Geometry of Deep Learning for Inverse Problems: A Signal Processing Perspective Jong Chul Ye, Ph.D Professor BISPL - BioImaging, Signal Processing, and Learning lab. KAIST, Korea 

Classical Learning vs Deep Learning Diagnosis Classical machine learning Deep learning (no feature engineering) Feature Engineering Esteva et al, Nature Medicine, (2019) 

Deep Learning for Inverse Problems Diagnosis Diagnosis & analysis New trend of deep learning: inverse problems 

4 

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

ANOTHER MYSTERY 

CNN Encoder-Decoder CNN for Inverse Problems 

CNN Encoder-Decoder CNN for Inverse Problems 

CNN Successful applications to various inverse problems Encoder-Decoder CNN for Inverse Problems 

Why Same Architecture Works for Different Inverse Problems ? 

Desiderate of Machine Learning 

Kernel Machines: Representer Theorem SVM, Kernel regressions RKHS (Reproducing Kernel Hilbert Space) 

Kernel Machines: Limitations Fixed during run time Non-adaptive RKHS • limited space (expressivity) • top-down definition 

Perceptron: Universal Approximation Theorem Space can be larger than kernel 

Perceptron: Limitations Fixed during run time * Expressivity still limited Can be exponentially large 

Frame Expansion • Linear representation à No learning • Expressive à eg. shift invariant space 

Compressed Sensing with Frame Thresholding: nonlinear à learning Limitations • Top-down model • transductive à non-inductive 

Data-driven model Adaptive expansion Expressivity Inductive model Learning Kernel Machine No No No Yes Yes Single layer perceptron Yes No No Yes Yes Frame No No Yes No No Compressed sensing + Frame No Yes Yes No Yes Summary of Classical Machine Learning 

Ye et al, SIAM J. Imaging Sciences, 2018 Deep Convolutional Framelets 

Convolution & Hankel Matrix Circular convolution (periodic boundary condition) 

: Non-local basis : Local basis Convolution Framelets: Non-local & Local basis Yin et al, SIAM J. Imaging Sciences, 2017 

Convolution Framelet Expansion Global and local basis Framelet coefficient Frame (dual) basis Yin et al, SIAM J. Imaging Sciences, 2017 

Convolution Framelet: Pros and Cons Convolution Framelet + Regularization Pro: data-driven model Cons: expressivity limited, non-inductive 

: Non-local basis : Local basis : Pooling : Convolution filters Convolution Framelets: Why So Special? Ye et al, SIAM J. Imaging Sciences, 2018 

: Pooling : Convolution filters Convolution Framelets: Why So Special? Ye et al, SIAM J. Imaging Sciences, 2018 

: Pooling : Convolution filters Convolution Framelets: Why So Special? Ye et al, SIAM J. Imaging Sciences, 2018 

: Pooling : Convolution filters Encoder: convolution pooling Convolution Framelets: Why So Special? Ye et al, SIAM J. Imaging Sciences, 2018 

: Pooling : Convolution filters Encoder: convolution pooling un-pooling convolution Decoder: Convolution Framelets: Why So Special? Ye et al, SIAM J. Imaging Sciences, 2018 

Single Resolution Network Architecture 

Multi-Resolution Network Architecture 

How Can We Restore Frame Representation? Vectorization of Feature Map 

Our Theoretical Findings y = X i hbi(x), xi˜ bi(x) AAACGXicbZDLSsNAFIYnXmu9RV26GSxCBSmJCroRim5cVrAXaEKYTCbt0MkkzEykIfQ13Pgqblwo4lJXvo3TNoK2/jDw851zOHN+P2FUKsv6MhYWl5ZXVktr5fWNza1tc2e3JeNUYNLEMYtFx0eSMMpJU1HFSCcRBEU+I21/cD2ut++JkDTmdypLiBuhHqchxUhp5JlWdunINPIohA5DvMcI9D1aHR4dw6EjpsBRlAU/3DMrVs2aCM4buzAVUKjhmR9OEOM0IlxhhqTs2lai3BwJRTEjo7KTSpIgPEA90tWWo4hIN59cNoKHmgQwjIV+XMEJ/T2Ro0jKLPJ1Z4RUX87WxvC/WjdV4YWbU56kinA8XRSmDKoYjmOCARUEK5Zpg7Cg+q8Q95FAWOkwyzoEe/bkedM6qdmnNev2rFK/KuIogX1wAKrABuegDm5AAzQBBg/gCbyAV+PReDbejPdp64JRzOyBPzI+vwEaXJ8Y Ye et al, SIIMS, 2018; Ye et al, ICML, 2019 

y = X i hbi(x), xi˜ bi(x) AAACGXicbZDLSsNAFIYnXmu9RV26GSxCBSmJCroRim5cVrAXaEKYTCbt0MkkzEykIfQ13Pgqblwo4lJXvo3TNoK2/jDw851zOHN+P2FUKsv6MhYWl5ZXVktr5fWNza1tc2e3JeNUYNLEMYtFx0eSMMpJU1HFSCcRBEU+I21/cD2ut++JkDTmdypLiBuhHqchxUhp5JlWdunINPIohA5DvMcI9D1aHR4dw6EjpsBRlAU/3DMrVs2aCM4buzAVUKjhmR9OEOM0IlxhhqTs2lai3BwJRTEjo7KTSpIgPEA90tWWo4hIN59cNoKHmgQwjIV+XMEJ/T2Ro0jKLPJ1Z4RUX87WxvC/WjdV4YWbU56kinA8XRSmDKoYjmOCARUEK5Zpg7Cg+q8Q95FAWOkwyzoEe/bkedM6qdmnNev2rFK/KuIogX1wAKrABuegDm5AAzQBBg/gCbyAV+PReDbejPdp64JRzOyBPzI+vwEaXJ8Y Our Theoretical Findings Ye et al, SIIMS, 2018; Ye et al, ICML, 2019 

y = X i hbi(x), xi˜ bi(x) AAACGXicbZDLSsNAFIYnXmu9RV26GSxCBSmJCroRim5cVrAXaEKYTCbt0MkkzEykIfQ13Pgqblwo4lJXvo3TNoK2/jDw851zOHN+P2FUKsv6MhYWl5ZXVktr5fWNza1tc2e3JeNUYNLEMYtFx0eSMMpJU1HFSCcRBEU+I21/cD2ut++JkDTmdypLiBuhHqchxUhp5JlWdunINPIohA5DvMcI9D1aHR4dw6EjpsBRlAU/3DMrVs2aCM4buzAVUKjhmR9OEOM0IlxhhqTs2lai3BwJRTEjo7KTSpIgPEA90tWWo4hIN59cNoKHmgQwjIV+XMEJ/T2Ro0jKLPJ1Z4RUX87WxvC/WjdV4YWbU56kinA8XRSmDKoYjmOCARUEK5Zpg7Cg+q8Q95FAWOkwyzoEe/bkedM6qdmnNev2rFK/KuIogX1wAKrABuegDm5AAzQBBg/gCbyAV+PReDbejPdp64JRzOyBPzI+vwEaXJ8Y Our Theoretical Findings Ye et al, SIIMS, 2018; Ye et al, ICML, 2019 

analysis basis y = X i hbi(x), xi˜ bi(x) AAACGXicbZDLSsNAFIYnXmu9RV26GSxCBSmJCroRim5cVrAXaEKYTCbt0MkkzEykIfQ13Pgqblwo4lJXvo3TNoK2/jDw851zOHN+P2FUKsv6MhYWl5ZXVktr5fWNza1tc2e3JeNUYNLEMYtFx0eSMMpJU1HFSCcRBEU+I21/cD2ut++JkDTmdypLiBuhHqchxUhp5JlWdunINPIohA5DvMcI9D1aHR4dw6EjpsBRlAU/3DMrVs2aCM4buzAVUKjhmR9OEOM0IlxhhqTs2lai3BwJRTEjo7KTSpIgPEA90tWWo4hIN59cNoKHmgQwjIV+XMEJ/T2Ro0jKLPJ1Z4RUX87WxvC/WjdV4YWbU56kinA8XRSmDKoYjmOCARUEK5Zpg7Cg+q8Q95FAWOkwyzoEe/bkedM6qdmnNev2rFK/KuIogX1wAKrABuegDm5AAzQBBg/gCbyAV+PReDbejPdp64JRzOyBPzI+vwEaXJ8Y Encoder Our Theoretical Findings Ye et al, SIIMS, 2018; Ye et al, ICML, 2019 

analysis basis synthesis basis y = X i hbi(x), xi˜ bi(x) AAACGXicbZDLSsNAFIYnXmu9RV26GSxCBSmJCroRim5cVrAXaEKYTCbt0MkkzEykIfQ13Pgqblwo4lJXvo3TNoK2/jDw851zOHN+P2FUKsv6MhYWl5ZXVktr5fWNza1tc2e3JeNUYNLEMYtFx0eSMMpJU1HFSCcRBEU+I21/cD2ut++JkDTmdypLiBuhHqchxUhp5JlWdunINPIohA5DvMcI9D1aHR4dw6EjpsBRlAU/3DMrVs2aCM4buzAVUKjhmR9OEOM0IlxhhqTs2lai3BwJRTEjo7KTSpIgPEA90tWWo4hIN59cNoKHmgQwjIV+XMEJ/T2Ro0jKLPJ1Z4RUX87WxvC/WjdV4YWbU56kinA8XRSmDKoYjmOCARUEK5Zpg7Cg+q8Q95FAWOkwyzoEe/bkedM6qdmnNev2rFK/KuIogX1wAKrABuegDm5AAzQBBg/gCbyAV+PReDbejPdp64JRzOyBPzI+vwEaXJ8Y Encoder Decoder Our Theoretical Findings Ye et al, SIIMS, 2018; Ye et al, ICML, 2019 

Linear Encoder-Decoder (ED) CNN Learned filters y = ˜ BB>x = X i hx, bi i˜ bi AAACJ3icbVBNSwMxEM36bf2qevQSLIIHKbsq6KUievGoaK3QrSWbzrah2eySzIpl8d948a94EVREj/4T03YP2joQePPePCbzgkQKg6775UxMTk3PzM7NFxYWl5ZXiqtr1yZONYcqj2WsbwJmQAoFVRQo4SbRwKJAQi3onvb12h1oI2J1hb0EGhFrKxEKztBSzeJRr+KjkC2gJ/Tk1sc4ofe0Qn2TRk1BfclUWwK936FBv9XDNndYqlksuWV3UHQceDkokbzOm8VXvxXzNAKFXDJj6p6bYCNjGgWX8FDwUwMJ413WhrqFikVgGtngzge6ZZkWDWNtn0I6YH87MhYZ04sCOxkx7JhRrU/+p9VTDA8bmVBJiqD4cFGYSoox7YdGW0IDR9mzgHEt7F8p7zDNONpoCzYEb/TkcXC9W/b2yu7Ffun4Mo9jjmyQTbJNPHJAjskZOSdVwskjeSZv5N15cl6cD+dzODrh5J518qec7x9ZY6R3 pooling un-pooling 

Linear E-D CNN w/ Skipped Connection more redundant expression Learned filters y = ˜ BB>x = X i hx, bi i˜ bi AAACJ3icbVBNSwMxEM36bf2qevQSLIIHKbsq6KUievGoaK3QrSWbzrah2eySzIpl8d948a94EVREj/4T03YP2joQePPePCbzgkQKg6775UxMTk3PzM7NFxYWl5ZXiqtr1yZONYcqj2WsbwJmQAoFVRQo4SbRwKJAQi3onvb12h1oI2J1hb0EGhFrKxEKztBSzeJRr+KjkC2gJ/Tk1sc4ofe0Qn2TRk1BfclUWwK936FBv9XDNndYqlksuWV3UHQceDkokbzOm8VXvxXzNAKFXDJj6p6bYCNjGgWX8FDwUwMJ413WhrqFikVgGtngzge6ZZkWDWNtn0I6YH87MhYZ04sCOxkx7JhRrU/+p9VTDA8bmVBJiqD4cFGYSoox7YdGW0IDR9mzgHEt7F8p7zDNONpoCzYEb/TkcXC9W/b2yu7Ffun4Mo9jjmyQTbJNPHJAjskZOSdVwskjeSZv5N15cl6cD+dzODrh5J518qec7x9ZY6R3 

Deep Convolutional Framelets x = ˜ BB>x = X i hx, bi i˜ bi AAACJ3icbVBNSwMxEM36bf2qevQSLIIHKbsq6EWRevGoaK3QXZdsOq2h2eySzEpL6b/x4l/xIqiIHv0npu0etDoQePPePCbzolQKg6776UxMTk3PzM7NFxYWl5ZXiqtr1ybJNIcqT2SibyJmQAoFVRQo4SbVwOJIQi1qnw702j1oIxJ1hd0Ugpi1lGgKztBSYfG4c+SjkA2gFVq59TFJaYceUd9kcSioL5lqSaCdHRoNWj1qc4elwmLJLbvDon+Bl4MSyes8LL74jYRnMSjkkhlT99wUgx7TKLiEfsHPDKSMt1kL6hYqFoMJesM7+3TLMg3aTLR9CumQ/enosdiYbhzZyZjhnRnXBuR/Wj3D5mHQEyrNEBQfLWpmkmJCB6HRhtDAUXYtYFwL+1fK75hmHG20BRuCN37yX3C9W/b2yu7FfunkMo9jjmyQTbJNPHJATsgZOSdVwskDeSKv5M15dJ6dd+djNDrh5J518qucr29XoqR2 Perfect reconstruction Ye et al, SIIMS 2018; Ye et al, ICML 2019 Frame conditions w skipped connection w/o skipped connection 

Deep Convolutional Framelets x = ˜ BB>x = X i hx, bi i˜ bi AAACJ3icbVBNSwMxEM36bf2qevQSLIIHKbsq6EWRevGoaK3QXZdsOq2h2eySzEpL6b/x4l/xIqiIHv0npu0etDoQePPePCbzolQKg6776UxMTk3PzM7NFxYWl5ZXiqtr1ybJNIcqT2SibyJmQAoFVRQo4SbVwOJIQi1qnw702j1oIxJ1hd0Ugpi1lGgKztBSYfG4c+SjkA2gFVq59TFJaYceUd9kcSioL5lqSaCdHRoNWj1qc4elwmLJLbvDon+Bl4MSyes8LL74jYRnMSjkkhlT99wUgx7TKLiEfsHPDKSMt1kL6hYqFoMJesM7+3TLMg3aTLR9CumQ/enosdiYbhzZyZjhnRnXBuR/Wj3D5mHQEyrNEBQfLWpmkmJCB6HRhtDAUXYtYFwL+1fK75hmHG20BRuCN37yX3C9W/b2yu7FfunkMo9jjmyQTbJNPHJATsgZOSdVwskDeSKv5M15dJ6dd+djNDrh5J518qucr29XoqR2 Perfect reconstruction Ye et al, SIAM J. Imaging Science, 2018 Frame conditions w skipped connection w/o skipped connection Frame Conditions for Pooling layers 

Deep Convolution Framelet: Pros and Cons Deep Convolutional Framelet + Regularization Pros: data-driven, expressive Cons: non-inductive, transductive 

Data-driven model Adaptive expansion Expressivity Inductive model Learning Kernel Machine No No No Yes Yes Single layer perceptron Yes No No Yes Yes Frame No No Yes No No Compressed sensing No Yes Yes No Yes Deep Convolutional Framelet + CS Yes Yes Yes No Yes Summary So Far 

Ye et al, ICML, 2019 How to make it inductive? Role of Nonlinearities 

Role of ReLUs? Generator for Multiple Expressions y = ˜ B(x)B(x)>x = X i hx, bi(x)i˜ bi(x) 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 ⌃l(x) = 2 6 6 6 4 1 0 · · · 0 0 2 · · · 0 . . . . . . ... . . . 0 0 · · · ml 3 7 7 7 5 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 Input dependent {0,1} matrix --> Input adaptivity 

Input Space Partitioning for Multiple Expressions A CNN performs automatic assignment of distinct linear representation depending on input 

Geometry of Manifold Partitioning 

Toy Examples: Two layer Perceptron with ReLUs 

A final exam question from BiS400@KAIST 

Expressivity of E-D CNN # of representation # of network elements 

Expressivity of E-D CNN # of representation # of network elements # of channel 

Expressivity of E-D CNN # of representation # of network elements # of channel Network depth 

Expressivity of E-D CNN # of representation # of network elements # of channel Network depth Skipped connection 

Inductive Bias in Partitioning X. Zhang and D. Wu, ICLR, 2020. 

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Data-driven model Adaptive expans ion Expressivity Inductive model Learning Kernel Machine No No No Yes Yes Single layer perceptron Yes No No Yes Yes Frame No No No No No Compressed sensing No Yes Yes No Yes Deep Convolutional F ramelet + CS Yes Yes Yes No Yes Deep Learning Yes Yes Yes Yes Yes Deep Learning as an Ultimate Learning Machine 

Ye Ye et al, ICML, 2019 More about ReLUs 

Lipschitz Continuity K = max p Kp, Kp = k ˜ B(zp)B(zp)>k2 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 z1 AAAB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0lU0GPRi8eK9gPaUDbbTbt0swm7E6GG/gQvHhTx6i/y5r9x2+agrQ8GHu/NMDMvSKQw6LrfTmFldW19o7hZ2tre2d0r7x80TZxqxhsslrFuB9RwKRRvoEDJ24nmNAokbwWjm6nfeuTaiFg94DjhfkQHSoSCUbTS/VPP65UrbtWdgSwTLycVyFHvlb+6/ZilEVfIJDWm47kJ+hnVKJjkk1I3NTyhbEQHvGOpohE3fjY7dUJOrNInYaxtKSQz9fdERiNjxlFgOyOKQ7PoTcX/vE6K4ZWfCZWkyBWbLwpTSTAm079JX2jOUI4toUwLeythQ6opQ5tOyYbgLb68TJpnVe+86t5dVGrXeRxFOIJjOAUPLqEGt1CHBjAYwDO8wpsjnRfn3fmYtxacfOYQ/sD5/AEPZo2k zp AAAB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0lU0GPRi8eK9gPaUDbbSbt0swm7G6GG/gQvHhTx6i/y5r9x2+agrQ8GHu/NMDMvSATXxnW/ncLK6tr6RnGztLW9s7tX3j9o6jhVDBssFrFqB1Sj4BIbhhuB7UQhjQKBrWB0M/Vbj6g0j+WDGSfoR3QgecgZNVa6f+olvXLFrbozkGXi5aQCOeq98le3H7M0QmmYoFp3PDcxfkaV4UzgpNRNNSaUjegAO5ZKGqH2s9mpE3JilT4JY2VLGjJTf09kNNJ6HAW2M6JmqBe9qfif10lNeOVnXCapQcnmi8JUEBOT6d+kzxUyI8aWUKa4vZWwIVWUGZtOyYbgLb68TJpnVe+86t5dVGrXeRxFOIJjOAUPLqEGt1CHBjAYwDO8wpsjnBfn3fmYtxacfOYQ/sD5/AFu4o3j Related to the generalizability Dependent on the Local Lipschitz 

Backpropagation (Backprop, BP) 

Key Step of BP Derivation: Backward Propagation 

Rectified Linear Unit (ReLU) 

ReLU makes back prop symmetric 

Implication to Optimization Positive semidefinite à with sufficient small step size, cost always decreases 

GEOMETRY DRIVEN DESIGN 

Which Domain is Good for Learning? Han et al, IEEE Trans. Medical Imaging (in press), 2019 Lee et al, MRM (in press), 2019 Han et al, Medical Physics, 2020 

Image Domain Learning is Essential? 73 Kravitz et al, Trends in Cognitive Sciences January 2013, Vol. 17, No. 1 

k-Space Deep Learning Han et al, IEEE Trans. Medical Imaging, 2019 Lee et al, MRM, 2019 

ALOHA CNN k-Space Deep Learning Han et al, IEEE TMI,2019 Jin et al, IEEE TCI,2016 ; Ye et al, IEEE TIT, 2017 

K-space Deep Learning (Radial R=6) Ground-truth Acceleration Image learning CS K-space learning Han et al, IEEE TMI , 2020 

K-space Deep Learning (Radial R=6) Ground-truth Acceleration Image learning CS K-space learning Han et al, IEEE TMI , 2020 

k-space Learning for EPI Ghost Correction Image domain loss L2 loss is calculated on the image domain k-space (with Ghost) e eo o … e e o o … ALOHA IFT e e o o … Coil 1 … coil P Coil 1 … coil P Coil 1 … coil P Neural network k-space (with Ghost) e e o o … e e o o … IFT e e o o … Coil 1 … coil P Coil 1 … coil P Coil 1 … coil P k-space learning Network Input Network Label 34 Lee et al, MRM (in press), 2019 

7T EPI result (R=2) ALOHA Ghost image Half ROI learning With Reference PEC-SENSE Proposed (Full ROI) GSR : 10.48% GSR : 9.71% GSR : 15.04% GSR : 8.80% GSR : 4.92% 49 Lee et al, MRM (in press), 2019 

DBP Domain Deep Learning Han et al, Medical Physics 46 (12), e855-e872, 2020 Han et al, IEEE TMI, 2020 Differentiated Backprojection 

• Ramp filtering • Back-projection • Differentiation • Back-projection • Hilbert transform Two Approaches for CT Reconstruction Zou, Y et al, PMB (2004). Backprojection Filtration (BPF) Filtered Backprojection (FBP) 

DBP Domain ROI Tomography Interior (ROI) Tomography à 2-D Deconvolution problem Han et al, Medical Physics, 2019 

DBP Domain Conebeam Artifact Removal Han et al, IEEE TMI, 2020 https://www.ndt.net/article/wcndt00/papers/idn730/idn730.htm Standard Method: FDK Algorithm 

DBP Domain Conebeam Artifact Removal Exact Factorizationà 2-D Deconvolution problem F. Dennerlein et al, IEEE TMI, 2008 

DBP Domain Conebeam Artifact Removal Han et al, IEEE TMI, 2020 

Unsupervised Wavelet Directional Learning Song et al. "Unsupervised Denoising for Satellite Imagery using Wavelet Subband CycleGAN." arXiv:2002.09847 (2020). 

Noise Patterns in Satellite Imagery 

Wavelet Directional Residual Learning 

Unsupervised Learning with CycleGAN Optimal transport geometry of cycleGAN @ SPACE Webinar, Tues, July 14th, 11:00am EST 

Agricultural area 

Cloud 

Ocean 

WHAT DOES A CNN LEARN? 

Classical vs. Deep Learning for Inverse Problems Diagnosis Classical Regularized Recon (basis engineering) Deep Recon (no basis engineering) Basis Engineering 

Questions?