BM3D-prGAMP: Compressive

Phase Retrieval Based on

BM3D Denoising

Chris Metzler, Richard Baraniuk

Rice University

Arian Maleki

Columbia University

Phase Retrieval

Applications:

• Crystallography

• Microscopy

• Ptychography

• Astronomical Imaging

• Compressive Imaging

Motivational Setup

Measurement Process

Scattering Medium

Reconstruction: Phase Retrieval

Phase Retrieval Alg

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]

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

Images are not Sparse

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]

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

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]

Denoisers as Black Boxes

Denoiser

Denoisers as Projections

C

Understanding D-prGAMP

C

Understanding D-prGAMP

Our prior on x

C

Understanding D-prGAMP

C

Understanding D-prGAMP

C

Understanding D-prGAMP

C

Understanding D-prGAMP

Onsager Correction:

• Where did it come from?• Approximation of message passing algorithm

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

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]

60% Under-sampled Gaussian Measurements

prGAMP (db4) BM3D-prGAMP

100% Masked Fourier Measurements

prGAMP (db4) BM3D-prGAMP

Performance Low Noise

Performance High Noise

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

D-prGAMP Summary

• Plug & play method to impose priors

• Imaging: BM3D > Wavelet Sparsity

• Efficient and scalable

• Robust to noise

• ½ as many measurements required