Low-Rank plus Sparse Reconstruction

Low-Rank plus Sparse Reconstruction

Tutorial Talk @ Sunday Education Session​, ​The 27th Annual Meeting of ISMRM, May 12th, 2019, Montreal, Canada

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Jong Chul Ye

May 12, 2019
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  1. Low-Rank plus Sparse Reconstruction Jong Chul Ye, Ph.D Endowed Chair

    Professor BISPL - BioImaging, Signal Processing, and Learning lab. Dept. Bio & Brain Engineering KAIST, Korea
  2. Speaker Name: Jong Chul Ye I have no financial interests

    or relationships to disclose with regard to the subject matter of this presentation. Declaration of Financial Interests or Relationships
  3. Unmet Needs in MRI q MR exam protocol : 30~60

    min/patient ü should increase the throughput of MR scanning q Cardiac imaging, fMRI ü Should improve temporal resolution q Various artifacts in real acquisition ü EPI, motion, hardware
  4. Classical approaches fast pulse sequence parallel/multiband imaging Sodickson et al,

    MRM, 1997; Pruessmann et al, MRM 1999; Griswold et al, MRM, 2002 Mansfield, JPC 1977; Ahn et al, TMI, 1986 Artifact correction Xiang QS et al., MRM, 2007 Poser BA et al., MRM, 2013
  5. RECON-BASED APPROACHES

  6. Signal Representation: Key to Recon x = X hx, e

    bn ibn <latexit sha1_base64="tCLG2nbXwywzwFFUoqMNNZJmas8=">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</latexit> Synthesis basis Analysis basis coefficients Concise representation comes from Adaptive basis
  7. GLM Basis Representation of fMRI = X i=1 nbn <latexit

    sha1_base64="hF8YSuka+Y36npfrys6bhoSxFsk=">AAACCHicbVDLSgMxFM34rPVVdenCYBFclRkVdFMounFZwT6gU4YkzbShSWZIMkIZZunGX3HjQhG3foI7/8ZMOwttPRA4nHMuuffgmDNtXPfbWVpeWV1bL22UN7e2d3Yre/ttHSWK0BaJeKS6GGnKmaQtwwyn3VhRJDCnHTy+yf3OA1WaRfLeTGLaF2goWcgIMlYKKkd1XyciSFndy3xMDQok9AUyIxymOAtsourW3CngIvEKUgUFmkHlyx9EJBFUGsKR1j3PjU0/RcowwmlW9hNNY0TGaEh7lkokqO6n00MyeGKVAQwjZZ80cKr+nkiR0HoisE3mO+p5Lxf/83qJCa/6KZNxYqgks4/ChEMTwbwVOGCKEsMnliCimN0VkhFSiBjbXdmW4M2fvEjaZzXvvObeXVQb10UdJXAIjsEp8MAlaIBb0AQtQMAjeAav4M15cl6cd+djFl1yipkD8AfO5w/SO5nZ</latexit> x <latexit sha1_base64="xEIMch3yuo7JxT4Wy1udoMzDhIk=">AAAB8XicbVDLSgMxFL1TX7W+qi7dBIvgqsyooMuiG5cV7APbUjLpnTY0kxmSjFiG/oUbF4q49W/c+Tdm2llo64HA4Zx7ybnHjwXXxnW/ncLK6tr6RnGztLW9s7tX3j9o6ihRDBssEpFq+1Sj4BIbhhuB7VghDX2BLX98k/mtR1SaR/LeTGLshXQoecAZNVZ66IbUjPwgfZr2yxW36s5AlomXkwrkqPfLX91BxJIQpWGCat3x3Nj0UqoMZwKnpW6iMaZsTIfYsVTSEHUvnSWekhOrDEgQKfukITP190ZKQ60noW8ns4R60cvE/7xOYoKrXsplnBiUbP5RkAhiIpKdTwZcITNiYgllitushI2ooszYkkq2BG/x5GXSPKt651X37qJSu87rKMIRHMMpeHAJNbiFOjSAgYRneIU3RzsvzrvzMR8tOPnOIfyB8/kD/eORHg==</latexit> GLM basis Regression coefficient
  8. Sparse Representation in CS bn <latexit sha1_base64="+PJYnVb53ACFuJdwoQMCxK7vOoI=">AAAB83icbVDLSsNAFL2pr1pfVZduBovgqiQq6LLoxmUF+4AmlMn0ph06mYSZiVBCf8ONC0Xc+jPu/BunbRbaemDgcM693DMnTAXXxnW/ndLa+sbmVnm7srO7t39QPTxq6yRTDFssEYnqhlSj4BJbhhuB3VQhjUOBnXB8N/M7T6g0T+SjmaQYxHQoecQZNVby/ZiaURjl4bQv+9WaW3fnIKvEK0gNCjT71S9/kLAsRmmYoFr3PDc1QU6V4UzgtOJnGlPKxnSIPUsljVEH+TzzlJxZZUCiRNknDZmrvzdyGms9iUM7Ocuol72Z+J/Xy0x0E+RcpplByRaHokwQk5BZAWTAFTIjJpZQprjNStiIKsqMraliS/CWv7xK2hd177LuPlzVGrdFHWU4gVM4Bw+uoQH30IQWMEjhGV7hzcmcF+fd+ViMlpxi5xj+wPn8AWXkkek=</latexit> hx, e bn

    i <latexit sha1_base64="U7NhedCxI11UMRr+85PC6B6cwUI=">AAACGXicbVDLSsNAFJ34rPVVdelmsAgupCQq6LLoxmUF+4AmhMnkph06mYSZiVpCf8ONv+LGhSIudeXfOG2z0NYDA4dzzmXuPUHKmdK2/W0tLC4tr6yW1srrG5tb25Wd3ZZKMkmhSROeyE5AFHAmoKmZ5tBJJZA44NAOBldjv30HUrFE3OphCl5MeoJFjBJtJL9iu5yIHgc3JrofRPnD6Ni9ZyFoxkPICxUHI1+4chL0K1W7Zk+A54lTkCoq0PArn26Y0CwGoSknSnUdO9VeTqRmlMOo7GYKUkIHpAddQwWJQXn55LIRPjRKiKNEmic0nqi/J3ISKzWMA5Mcr6pmvbH4n9fNdHTh5UykmQZBpx9FGcc6weOacMgkUM2HhhAqmdkV0z6RhGpTZtmU4MyePE9aJzXntGbfnFXrl0UdJbSPDtARctA5qqNr1EBNRNEjekav6M16sl6sd+tjGl2wipk99AfW1w9mz6HH</latexit> x <latexit sha1_base64="774qhuNAFXKctSHUINibxc5Dim4=">AAAB8nicbVBNS8NAFHypX7V+VT16WSyCp5KooMeiF48VbC20oWy2m3bpZhN2X8QS+jO8eFDEq7/Gm//GTZuDtg4sDDPvsfMmSKQw6LrfTmlldW19o7xZ2dre2d2r7h+0TZxqxlsslrHuBNRwKRRvoUDJO4nmNAokfwjGN7n/8Mi1EbG6x0nC/YgOlQgFo2ilbi+iOArC7GlK+tWaW3dnIMvEK0gNCjT71a/eIGZpxBUySY3pem6CfkY1Cib5tNJLDU8oG9Mh71qqaMSNn80iT8mJVQYkjLV9CslM/b2R0ciYSRTYyTyiWfRy8T+vm2J45WdCJSlyxeYfhakkGJP8fjIQmjOUE0so08JmJWxENWVoW6rYErzFk5dJ+6zundfdu4ta47qoowxHcAyn4MElNOAWmtACBjE8wyu8Oei8OO/Ox3y05BQ7h/AHzucPV6aRSA==</latexit> X n <latexit sha1_base64="eQZvkOUKW8DFp/whBQaQiuX1XSc=">AAAB7XicbVDLSgNBEOyNrxhfUY9eBoPgKexqQI9BLx4jmAckS5idzCZj5rHMzAphyT948aCIV//Hm3/jJNmDJhY0FFXddHdFCWfG+v63V1hb39jcKm6Xdnb39g/Kh0cto1JNaJMornQnwoZyJmnTMstpJ9EUi4jTdjS+nfntJ6oNU/LBThIaCjyULGYEWye1eiYVfdkvV/yqPwdaJUFOKpCj0S9/9QaKpIJKSzg2phv4iQ0zrC0jnE5LvdTQBJMxHtKuoxILasJsfu0UnTllgGKlXUmL5urviQwLYyYicp0C25FZ9mbif143tfF1mDGZpJZKslgUpxxZhWavowHTlFg+cQQTzdytiIywxsS6gEouhGD55VXSuqgGl1X/vlap3+RxFOEETuEcAriCOtxBA5pA4BGe4RXePOW9eO/ex6K14OUzx/AH3ucPtu+PNg==</latexit> basis Wavelet basis Learned Dictionary Sparse coefficient
  9. Sparse Representation in CS k-space

  10. = X i=1 nbn <latexit sha1_base64="hF8YSuka+Y36npfrys6bhoSxFsk=">AAACCHicbVDLSgMxFM34rPVVdenCYBFclRkVdFMounFZwT6gU4YkzbShSWZIMkIZZunGX3HjQhG3foI7/8ZMOwttPRA4nHMuuffgmDNtXPfbWVpeWV1bL22UN7e2d3Yre/ttHSWK0BaJeKS6GGnKmaQtwwyn3VhRJDCnHTy+yf3OA1WaRfLeTGLaF2goWcgIMlYKKkd1XyciSFndy3xMDQok9AUyIxymOAtsourW3CngIvEKUgUFmkHlyx9EJBFUGsKR1j3PjU0/RcowwmlW9hNNY0TGaEh7lkokqO6n00MyeGKVAQwjZZ80cKr+nkiR0HoisE3mO+p5Lxf/83qJCa/6KZNxYqgks4/ChEMTwbwVOGCKEsMnliCimN0VkhFSiBjbXdmW4M2fvEjaZzXvvObeXVQb10UdJXAIjsEp8MAlaIBb0AQtQMAjeAav4M15cl6cd+djFl1yipkD8AfO5w/SO5nZ</latexit> Low-rank Representation of Dynamic

    MRI x <latexit sha1_base64="xEIMch3yuo7JxT4Wy1udoMzDhIk=">AAAB8XicbVDLSgMxFL1TX7W+qi7dBIvgqsyooMuiG5cV7APbUjLpnTY0kxmSjFiG/oUbF4q49W/c+Tdm2llo64HA4Zx7ybnHjwXXxnW/ncLK6tr6RnGztLW9s7tX3j9o6ihRDBssEpFq+1Sj4BIbhhuB7VghDX2BLX98k/mtR1SaR/LeTGLshXQoecAZNVZ66IbUjPwgfZr2yxW36s5AlomXkwrkqPfLX91BxJIQpWGCat3x3Nj0UqoMZwKnpW6iMaZsTIfYsVTSEHUvnSWekhOrDEgQKfukITP190ZKQ60noW8ns4R60cvE/7xOYoKrXsplnBiUbP5RkAhiIpKdTwZcITNiYgllitushI2ooszYkkq2BG/x5GXSPKt651X37qJSu87rKMIRHMMpeHAJNbiFOjSAgYRneIU3RzsvzrvzMR8tOPnOIfyB8/kD/eORHg==</latexit> PCA basis = low-rank basis basis X <latexit sha1_base64="mRxQ3nFrUAg1mos1/MJ2As7dETY=">AAAB63icbVDLSgNBEOyNrxhfUY9eBoPgKeyqoMegF48RzAOSJcxOZpMhM7PLPISw5Be8eFDEqz/kzb9xNtmDJhY0FFXddHdFKWfa+P63V1pb39jcKm9Xdnb39g+qh0dtnVhFaIskPFHdCGvKmaQtwwyn3VRRLCJOO9HkLvc7T1RplshHM01pKPBIspgRbHKpr60YVGt+3Z8DrZKgIDUo0BxUv/rDhFhBpSEca90L/NSEGVaGEU5nlb7VNMVkgke056jEguowm986Q2dOGaI4Ua6kQXP190SGhdZTEblOgc1YL3u5+J/Xsya+CTMmU2uoJItFseXIJCh/HA2ZosTwqSOYKOZuRWSMFSbGxVNxIQTLL6+S9kU9uKz7D1e1xm0RRxlO4BTOIYBraMA9NKEFBMbwDK/w5gnvxXv3PhatJa+YOYY/8D5/ADKVjlU=</latexit>             coefficient
  11. Motion Guided Low-Rank MRI Singular value : Low-rank Yoon et

    al, TMI, 2014
  12. T -T 0 n1 - n1 0 Structured Low-Rank MRI

    (ALOHA) = 0 RankH(f) = k Lee et al, MRM, 2016; Jin et al, TCI, 2016; Jin et al, MRM, 2017; Ye et al, TIT, 2017; Ongie, SIIMS, 2017; Haldar, TMI, 2015
  13. ALOHA for Parallel MRI Lee et al, MRM, 2016; Jin

    et al, TCI 2017
  14. Parallel MRI Jin et al, TCI, 2017

  15. ALOHA for Dynamic MRI Lee et al, MRM, 2016; Jin

    et al, TCI, 2017
  16. Six fold (x6) down sampling # of coils =4 Dynamic

    MRI Jin et al, TCI, 2017
  17. ALOHA, Signal Representation? Yes ! Jin et al, TIP, 2015

  18. LOW RANK+SPARSE SIGNAL REPRESENTATION

  19. Motivation q Sparse representation, low rank ü Sparse combination of

    analytic basis or subspace ü Top-down mathematical modeling q Real world signal ü Many outliers from the global signal modeling à IS THERE SIGNAL REPRESENTATION TO DEAL WITH OUTLIERS ?
  20. Low Rank plus Sparse (L+S) MRI + k t Otazo

    R et al. MRM 2015 L S Courtesy of Ricardo Otazo
  21. L S k t Low Rank plus Sparse (L+S) MRI

    Otazo R et al. MRM 2015 Courtesy of Ricardo Otazo
  22. L+S time-resolved MRA (4D imaging) • 7.5-fold acceleration – ky

    -kz -t random undersampling S L L+S CS CS uses conventional data subtraction from pre-contrast reference Automatic and improved background suppression Otazo R et al. MRM 2015 Courtesy of Ricardo Otazo
  23. L+S abdominal radial DCE-MRI (4D imaging) S • Continuous golden-angle

    radial acquisition • Only 8 spokes/temporal frame – 48-fold acceleration L L+S Compressed sensing (S-only) Otazo R et al. MRM 2015 Courtesy of Ricardo Otazo
  24. Motion-guided L+S model W1 W2 W3 W =[ ] (frame-by-frame

    motion operator) L+S = W(M) Courtesy of Ricardo Otazo
  25. Free-breathing 8-fold accelerated cardiac perfusion Siemens 3T scanner, TurboFLASH sequence

    with 8-fold acceleration Temporal resolution = 60ms, Spatial resolution = 1.7x1.7 mm2 L+S L S Standard Motion-guided Otazo R et al. ISMRM 2014 Courtesy of Ricardo Otazo
  26. L+S L S Standard Motion-guided Free-breathing 8-fold accelerated cardiac perfusion

    Siemens 3T scanner, TurboFLASH sequence with 8-fold acceleration Temporal resolution = 60ms, Spatial resolution = 1.7x1.7 mm2 Otazo R et al. ISMRM 2014 Courtesy of Ricardo Otazo
  27. * Sparse outlier is still sparse in weighted Hankel matrix

    Structured L+S Representation: Robust ALOHA Jin et al, MRM, 2017
  28. Some MR artifacts are sparsely represented in k-space Application: MR

    Artifacts Removal Jin et al, MRM, 2017
  29. Robust ALOHA for Artifact Removal K-space weighting Jin et al,

    MRM, 2017; Jin et al, TIP, 2018
  30. Method Flowchart Jin et al, MRM, 2017

  31. High intensity Spike noise Low intensity Spike noise (low frequency

    region) High intensity Spike noise with down sampling (x5) Numerical Study Jin et al, MRM, 2017
  32. Singular value distribution of Hankel matrix Jin et al, MRM,

    2017
  33. 2-D herringbone artifacts Jin et al, MRM, 2017

  34. Instructed motion (3 times) Motion artifact Jin et al, MRM,

    2017
  35. Motion artifact Jin et al, MRM, 2017

  36. Zipper artifact Jin et al, MRM, 2017

  37. 2-D herringbone (in-vivo) before after Jin et al, MRM, 2017

  38. 2-D herringbone (in-vivo) Jin et al, MRM, 2017

  39. L+S Representation for MRI q More realistic signal representation in

    MRI by considering outliers q More flexible model than CS, low rank q Applications: ü accelerated MRI, artifact correction q Limitation ü Computational complexity ü Still top-down basis engineering à IS THERE ULTIMATE SIGNAL REPRESENTATION ?
  40. Ultimate Signal Representation for MR ? = <latexit sha1_base64="2wsinhV7OEj9020G2B+xBypL2+k=">AAAB6HicbVBNS8NAEJ34WetX1aOXxSJ4KokKehGKXjy2YD+gDWWznbRrN5uwuxFK6C/w4kERr/4kb/4bt20O2vpg4PHeDDPzgkRwbVz321lZXVvf2CxsFbd3dvf2SweHTR2nimGDxSJW7YBqFFxiw3AjsJ0opFEgsBWM7qZ+6wmV5rF8MOME/YgOJA85o8ZK9ZteqexW3BnIMvFyUoYctV7pq9uPWRqhNExQrTuemxg/o8pwJnBS7KYaE8pGdIAdSyWNUPvZ7NAJObVKn4SxsiUNmam/JzIaaT2OAtsZUTPUi95U/M/rpCa89jMuk9SgZPNFYSqIicn0a9LnCpkRY0soU9zeStiQKsqMzaZoQ/AWX14mzfOKd1Fx65fl6m0eRwGO4QTOwIMrqMI91KABDBCe4RXenEfnxXl3PuatK04+cwR/4Hz+AI13jMM=</latexit> x

    = X hx, e bn ibn <latexit sha1_base64="tCLG2nbXwywzwFFUoqMNNZJmas8=">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</latexit> basis 1 1 x coefficient
  41. = <latexit sha1_base64="2wsinhV7OEj9020G2B+xBypL2+k=">AAAB6HicbVBNS8NAEJ34WetX1aOXxSJ4KokKehGKXjy2YD+gDWWznbRrN5uwuxFK6C/w4kERr/4kb/4bt20O2vpg4PHeDDPzgkRwbVz321lZXVvf2CxsFbd3dvf2SweHTR2nimGDxSJW7YBqFFxiw3AjsJ0opFEgsBWM7qZ+6wmV5rF8MOME/YgOJA85o8ZK9ZteqexW3BnIMvFyUoYctV7pq9uPWRqhNExQrTuemxg/o8pwJnBS7KYaE8pGdIAdSyWNUPvZ7NAJObVKn4SxsiUNmam/JzIaaT2OAtsZUTPUi95U/M/rpCa89jMuk9SgZPNFYSqIicn0a9LnCpkRY0soU9zeStiQKsqMzaZoQ/AWX14mzfOKd1Fx65fl6m0eRwGO4QTOwIMrqMI91KABDBCe4RXenEfnxXl3PuatK04+cwR/4Hz+AI13jMM=</latexit> x = X hx, e bn ibn

    <latexit sha1_base64="tCLG2nbXwywzwFFUoqMNNZJmas8=">AAACOnicbVDLSsNAFJ34rPVVdelmsAgupCQq6EYounHZgn1AU8pkctMOnUzCzEQtod/lxq9w58KNC0Xc+gFO2oBaPTBwOOdc5t7jxZwpbdtP1tz8wuLScmGluLq2vrFZ2tpuqiiRFBo04pFse0QBZwIammkO7VgCCT0OLW94mfmtG5CKReJaj2LohqQvWMAo0UbqlepuSPTAC9K7MT53VRJilxPR54C/jUPs3jIfNOM+pLmMvXFPuPJ3NNNwr1S2K/YE+C9xclJGOWq90qPrRzQJQWjKiVIdx451NyVSM8phXHQTBTGhQ9KHjqGChKC66eT0Md43io+DSJonNJ6oPydSEio1Cj2TzJZUs14m/ud1Eh2cdVMm4kSDoNOPgoRjHeGsR+wzCVTzkSGESmZ2xXRAJKHatF00JTizJ/8lzaOKc1yx6yfl6kVeRwHtoj10gBx0iqroCtVQA1F0j57RK3qzHqwX6936mEbnrHxmB/2C9fkFVQWu8g==</latexit> basis 1 1 x coefficient x = X hx, e bn(x)ibn(x) <latexit sha1_base64="erKoPFMSsbyGoza+nkJyT8tbnTk=">AAACdHicfVFNa9tAEF0pTeu4SeM2hx7awzbG4EIxUltoL4GQXnJ0of4Ay5jVauQsWa3E7qi1EfoF+Xe55WfkknNWjsC1XTqw8Hhv3szsTJhJYdDz7hx379n+8xeNg+bLw6NXx63Xb4YmzTWHAU9lqschMyCFggEKlDDONLAklDAKr39U+ug3aCNS9QuXGUwTNlciFpyhpWatmyBheBXGxaKkZ4HJExpIpuYS6Fr4RIM/IgIUMoKipmlYzlSAsMDVDIWGqCy6a8/HMtCbdSoD/a9j1mp7PW8VdBf4NWiTOvqz1m0QpTxPQCGXzJiJ72U4LZhGwSWUzSA3kDF+zeYwsVCxBMy0WLUvaccyEY1TbZ9CumL/dhQsMWaZhDazmtFsaxX5L22SY/x9WgiV5QiKPzWKc0kxpdUFaCQ0cJRLCxjXws5K+RXTjKO9U9Muwd/+8i4Yfu75X3rez6/t84t6HQ3yjpySLvHJN3JOLkmfDAgn985bhzofnAf3vdt2O0+prlN7TshGuL1HHSHCMQ==</latexit> Ideal basis should be adaptive to the input in real-time Ultimate Signal Representation for MR ?
  42. Deep Learning toward an Ultimate Signal Representation y = X

    i h ,bi(x)ie bi(x) <latexit sha1_base64="wvdFNgdWBgyp03OsXJvyc2GFH4c=">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</latexit> • Deep learning is a signal representation with automatic input adaptivity. • Extension of classical regression, CS, L+S, ALOHA, etc • More training data gives better representation à Plenary Talk on 11:00-11:20, Tues, 14th May
  43. math Thank You