BM3D-prGAMP: Compressive Phase Retrieval Based …cam6.web.rice.edu › talks ›...
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BM3D-prGAMP: Compressive
Phase Retrieval Based on
BM3D Denoising
Chris Metzler, Richard Baraniuk
Rice University
Arian Maleki
Columbia University
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Phase Retrieval
Applications:
• Crystallography
• Microscopy
• Ptychography
• Astronomical Imaging
• Compressive Imaging
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Motivational Setup
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Measurement Process
Scattering Medium
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Reconstruction: Phase Retrieval
Phase Retrieval Alg
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Phase Retrieval:
Conventional:• Gerchberg [Gerchberg 72]
• Fienup [Fienup 78]
• Griffin-Lim [Griffin and Lim 84]
• PhaseLift [Candes and Eldar 15]
• PhaseCut [Waldspurger and Mallat 15]
• WirtingerFlow [Candes and Soltanokotabi
15]
• prVBEM [Drémeau and Krzakala 14]
• prGAMP [Schniter and Rangan 15]
Compressive:
• CPR [Moravec et al. 07]
• Sparse-Fienup [Mukherjee and
Seelamantula 14]
• GESPAR [Shechtman et al. 14]
• CPRL [Ohlsson et al. 12]
• TSPR [Jaganathan et al. 13]
• prGAMP [Schniter and Rangan 15]
• prSAMP [Rajaie et al. 2016]
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Phase Retrieval:
Conventional:• Gerchberg [Gerchberg 72]
• Fienup [Fienup 78]
• Griffin-Lim [Griffin and Lim 84]
• PhaseLift [Candes and Eldar 15]
• PhaseCut [Waldspurger and Mallat 15]
• WirtingerFlow [Candes and Soltanokotabi
15]
• prVBEM [Drémeau and Krzakala 14]
• prGAMP [Schniter and Rangan 15]
Compressive:
• CPR [Moravec et al. 07]
• Sparse-Fienup [Mukherjee and
Seelamantula 14]
• GESPAR [Shechtman et al. 14]
• CPRL [Ohlsson et al. 12]
• TSPR [Jaganathan et al. 13]
• prGAMP [Schniter and Rangan 15]
• prSAMP [Rajaie et al. 2016]
Sparsity
and
other
simple
priors
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Images are not Sparse
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Insight: Denoisers Impose Priors
• Gaussian Kernel• Smooth
• Soft Wavelet Thresholding [Donoho and Johnstone 94]
• Wavelet Sparse
• BLS-GSM [Portilla et al. 03]
• Coefficients follow GMM
• NLM [Baudes et al. 05]
• Correlated structures
• BM3D [Dabov et al. 07]
• Group-sparse in DCT/Wavelet representation
To solve…
• Super-resolution [Danielyan et al.
2010]
• Compressive Sensing [Danielyan
et al. 2010]
• Tomography [Venkatakrishnan et al.
2014]
• Deblurring [Heide et al. 2014]
• Inpainting [Sreehari et al. 2015]
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This Talk
• Use denoisers to solve compressive phase retrieval
• Demonstrate state-of-the-art performance• Comparable run-times
• Robust and stable
• ½ as many measurements required
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The Evolution of D-prGAMP
Approximate Message Passing (AMP)
Iterative Shrinkage/Thresholding (IST)
Generalized AMP (GAMP)
Phase Retrieval GAMP (prGAMP)
Denoising-based prGAMP (D-prGAMP)
[Blumensath and Davies 09]
[Donoho et al. 09]
[Rangan 10]
[Schnitter and Rangan 15]
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Denoisers as Black Boxes
Denoiser
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Denoisers as Projections
C
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Understanding D-prGAMP
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C
Understanding D-prGAMP
Our prior on x
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C
Understanding D-prGAMP
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C
Understanding D-prGAMP
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C
Understanding D-prGAMP
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C
Understanding D-prGAMP
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Onsager Correction:
• Where did it come from?• Approximation of message passing algorithm
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Onsager Correction:
• Where did it come from?• Approximation of message passing algorithm
• Why does it help?• st stores residuals over many iterations (momentum)
• Corrects for bias in denoiser solutions
• Makes errors uncorrelated (Gaussian) and thus easy to remove
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Onsager Correction:
• Where did it come from?• Approximation of message passing algorithm
• Why does it help?• st stores residuals over many iterations (momentum)
• Corrects for bias in denoiser solutions
• Makes errors uncorrelated (Gaussian) and thus easy to remove
• How is it calculated?• Approximation from Monte Carlo SURE [Ramani et al. 08]
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60% Under-sampled Gaussian Measurements
prGAMP (db4) BM3D-prGAMP
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100% Masked Fourier Measurements
prGAMP (db4) BM3D-prGAMP
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Performance Low Noise
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Performance High Noise
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Computation Times
0
5
10
15
20
25
30
35
40
45
50
40% 80% 100% 200% 400% 600%
Min
ute
s
Sampling Rate
prGAMP Conventional
prGAMP Compressive
BM3D-prGAMP
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D-prGAMP Summary
• Plug & play method to impose priors
• Imaging: BM3D > Wavelet Sparsity
• Efficient and scalable
• Robust to noise
• ½ as many measurements required