Jong Chul Ye Deep Convolutional Framelets: A general deep learning framework for inverse problems Bio-Imaging, Signal Processing, & Learning (BISPL) Dept. Bio & Brain Engineering Dept. Mathematical Sciences KAIST, Korea 

• Successful demonstration of deep learning for various image reconstruction problems – Low-dose x-ray CT (Kang et al, Chen et al, Wolterink et al, Ye et al) – Sparse view CT (Jin et al, Han et al, Adler et al) – Interior tomography (Han et al) – Stationary CT for baggage inspection (Han et al) – CS-MRI (Hammernik et al, Schlemper et al, Yang et al, Lee et al, Zhu et al) – US imaging (Yoon et al ) – Diffuse optical tomography (Yoo et al) – Elastic tomography (Yoo et al) – Optical diffraction tomography (Kamilov et al) – etc • Advantages – Very fast reconstruction time – Significantly improved results Deep Learning for Inverse Problems 

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WHY DEEP LEARNING WORKS FOR RECON ? DOES IT CREATE ANY ARTIFICIAL FEATURES ? 

Too Simple to Analyze..? Convolution & pooling à stone age tools of signal processing What do they do ? 

Paradox and Mysteries Residual Network Clean image Standard Network Zhang, K., et al, IEEE TIP, 2017. 

Dark Age of Applied Mathematics ? 

• What is the role of the nonlinearity such as rectified linear unit (ReLU) ? • Why do we need a pooling and unpooling in some architectures ? • Why do some networks need fully connected layers whereas the others do not ? • What is the role of by-pass connection or residual network ? • What is the role of the filter channels in convolutional layer ? Many Mysteries… 

Our Proposal: Deep Learning == Deep Convolutional Framelets • Ye et al, “Deep convolutional framelets: A general deep learning framework for inverse problems”, SIAM Journal Imaging Sciences, 11(2), 991-1048, 2018. 

Matrix Representation of CNN Figure courtesy of Shoieb et al, 2016 

Hankel Matrix: Linear Lifting to Higher Dimensional Space 

Why we are excited about Hankel matrix ? T -T 0 n1 -n1 0 * FRI Sampling theory (Vetterlie et al) and compressed sensing 

1 2 3 4 5 6 7 8 9 -1 0 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 12 2 3 4 5 6 7 8 9 12 13 3 4 5 6 7 8 9 10 10 10 10 0 11 11 11 1 2 3 4 5 Finite length convolution Matrix Representation * ALOHA : Annihilating filter based LOw rank Hankel matrix Approach * Jin KH et al. IEEE TCI, 2016 * Jin KH et al.,IEEE TIP, 2015 * Ye JC et al. IEEE TIT, 2016 Annihilating filter-based low-rank Hankel matrix 

Missing elements can be found by low rank Hankel structured matrix completion Nuclear norm Projection on sampling positions min m kH(m)k⇤ subject to P⌦(b) = P⌦(f) RankH(f) = k * Jin KH et al IEEE TCI, 2016 * Jin KH et al.,IEEE TIP, 2015 * Ye JC et al., IEEE TIT, 2016 m Annihilating filter-based low-rank Hankel matrix 

18. APR. 2015. 18 * Image Inpainting Results Jin et al, IEEE TIP, 2015 

19 Jin et al, IEEE TIP, 2015 

Key Observation Data-Driven Hankel matrix decomposition => Deep Learning • Ye et al, “Deep convolutional framelets: A general deep learning framework for inverse problems”, SIAM Journal Imaging Sciences, 11(2), 991-1048, 2018. 

Hd(f) = U⌃V T : Non-local basis : Local basis Convolution Framelets (Yin et al; 2017) > = I > = I Hd(f) 

Hd(f) Hd(f) = ˜ T ˜ T C C = T Hd(f) C = T (f ~ ) Encoder: ˜ T = I ˜ = PR(V ) Hd(f) = U⌃V T Unlifting: f = (˜C) ~ ⌧(˜ ) : Non-local basis : Local basis : Frame condition : rank condition convolution pooling un-pooling convolution : User-defined pooling : Learnable filters Hpi (gi) = X k,l [Ci]kl e Bkl i Decoder: Deep Convolutional Framelets (Y, Han, Cha; 2018) 

Single Resolution Network Architecture 

Multi-Resolution Network Architecture 

Conic fi [Ci]kl 0 Hpi (gi) = X k,l [Ci]kl e Bkl i Hpi (fi) ' Linear Lifting Geometry of CNN gi Linear Un-lifting Ci(fi) Ci(fi) 0 i ⇣i ⇣ e i ⌘ 

fi [Ci]kl 0 Hpi (gi) = X k,l [Ci]kl e Bkl i Hpi (fi) ' Lifting Geometry of Residual CNN Ci(fi) Ci(fi) 0 i ⇣i ⇣ e i ⌘ gi Un-lifting 

Deep CNN Lifting Un-lifting Conic Lifting Un-lifting Lifting Un-lifting 

fi Nonlinear Lifting to Feature space Comparison with Kernel PCA gi Nonlinear Pre-Image calculation (fi) AAAB+nicbVBNS8NAFHzxs9avWI9egkWol5KIoN6KXjxWMLbQhLDZbtqlm03Y3Ygl5K948aDi1V/izX/jps1BWwcWhpn3eLMTpoxKZdvfxsrq2vrGZm2rvr2zu7dvHjQeZJIJTFycsET0QyQJo5y4iipG+qkgKA4Z6YWTm9LvPRIhacLv1TQlfoxGnEYUI6WlwGx43TFteTFS4zDKoyKgp4HZtNv2DNYycSrShArdwPzyhgnOYsIVZkjKgWOnys+RUBQzUtS9TJIU4QkakYGmHMVE+vkse2GdaGVoRYnQjytrpv7eyFEs5TQO9WQZUi56pfifN8hUdOnnlKeZIhzPD0UZs1RilUVYQyoIVmyqCcKC6qwWHiOBsNJ11XUJzuKXl4l71r5qO3fnzc511UYNjuAYWuDABXTgFrrgAoYneIZXeDMK48V4Nz7moytGtXMIf2B8/gD8DZP1 AAAB+nicbVBNS8NAFHzxs9avWI9egkWol5KIoN6KXjxWMLbQhLDZbtqlm03Y3Ygl5K948aDi1V/izX/jps1BWwcWhpn3eLMTpoxKZdvfxsrq2vrGZm2rvr2zu7dvHjQeZJIJTFycsET0QyQJo5y4iipG+qkgKA4Z6YWTm9LvPRIhacLv1TQlfoxGnEYUI6WlwGx43TFteTFS4zDKoyKgp4HZtNv2DNYycSrShArdwPzyhgnOYsIVZkjKgWOnys+RUBQzUtS9TJIU4QkakYGmHMVE+vkse2GdaGVoRYnQjytrpv7eyFEs5TQO9WQZUi56pfifN8hUdOnnlKeZIhzPD0UZs1RilUVYQyoIVmyqCcKC6qwWHiOBsNJ11XUJzuKXl4l71r5qO3fnzc511UYNjuAYWuDABXTgFrrgAoYneIZXeDMK48V4Nz7moytGtXMIf2B8/gD8DZP1 AAAB+nicbVBNS8NAFHzxs9avWI9egkWol5KIoN6KXjxWMLbQhLDZbtqlm03Y3Ygl5K948aDi1V/izX/jps1BWwcWhpn3eLMTpoxKZdvfxsrq2vrGZm2rvr2zu7dvHjQeZJIJTFycsET0QyQJo5y4iipG+qkgKA4Z6YWTm9LvPRIhacLv1TQlfoxGnEYUI6WlwGx43TFteTFS4zDKoyKgp4HZtNv2DNYycSrShArdwPzyhgnOYsIVZkjKgWOnys+RUBQzUtS9TJIU4QkakYGmHMVE+vkse2GdaGVoRYnQjytrpv7eyFEs5TQO9WQZUi56pfifN8hUdOnnlKeZIhzPD0UZs1RilUVYQyoIVmyqCcKC6qwWHiOBsNJ11XUJzuKXl4l71r5qO3fnzc511UYNjuAYWuDABXTgFrrgAoYneIZXeDMK48V4Nz7moytGtXMIf2B8/gD8DZP1 AAAB+nicbVBNS8NAFHzxs9avWI9egkWol5KIoN6KXjxWMLbQhLDZbtqlm03Y3Ygl5K948aDi1V/izX/jps1BWwcWhpn3eLMTpoxKZdvfxsrq2vrGZm2rvr2zu7dvHjQeZJIJTFycsET0QyQJo5y4iipG+qkgKA4Z6YWTm9LvPRIhacLv1TQlfoxGnEYUI6WlwGx43TFteTFS4zDKoyKgp4HZtNv2DNYycSrShArdwPzyhgnOYsIVZkjKgWOnys+RUBQzUtS9TJIU4QkakYGmHMVE+vkse2GdaGVoRYnQjytrpv7eyFEs5TQO9WQZUi56pfifN8hUdOnnlKeZIhzPD0UZs1RilUVYQyoIVmyqCcKC6qwWHiOBsNJ11XUJzuKXl4l71r5qO3fnzc511UYNjuAYWuDABXTgFrrgAoYneIZXeDMK48V4Nz7moytGtXMIf2B8/gD8DZP1 C = 1 N N X i=1 (fi) >(fi) 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 PCA of • Nonlinear lifting & unlifting • Deterministic kernel • Difficulty in multilevel extension 

APPLICATION-DRIVEN EVIDENCES 

Problem of U-net Pooling does NOT satisfy the frame condition JC Ye et al, SIAM Journal Imaging Sciences, 2018 Y. Han et al, TMI, 2018. ext > ext = I + > 6= I 

Improving U-net using Deep Conv Framelets • Dual Frame U-net • Tight Frame U-net JC Ye et al, SIAM Journal Imaging Sciences, 2018 Y. Han and J. C. Ye, TMI, 2018 

U-Net versus Dual Frame U-Net 

Tight-Frame U-Net JC Ye et al, SIAM Journal Imaging Sciences, 2018 

Denoising: U-Net vs. Tight-Frame U-Net 

Inpainting: U-Net vs. Tight-Frame U-Net 

Low-Dose CT • To reduce the radiation exposure, sparse-view CT, low-dose CT and interior tomography. Sparse-view CT (Down-sampled View) Low-dose CT (Reduced X-ray dose) Interior Tomography (Truncated FOV) 

WavResNet for low-dose CT (Kang, Yoo, Y; 2017) 

Signal Boosting with Multiple Framelet Expansions 

WavResNet results (Kang et al, TMI, 2018) 

WavResNet results (Kang et al, TMI, 2018) 

MBIR Our latest Result C D WavResNet results 

K-space Deep Learning for Accelerated MRI Han, Y., & Ye, J. C. (2018). k-Space Deep Learning for Accelerated MRI. arXiv preprint arXiv:1805.03779. Conventional Image Domain Learning 

K-space Deep Learning for Accelerated MRI Deep Neural Network IFT Han, Y., & Ye, J. C. (2018). k-Space Deep Learning for Accelerated MRI. arXiv preprint arXiv:1805.03779. Proposed k-space Deep Learning ALOHA : k-space interpolation à k-space interpolation using deep learning ? Yes 

ALOHA for Compressed Sensing MRI ALOHA: Annihilating filter-based low-rank Hankel matrix approach • Jin KH et al IEEE TCI, 2016 • Lee et al, MRM, 2015 

Single coil static MRI 

Parallel MRI 

ALOHA CNN 

k-Space Deep Learning for Accelerated MRI Han et al, arXiv:1805.03779 ~3dB gain 

Improved Time-Resolved MRA using k-Space Deep Learning Eunju Cha , Eung Yeop Kim and Jong Chul Ye 1 1 2 Dept. of Bio and Brain Engineering, KAIST, Dept. of Radiology, Gachon University Gil Mdeical Center 1 2 Motivation Ø To cover k-space data at different rate Ø Regular sampling pattern following view sharing of several temporal frames • Reconstruction using GRAPPA TWIST Fixed spatial resolution Limited temporal resolution How to reconstruct? 

Improved Time-Resolved MRA using k-Space Deep Learning Research Goal Ø To improve temporal resolution of TWIST imaging using deep k-space learning Ø To generate multiple reconstruction results with various spatial and temporal resolution using one network VS = 5 VS = 2 CNN 

k-Space Deep Learning for Parallel MRI Cha et al, arXiv:1806.00806 

WHAT IF WE DON’T HAVE REFERENCE ? 

Semi-Supervised Learning for low-dose CT 54 • Multiphase Cardiac CT denoising – Phase 1, 2: low-dose, Phase 3 ~ 10: normal dose – Goal: dynamic changes of heart structure – No reference available Kang et al, arXiv:1806.09748 

55 • Cardiac CT denoising – Cycle Consistent Adversarial Denoising Network for Multiphase Coronary CT Angiography Semi-supervised Learning using Cyclic-GAN 

56 Input: phase 1 Denoised output Target: phase 8 Input- output 

fbp admire recon 139 fbp-target input - admire input - recon AMC002_20180903 FBP(Phase1) FBP(Phase8) RECON ADMIRE Phase1 - ADMIRE Phase1 - RECON 

• Figures from internet 9 View CT for Baggage Screening 

9 View CT for Baggage Screening 

1st view 2nd view 3rd view 4th view 5th view 6th view 7th view 8th view 9th view 

Semi-Supervised High Resolution View Synthesis Han et al, arXiv preprint arXiv:1712.10248, (2017). CT meeting 2018 

Semi-Supervised High Resolution View Synthesis 128 256 64 128 6 4 256 256 512 512 512 1024 512 1024 512 256 512 256128 • Key idea • Training with measured views • Inference with non-measured views 

FBP 

TV 

Ours 

Summary • Deep learning for inverse problems • Significant performance gain • Has becomes mainstream topics • Deep convolutional framelets: • A new mathematical tool for understanding deep neural network for inverse problems • Biomedical image reconstruction • Key application for machine learning • Semi-supervised learning • New opportunities 

math Muchas gracias !