ee123 digital signal processingee123/sp16/notes/lecture...m. lustig, eecs uc berkeley sensing matrix...
TRANSCRIPT
![Page 1: EE123 Digital Signal Processingee123/sp16/Notes/Lecture...M. Lustig, EECS UC Berkeley sensing matrix Compressed Sensing ! ! (Candes, Romber, Tao 2006; Donoho 2006) • x∈ℜN is](https://reader036.vdocuments.us/reader036/viewer/2022071106/5fe01b06d8c13357ac142132/html5/thumbnails/1.jpg)
M. Lustig, EECS UC Berkeley
EE123Digital Signal Processing
Lecture 27Compressed Sensing II
![Page 2: EE123 Digital Signal Processingee123/sp16/Notes/Lecture...M. Lustig, EECS UC Berkeley sensing matrix Compressed Sensing ! ! (Candes, Romber, Tao 2006; Donoho 2006) • x∈ℜN is](https://reader036.vdocuments.us/reader036/viewer/2022071106/5fe01b06d8c13357ac142132/html5/thumbnails/2.jpg)
M. Lustig, EECS UC Berkeley
From Samples to Measurements
• Shanon-Nyquist sampling–Worst case for ANY bandlimited data
• Compressive sampling (CS)! “Sparse signals statistics can be recovered from a
small number of non-adaptive linear measurements”
–Integrated sensing, compression and processing.–Based on concepts of incoherency between signal and
measurements
![Page 3: EE123 Digital Signal Processingee123/sp16/Notes/Lecture...M. Lustig, EECS UC Berkeley sensing matrix Compressed Sensing ! ! (Candes, Romber, Tao 2006; Donoho 2006) • x∈ℜN is](https://reader036.vdocuments.us/reader036/viewer/2022071106/5fe01b06d8c13357ac142132/html5/thumbnails/3.jpg)
M. Lustig, EECS UC Berkeley
• x∈ℜN is a signal • Make N linear measurements
N N
Traditional Sensing
xy
=
Φ
sensing matrix
NxN
Desktop scanner/ digital camera sensing
![Page 4: EE123 Digital Signal Processingee123/sp16/Notes/Lecture...M. Lustig, EECS UC Berkeley sensing matrix Compressed Sensing ! ! (Candes, Romber, Tao 2006; Donoho 2006) • x∈ℜN is](https://reader036.vdocuments.us/reader036/viewer/2022071106/5fe01b06d8c13357ac142132/html5/thumbnails/4.jpg)
M. Lustig, EECS UC Berkeley
• x∈ℜN is a signal • Make N linear measurements
N N
Traditional Sensing
xy
=
Φ
sensing matrix
NxN
MRI Fourier Imaging
![Page 5: EE123 Digital Signal Processingee123/sp16/Notes/Lecture...M. Lustig, EECS UC Berkeley sensing matrix Compressed Sensing ! ! (Candes, Romber, Tao 2006; Donoho 2006) • x∈ℜN is](https://reader036.vdocuments.us/reader036/viewer/2022071106/5fe01b06d8c13357ac142132/html5/thumbnails/5.jpg)
M. Lustig, EECS UC Berkeley
• x∈ℜN is a signal • Make N linear measurements
N N
Traditional Sensing
xy
=
Φ
sensing matrix
NxN
A “good” sensing matrix is orthogonal
Φ* Φ = I
Arbitrary sensing
![Page 6: EE123 Digital Signal Processingee123/sp16/Notes/Lecture...M. Lustig, EECS UC Berkeley sensing matrix Compressed Sensing ! ! (Candes, Romber, Tao 2006; Donoho 2006) • x∈ℜN is](https://reader036.vdocuments.us/reader036/viewer/2022071106/5fe01b06d8c13357ac142132/html5/thumbnails/6.jpg)
M. Lustig, EECS UC Berkeley
sensing matrix
Compressed Sensing ! ! (Candes, Romber, Tao 2006; Donoho 2006)
• x∈ℜN is a K-sparse signal (K<<N)• Make M (K<M<<N) incoherent linear projections
x
=
Φ
MxNK
sensing matrix
A “good” compressed sensing matrix is incoherent i.e, approximately orthogonal
Φ* Φ ≈ I
Incoherency can preserve information
M
y
![Page 7: EE123 Digital Signal Processingee123/sp16/Notes/Lecture...M. Lustig, EECS UC Berkeley sensing matrix Compressed Sensing ! ! (Candes, Romber, Tao 2006; Donoho 2006) • x∈ℜN is](https://reader036.vdocuments.us/reader036/viewer/2022071106/5fe01b06d8c13357ac142132/html5/thumbnails/7.jpg)
M. Lustig, EECS UC Berkeley
CS recovery
• Given y = Φxfind x
• But there’s hope, x is sparse!
Under-determined
=
Φ xy
![Page 8: EE123 Digital Signal Processingee123/sp16/Notes/Lecture...M. Lustig, EECS UC Berkeley sensing matrix Compressed Sensing ! ! (Candes, Romber, Tao 2006; Donoho 2006) • x∈ℜN is](https://reader036.vdocuments.us/reader036/viewer/2022071106/5fe01b06d8c13357ac142132/html5/thumbnails/8.jpg)
M. Lustig, EECS UC Berkeley
CS recovery
• Given y = Φxfind x
• But there’s hope, x is sparse!
Under-determined
![Page 9: EE123 Digital Signal Processingee123/sp16/Notes/Lecture...M. Lustig, EECS UC Berkeley sensing matrix Compressed Sensing ! ! (Candes, Romber, Tao 2006; Donoho 2006) • x∈ℜN is](https://reader036.vdocuments.us/reader036/viewer/2022071106/5fe01b06d8c13357ac142132/html5/thumbnails/9.jpg)
M. Lustig, EECS UC Berkeley
CS recovery
• Given y = Φxfind x
• But there’s hope, x is sparse!
Under-determined
minimize ||x||2 s.t. y = Φx
WRONG!
![Page 10: EE123 Digital Signal Processingee123/sp16/Notes/Lecture...M. Lustig, EECS UC Berkeley sensing matrix Compressed Sensing ! ! (Candes, Romber, Tao 2006; Donoho 2006) • x∈ℜN is](https://reader036.vdocuments.us/reader036/viewer/2022071106/5fe01b06d8c13357ac142132/html5/thumbnails/10.jpg)
M. Lustig, EECS UC Berkeley
CS recovery
• Given y = Φxfind x
• But there’s hope, x is sparse!
Under-determined
minimize ||x||0 s.t. y = Φx
HARD!
![Page 11: EE123 Digital Signal Processingee123/sp16/Notes/Lecture...M. Lustig, EECS UC Berkeley sensing matrix Compressed Sensing ! ! (Candes, Romber, Tao 2006; Donoho 2006) • x∈ℜN is](https://reader036.vdocuments.us/reader036/viewer/2022071106/5fe01b06d8c13357ac142132/html5/thumbnails/11.jpg)
M. Lustig, EECS UC Berkeley
CS recovery
• Given y = Φxfind x
• But there’s hope, x is sparse!
Under-determined
minimize ||x||1 s.t. y = Φx
need M ≈ K log(N) <<NSolved by linear-programming
![Page 12: EE123 Digital Signal Processingee123/sp16/Notes/Lecture...M. Lustig, EECS UC Berkeley sensing matrix Compressed Sensing ! ! (Candes, Romber, Tao 2006; Donoho 2006) • x∈ℜN is](https://reader036.vdocuments.us/reader036/viewer/2022071106/5fe01b06d8c13357ac142132/html5/thumbnails/12.jpg)
M. Lustig, EECS UC Berkeley
minimum ||x||1 minimum ||x||2
Geometric Interpretation
domain of sparse signals
![Page 13: EE123 Digital Signal Processingee123/sp16/Notes/Lecture...M. Lustig, EECS UC Berkeley sensing matrix Compressed Sensing ! ! (Candes, Romber, Tao 2006; Donoho 2006) • x∈ℜN is](https://reader036.vdocuments.us/reader036/viewer/2022071106/5fe01b06d8c13357ac142132/html5/thumbnails/13.jpg)
M. Lustig, EECS UC Berkeley
A non-linear sampling theorem
• f∈CN supported on a set Ω in Fourier• Shannon:
– Ω is known connected set, size B– Exact recovery from B equispaced time samples– Linear reconstruction by sinc interpolation
• Non-linear sampling theorem– Ω is an arbitrary, unknown set of size B– Exact recovery from ~ B logN (almost) arbitrary placed samples– Nonlinear reconstruction by convex programming
![Page 14: EE123 Digital Signal Processingee123/sp16/Notes/Lecture...M. Lustig, EECS UC Berkeley sensing matrix Compressed Sensing ! ! (Candes, Romber, Tao 2006; Donoho 2006) • x∈ℜN is](https://reader036.vdocuments.us/reader036/viewer/2022071106/5fe01b06d8c13357ac142132/html5/thumbnails/14.jpg)
M. Lustig, EECS UC Berkeley
Practicality of CS
• Can such sensing system exist in practice?
Fourier matrix
![Page 15: EE123 Digital Signal Processingee123/sp16/Notes/Lecture...M. Lustig, EECS UC Berkeley sensing matrix Compressed Sensing ! ! (Candes, Romber, Tao 2006; Donoho 2006) • x∈ℜN is](https://reader036.vdocuments.us/reader036/viewer/2022071106/5fe01b06d8c13357ac142132/html5/thumbnails/15.jpg)
M. Lustig, EECS UC Berkeley
Practicality of CS
• Can such sensing system exist in practice?
Fourier matrix
![Page 16: EE123 Digital Signal Processingee123/sp16/Notes/Lecture...M. Lustig, EECS UC Berkeley sensing matrix Compressed Sensing ! ! (Candes, Romber, Tao 2006; Donoho 2006) • x∈ℜN is](https://reader036.vdocuments.us/reader036/viewer/2022071106/5fe01b06d8c13357ac142132/html5/thumbnails/16.jpg)
M. Lustig, EECS UC Berkeley
Practicality of CS
• Can such sensing system exist in practice?
Fourier matrix
![Page 17: EE123 Digital Signal Processingee123/sp16/Notes/Lecture...M. Lustig, EECS UC Berkeley sensing matrix Compressed Sensing ! ! (Candes, Romber, Tao 2006; Donoho 2006) • x∈ℜN is](https://reader036.vdocuments.us/reader036/viewer/2022071106/5fe01b06d8c13357ac142132/html5/thumbnails/17.jpg)
M. Lustig, EECS UC Berkeley
• Can such sensing system exist in practice?
• Randomly undersampled Fourier is incoherent
Φ* Φ ≈ I
Practicality of CS
=
• MRI samples in the Fourier domain!
![Page 18: EE123 Digital Signal Processingee123/sp16/Notes/Lecture...M. Lustig, EECS UC Berkeley sensing matrix Compressed Sensing ! ! (Candes, Romber, Tao 2006; Donoho 2006) • x∈ℜN is](https://reader036.vdocuments.us/reader036/viewer/2022071106/5fe01b06d8c13357ac142132/html5/thumbnails/18.jpg)
M. Lustig, EECS UC Berkeley
![Page 19: EE123 Digital Signal Processingee123/sp16/Notes/Lecture...M. Lustig, EECS UC Berkeley sensing matrix Compressed Sensing ! ! (Candes, Romber, Tao 2006; Donoho 2006) • x∈ℜN is](https://reader036.vdocuments.us/reader036/viewer/2022071106/5fe01b06d8c13357ac142132/html5/thumbnails/19.jpg)
Intuitive example of CS
![Page 20: EE123 Digital Signal Processingee123/sp16/Notes/Lecture...M. Lustig, EECS UC Berkeley sensing matrix Compressed Sensing ! ! (Candes, Romber, Tao 2006; Donoho 2006) • x∈ℜN is](https://reader036.vdocuments.us/reader036/viewer/2022071106/5fe01b06d8c13357ac142132/html5/thumbnails/20.jpg)
Intuitive example of CS
FFT
Nyquistsampling
![Page 21: EE123 Digital Signal Processingee123/sp16/Notes/Lecture...M. Lustig, EECS UC Berkeley sensing matrix Compressed Sensing ! ! (Candes, Romber, Tao 2006; Donoho 2006) • x∈ℜN is](https://reader036.vdocuments.us/reader036/viewer/2022071106/5fe01b06d8c13357ac142132/html5/thumbnails/21.jpg)
Intuitive example of CS
FFT
sub-Nyquistequispaced
![Page 22: EE123 Digital Signal Processingee123/sp16/Notes/Lecture...M. Lustig, EECS UC Berkeley sensing matrix Compressed Sensing ! ! (Candes, Romber, Tao 2006; Donoho 2006) • x∈ℜN is](https://reader036.vdocuments.us/reader036/viewer/2022071106/5fe01b06d8c13357ac142132/html5/thumbnails/22.jpg)
Intuitive example of CS
FFT
sub-Nyquist
Ambiguity
![Page 23: EE123 Digital Signal Processingee123/sp16/Notes/Lecture...M. Lustig, EECS UC Berkeley sensing matrix Compressed Sensing ! ! (Candes, Romber, Tao 2006; Donoho 2006) • x∈ℜN is](https://reader036.vdocuments.us/reader036/viewer/2022071106/5fe01b06d8c13357ac142132/html5/thumbnails/23.jpg)
Intuitive example of CS
FFT
sub-Nyquistrandom
![Page 24: EE123 Digital Signal Processingee123/sp16/Notes/Lecture...M. Lustig, EECS UC Berkeley sensing matrix Compressed Sensing ! ! (Candes, Romber, Tao 2006; Donoho 2006) • x∈ℜN is](https://reader036.vdocuments.us/reader036/viewer/2022071106/5fe01b06d8c13357ac142132/html5/thumbnails/24.jpg)
M. Lustig, EECS UC Berkeley
![Page 25: EE123 Digital Signal Processingee123/sp16/Notes/Lecture...M. Lustig, EECS UC Berkeley sensing matrix Compressed Sensing ! ! (Candes, Romber, Tao 2006; Donoho 2006) • x∈ℜN is](https://reader036.vdocuments.us/reader036/viewer/2022071106/5fe01b06d8c13357ac142132/html5/thumbnails/25.jpg)
Intuitive example of CS
FFT
sub-NyquistLooks like
“random noise”
![Page 26: EE123 Digital Signal Processingee123/sp16/Notes/Lecture...M. Lustig, EECS UC Berkeley sensing matrix Compressed Sensing ! ! (Candes, Romber, Tao 2006; Donoho 2006) • x∈ℜN is](https://reader036.vdocuments.us/reader036/viewer/2022071106/5fe01b06d8c13357ac142132/html5/thumbnails/26.jpg)
Intuitive example of CS
FFT
sub-NyquistBut it’s not
noise!
![Page 27: EE123 Digital Signal Processingee123/sp16/Notes/Lecture...M. Lustig, EECS UC Berkeley sensing matrix Compressed Sensing ! ! (Candes, Romber, Tao 2006; Donoho 2006) • x∈ℜN is](https://reader036.vdocuments.us/reader036/viewer/2022071106/5fe01b06d8c13357ac142132/html5/thumbnails/27.jpg)
M. Lustig, EECS UC Berkeley
![Page 28: EE123 Digital Signal Processingee123/sp16/Notes/Lecture...M. Lustig, EECS UC Berkeley sensing matrix Compressed Sensing ! ! (Candes, Romber, Tao 2006; Donoho 2006) • x∈ℜN is](https://reader036.vdocuments.us/reader036/viewer/2022071106/5fe01b06d8c13357ac142132/html5/thumbnails/28.jpg)
M. Lustig, EECS UC Berkeley
![Page 29: EE123 Digital Signal Processingee123/sp16/Notes/Lecture...M. Lustig, EECS UC Berkeley sensing matrix Compressed Sensing ! ! (Candes, Romber, Tao 2006; Donoho 2006) • x∈ℜN is](https://reader036.vdocuments.us/reader036/viewer/2022071106/5fe01b06d8c13357ac142132/html5/thumbnails/29.jpg)
Intuitive example of CS
FFT
Recovery
Example inspired by Donoho et. Al, 2007
![Page 30: EE123 Digital Signal Processingee123/sp16/Notes/Lecture...M. Lustig, EECS UC Berkeley sensing matrix Compressed Sensing ! ! (Candes, Romber, Tao 2006; Donoho 2006) • x∈ℜN is](https://reader036.vdocuments.us/reader036/viewer/2022071106/5fe01b06d8c13357ac142132/html5/thumbnails/30.jpg)
M. Lustig, EECS UC Berkeley
![Page 31: EE123 Digital Signal Processingee123/sp16/Notes/Lecture...M. Lustig, EECS UC Berkeley sensing matrix Compressed Sensing ! ! (Candes, Romber, Tao 2006; Donoho 2006) • x∈ℜN is](https://reader036.vdocuments.us/reader036/viewer/2022071106/5fe01b06d8c13357ac142132/html5/thumbnails/31.jpg)
Question!
• What if this was the signal?• Would CS still work?
sub-Nyquistrandom sub-Nyquist
![Page 32: EE123 Digital Signal Processingee123/sp16/Notes/Lecture...M. Lustig, EECS UC Berkeley sensing matrix Compressed Sensing ! ! (Candes, Romber, Tao 2006; Donoho 2006) • x∈ℜN is](https://reader036.vdocuments.us/reader036/viewer/2022071106/5fe01b06d8c13357ac142132/html5/thumbnails/32.jpg)
Domains in Compressed Sensing
Signal
SparseDomain
SamplingDomain
Not Sparse!
Sparse! incoherent
![Page 33: EE123 Digital Signal Processingee123/sp16/Notes/Lecture...M. Lustig, EECS UC Berkeley sensing matrix Compressed Sensing ! ! (Candes, Romber, Tao 2006; Donoho 2006) • x∈ℜN is](https://reader036.vdocuments.us/reader036/viewer/2022071106/5fe01b06d8c13357ac142132/html5/thumbnails/33.jpg)
MRI
Signal
Not Sparse!
Sparse! incoherent
SamplingDomainSignal
SparseDomain
![Page 34: EE123 Digital Signal Processingee123/sp16/Notes/Lecture...M. Lustig, EECS UC Berkeley sensing matrix Compressed Sensing ! ! (Candes, Romber, Tao 2006; Donoho 2006) • x∈ℜN is](https://reader036.vdocuments.us/reader036/viewer/2022071106/5fe01b06d8c13357ac142132/html5/thumbnails/34.jpg)
Acquired Data
Sparse “denoising”
Compressed SensingReconstruction
![Page 35: EE123 Digital Signal Processingee123/sp16/Notes/Lecture...M. Lustig, EECS UC Berkeley sensing matrix Compressed Sensing ! ! (Candes, Romber, Tao 2006; Donoho 2006) • x∈ℜN is](https://reader036.vdocuments.us/reader036/viewer/2022071106/5fe01b06d8c13357ac142132/html5/thumbnails/35.jpg)
Acquired Data
Sparse “denoising”
Compressed SensingReconstruction
![Page 36: EE123 Digital Signal Processingee123/sp16/Notes/Lecture...M. Lustig, EECS UC Berkeley sensing matrix Compressed Sensing ! ! (Candes, Romber, Tao 2006; Donoho 2006) • x∈ℜN is](https://reader036.vdocuments.us/reader036/viewer/2022071106/5fe01b06d8c13357ac142132/html5/thumbnails/36.jpg)
Acquired Data
Sparse “denoising”Undersampled Final Image
Compressed SensingReconstruction
Tutorial & code available at http://www.mlustig.com
![Page 37: EE123 Digital Signal Processingee123/sp16/Notes/Lecture...M. Lustig, EECS UC Berkeley sensing matrix Compressed Sensing ! ! (Candes, Romber, Tao 2006; Donoho 2006) • x∈ℜN is](https://reader036.vdocuments.us/reader036/viewer/2022071106/5fe01b06d8c13357ac142132/html5/thumbnails/37.jpg)
6 year old male abdomen. Fine structures (arrows) are buried in noise (artifactual + noise amplification) and are recovered by CS with L1-wavelets. x8 acceleration
liver
Linear Reconstruction Compressed sensing
portal vein
hepatic vein
![Page 38: EE123 Digital Signal Processingee123/sp16/Notes/Lecture...M. Lustig, EECS UC Berkeley sensing matrix Compressed Sensing ! ! (Candes, Romber, Tao 2006; Donoho 2006) • x∈ℜN is](https://reader036.vdocuments.us/reader036/viewer/2022071106/5fe01b06d8c13357ac142132/html5/thumbnails/38.jpg)
6 year old male abdomen. Fine structures (arrows) are buried in noise (artifactual + noise amplification) and are recovered by CS with L1-wavelets.
liver
Linear Reconstruction Compressed sensing
portal veinnot seen
Zoom Zoom
Hepatic vein
bile duct
pancreatic duct
![Page 39: EE123 Digital Signal Processingee123/sp16/Notes/Lecture...M. Lustig, EECS UC Berkeley sensing matrix Compressed Sensing ! ! (Candes, Romber, Tao 2006; Donoho 2006) • x∈ℜN is](https://reader036.vdocuments.us/reader036/viewer/2022071106/5fe01b06d8c13357ac142132/html5/thumbnails/39.jpg)
Back to Results6yearold8-foldacceleration16secondscan0.875mmin-plane1.6slicethickness
![Page 40: EE123 Digital Signal Processingee123/sp16/Notes/Lecture...M. Lustig, EECS UC Berkeley sensing matrix Compressed Sensing ! ! (Candes, Romber, Tao 2006; Donoho 2006) • x∈ℜN is](https://reader036.vdocuments.us/reader036/viewer/2022071106/5fe01b06d8c13357ac142132/html5/thumbnails/40.jpg)
Principles of Magnetic Resonance Imaging
EE c225E / BIOE c265
Shameless Promotion
Spring 2016
![Page 41: EE123 Digital Signal Processingee123/sp16/Notes/Lecture...M. Lustig, EECS UC Berkeley sensing matrix Compressed Sensing ! ! (Candes, Romber, Tao 2006; Donoho 2006) • x∈ℜN is](https://reader036.vdocuments.us/reader036/viewer/2022071106/5fe01b06d8c13357ac142132/html5/thumbnails/41.jpg)
Other Applications
• Compressive Imaging• Medical Imaging• Analog to information conversion• Biosensing• Geophysical Data Analysis• Compressive Radar• Astronomy• Communications• More ......
![Page 42: EE123 Digital Signal Processingee123/sp16/Notes/Lecture...M. Lustig, EECS UC Berkeley sensing matrix Compressed Sensing ! ! (Candes, Romber, Tao 2006; Donoho 2006) • x∈ℜN is](https://reader036.vdocuments.us/reader036/viewer/2022071106/5fe01b06d8c13357ac142132/html5/thumbnails/42.jpg)
Resources• CS + parallel imaging matlab code, examples
http://www.eecs.berkeley.edu/~mlustig/software/
• Rice University CS page: papers, tutorials, codes, ….http://www.dsp.ece.rice.edu/cs/
• IEEE Signal Processing Magazine, special issue on compressive sampling 2008;25(2)
• March 2010 Issue Wired Magazine: “Filling the Blanks”
• Igor Caron Blog: http://nuit-blanche.blogspot.com/
הברהדותThank you!