/YoninaEldar/xampling_top.html Y. C. Eldar and G. Kutyniok, "Compressed Sensing: Theory and Applications", Cambridge University Press, 2012 Y. C. Eldar, “Sampling Theory: Beyond Bandlimited Systems", Cambridge University Press, 2015
  78. 78. 78 SAMPL Lab Website www.sampl.technion.ac.il
  79. 79. 79 Thank you