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Learning to Solve Inverse
Problems in Imaging
Rebecca Willett, University of Chicago
Greg Ongie,

UChicago
Davis Gilton, 

UW-Madison
β
Inverse problems in imaging
• Inpainting
• Deblurring
• Superresolution
• Compressed
Sensing
• MRI
• Radar
Observe: y = Xβ + ε
Goal: Recover β from y
y
Classical approach: Tikhonov regularization (1943)
• Example: deblurring
• Least squares solution:
Wildly	Different	Solutions
Blurred	image	with	noiseBlurred	image
Deblurred the	“naïve”	way Deblurred the	“naïve”	way
ˆβ = (X X) 1
X y<latexit 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Wildly different solutions
Classical approach: Tikhonov regularization (1943)
• Example: deblurring
• Least squares solution:
Wildly	Different	Solutions
Blurred	image	with	noiseBlurred	image
Deblurred the	“naïve”	way Deblurred the	“naïve”	way
• Tikhonov regularization

(aka “ridge regression”)
better conditioned; suppresses noise
ˆβ = arg min
β
y Xβ 2
2 + λ β 2
2
=(X X + λI) 1
X y<latexit 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ˆβ = (X X) 1
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Wildly different solutions
Classical approach: Tikhonov regularization (1943)
• Example: deblurring
• Least squares solution:
Wildly	Different	Solutions
Blurred	image	with	noiseBlurred	image
Deblurred the	“naïve”	way Deblurred the	“naïve”	way
• Tikhonov regularization

(aka “ridge regression”)
better conditioned; suppresses noise
ˆβ = arg min
β
y Xβ 2
2 + λ β 2
2
=(X X + λI) 1
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ˆβ = (X X) 1
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Wildly different solutions
Tikhonov regularization
Geometric models of images
Total variation
Noisy
Patches
Denoised
Patches
Patch
Denoising
Combine to
estimate
denoised
pixel
Patch subspaces and manifolds
(Wavelet) sparsity
Regularization in inverse problems
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sha1_base64="3qRIWc7pIXZR0YKMqzYlnbgFQBU=">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</latexit>
ˆβ = arg min
β
y Xβ 2
2 + r(β)
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Regularization in inverse problems
y ˆβ<latexit 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ˆβ = arg min
β
y Xβ 2
2 + r(β)
<latexit 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Classical: r(β) is a pre-defined
smoothness-promoting regularizer 

(e.g. Tikhinov or ridge estimation)
Geometric: r(β) reflects image geometry 

(e.g. sparsity, patch redundancy, total variation)
Learned: use training data to learn r(β)
Examples in recent literature
• Deep CNN’s for signal recovery
• Compressed sensing with GANs
• Unrolled algorithms for solving inverse problems
• Deep proximal gradient descent nets
• Deep ADMM nets
• Deep half-quadratic splitting
• Deep primal-dual nets
Chen, Yu, Pock, 2015
Sun, Li, Xu, 2016
Chang, Li, Poczos, Kumar, Sankaranarayanan, 2017
Adler and Öktem, 2018
Bora, Jalal, Price, Dimakis, 2017
Mousavi and Baraniuk, 2017
Dong, Loy, He, Tang, 2014
Mardani et al, 2018
Jin, McCann, Froustey, Unser, 2017
Zhang, Zuo, Gu, Zhang, 2017
Ye, Han, Cha, 2018
Classes of methods
Model Agnostic 

(Ignore X)
Decoupled 

(First learn, then reconstruct)
Unrolled Optimization
Neumann Networks 

(this talk!)
X-1~
raw data
(low resolution image)
approximate
high-resolution
image
Train deep CNN
to remove artifacts
reconstruction
blurry/blocky
artifacts due to
re-scaling
bicubic
interpolation
Pictures from: http://webdav.tuebingen.mpg.de/pixel/enhancenet/
Super-resolution with CNNs
Model Agnostic 

(Ignore X)
Classes of methods
Model Agnostic 

(Ignore X)
Decoupled 

(First learn, then reconstruct)
Unrolled Optimization
Neumann Networks 

(this talk!)
GANs for inverse problems
y ˆβ<latexit 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ˆβ = arg min
β
y Xβ 2
2 + r(β)
<latexit 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sha1_base64="/+r4aBrVfxqpWL/MVPFAYn0tVFs=">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</latexit><latexit sha1_base64="/+r4aBrVfxqpWL/MVPFAYn0tVFs=">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</latexit><latexit 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r(β) =
0, β on image manifold
, otherwise<latexit 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“Good” image on manifold
“Bad” image off manifold
Decoupled 

(First learn, then
reconstruct)
Solving Inverse Problems in Imaging
Solving Inverse Problems in Imaging
Solving Inverse Problems in Imaging
Solving Inverse Problems in Imaging
Solving Inverse Problems in Imaging
Solving Inverse Problems in Imaging
Solving Inverse Problems in Imaging
Solving Inverse Problems in Imaging
Solving Inverse Problems in Imaging
Solving Inverse Problems in Imaging
Solving Inverse Problems in Imaging
Solving Inverse Problems in Imaging
Solving Inverse Problems in Imaging
Solving Inverse Problems in Imaging
Solving Inverse Problems in Imaging
Solving Inverse Problems in Imaging
Solving Inverse Problems in Imaging
Solving Inverse Problems in Imaging
Solving Inverse Problems in Imaging
Solving Inverse Problems in Imaging
Solving Inverse Problems in Imaging
Solving Inverse Problems in Imaging
Solving Inverse Problems in Imaging
Solving Inverse Problems in Imaging
Solving Inverse Problems in Imaging
Solving Inverse Problems in Imaging
Solving Inverse Problems in Imaging
Solving Inverse Problems in Imaging
Solving Inverse Problems in Imaging
Solving Inverse Problems in Imaging
Solving Inverse Problems in Imaging
Solving Inverse Problems in Imaging
Solving Inverse Problems in Imaging
Solving Inverse Problems in Imaging
Solving Inverse Problems in Imaging
Solving Inverse Problems in Imaging
Solving Inverse Problems in Imaging
Solving Inverse Problems in Imaging
Solving Inverse Problems in Imaging
Solving Inverse Problems in Imaging
Solving Inverse Problems in Imaging
Solving Inverse Problems in Imaging
Solving Inverse Problems in Imaging
Solving Inverse Problems in Imaging
Solving Inverse Problems in Imaging
Solving Inverse Problems in Imaging
Solving Inverse Problems in Imaging

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Solving Inverse Problems in Imaging

  • 1. Learning to Solve Inverse Problems in Imaging Rebecca Willett, University of Chicago Greg Ongie,
 UChicago Davis Gilton, 
 UW-Madison
  • 2. β Inverse problems in imaging • Inpainting • Deblurring • Superresolution • Compressed Sensing • MRI • Radar Observe: y = Xβ + ε Goal: Recover β from y y
  • 3. Classical approach: Tikhonov regularization (1943) • Example: deblurring • Least squares solution: Wildly Different Solutions Blurred image with noiseBlurred image Deblurred the “naïve” way Deblurred the “naïve” way ˆβ = (X X) 1 X y<latexit 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Wildly different solutions
  • 4. Classical approach: Tikhonov regularization (1943) • Example: deblurring • Least squares solution: Wildly Different Solutions Blurred image with noiseBlurred image Deblurred the “naïve” way Deblurred the “naïve” way • Tikhonov regularization
 (aka “ridge regression”) better conditioned; suppresses noise ˆβ = arg min β y Xβ 2 2 + λ β 2 2 =(X X + λI) 1 X y<latexit 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ˆβ = (X X) 1 X y<latexit 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Wildly different solutions
  • 5. Classical approach: Tikhonov regularization (1943) • Example: deblurring • Least squares solution: Wildly Different Solutions Blurred image with noiseBlurred image Deblurred the “naïve” way Deblurred the “naïve” way • Tikhonov regularization
 (aka “ridge regression”) better conditioned; suppresses noise ˆβ = arg min β y Xβ 2 2 + λ β 2 2 =(X X + λI) 1 X y<latexit 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Wildly different solutions Tikhonov regularization
  • 6. Geometric models of images Total variation Noisy Patches Denoised Patches Patch Denoising Combine to estimate denoised pixel Patch subspaces and manifolds (Wavelet) sparsity
  • 7. Regularization in inverse problems y ˆβ<latexit 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ˆβ = arg min β y Xβ 2 2 + r(β) <latexit 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sha1_base64="/+r4aBrVfxqpWL/MVPFAYn0tVFs=">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</latexit><latexit sha1_base64="/+r4aBrVfxqpWL/MVPFAYn0tVFs=">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</latexit><latexit 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  • 8. Regularization in inverse problems y ˆβ<latexit 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ˆβ = arg min β y Xβ 2 2 + r(β) <latexit 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Classical: r(β) is a pre-defined smoothness-promoting regularizer 
 (e.g. Tikhinov or ridge estimation) Geometric: r(β) reflects image geometry 
 (e.g. sparsity, patch redundancy, total variation) Learned: use training data to learn r(β)
  • 9. Examples in recent literature • Deep CNN’s for signal recovery • Compressed sensing with GANs • Unrolled algorithms for solving inverse problems • Deep proximal gradient descent nets • Deep ADMM nets • Deep half-quadratic splitting • Deep primal-dual nets Chen, Yu, Pock, 2015 Sun, Li, Xu, 2016 Chang, Li, Poczos, Kumar, Sankaranarayanan, 2017 Adler and Öktem, 2018 Bora, Jalal, Price, Dimakis, 2017 Mousavi and Baraniuk, 2017 Dong, Loy, He, Tang, 2014 Mardani et al, 2018 Jin, McCann, Froustey, Unser, 2017 Zhang, Zuo, Gu, Zhang, 2017 Ye, Han, Cha, 2018
  • 10. Classes of methods Model Agnostic 
 (Ignore X) Decoupled 
 (First learn, then reconstruct) Unrolled Optimization Neumann Networks 
 (this talk!)
  • 11. X-1~ raw data (low resolution image) approximate high-resolution image Train deep CNN to remove artifacts reconstruction blurry/blocky artifacts due to re-scaling bicubic interpolation Pictures from: http://webdav.tuebingen.mpg.de/pixel/enhancenet/ Super-resolution with CNNs Model Agnostic 
 (Ignore X)
  • 12. Classes of methods Model Agnostic 
 (Ignore X) Decoupled 
 (First learn, then reconstruct) Unrolled Optimization Neumann Networks 
 (this talk!)
  • 13. GANs for inverse problems y ˆβ<latexit 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ˆβ = arg min β y Xβ 2 2 + r(β) <latexit 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r(β) = 0, β on image manifold , otherwise<latexit 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“Good” image on manifold “Bad” image off manifold Decoupled 
 (First learn, then reconstruct)