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Thomas Oberlin

Thomas Oberlin

(ISAE-SUPAERO, ANITI, IRIT)

https://s3-seminar.github.io/seminars/thomas-oberlin

Title — Regularization via deep generative models: an analysis point of view

Abstract — In this talk, we present a new way of regularizing an inverse problem in imaging (e.g., deblurring or inpainting) by means of a deep generative neural network. Compared to end-to-end models, such approaches seem particularly interesting since the same network can be used for many different problems and experimental conditions, as soon as the generative model is suited to the data. Previous works proposed to use a synthesis framework, where the estimation is performed on the latent vector, the solution being obtained afterwards via the decoder. Instead, we propose an analysis formulation where we directly optimize the image itself and penalize the latent vector. We illustrate the interest of such a formulation by running experiments of inpainting, deblurring and super-resolution. In many cases our technique achieves a clear improvement of the performance and seems to be more robust, in particular with respect to initialization.

Biography — Thomas Oberlin holds a Ph.D. in applied mathematics from the University of Grenoble. In 2014, he was a postdoctoral fellow in signal processing and medical imaging at Inria Rennes, before joining as an Assistant Professor INP Toulouse - ENSEEIHT and the IRIT Laboratory, at Université de Toulouse. Since 2019, he is an Assistant/Associate Professor of Image Processing and Machine Learning at ISAE-SUPAERO and member of IA institute ANITI. His research interests include hyperspectral/spectral imaging, latent factor models, data-driven regularization of inverse problems, and time-frequency representations.

S³ Seminar

June 25, 2021
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  1. Regularization via deep generative models: an
    analysis point of view
    Thomas Oberlin and Mathieu Verm
    ISAE-SUPAERO, Université de Toulouse, ANITI, IRIT
    Séminaire L2S
    June 25th, 2021
    T. Oberlin Regularization via deep generative models 1 / 28

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  2. Outline
    1. Introduction : inverse problems
    2. Deep neural networks for inverse problems
    3. Analysis vs synthesis
    4. Experiments
    5. Conclusion
    T. Oberlin Regularization via deep generative models 2 / 28

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  3. Inverse problems in imaging
    Forward model of image reconstruction :
    y = Ax∗ + n, (1)
    Inverse problem (variational formulation)
    ˆ
    x = arg min
    x
    1
    2
    Ax − y 2
    2
    + λϕ(x) (2)
    Examples
    Inpainting (A is a mask)
    Deblurring (A is Toeplitz)
    Tomography (A computes radial projections)
    Compressed sensing (A satisfies conditions such as RIP)
    etc
    T. Oberlin Regularization via deep generative models 3 / 28

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  4. Priors and regularizers
    Standard penalties or priors
    Sparsity : 0
    , 1
    , greedy algorithms, etc
    Sparsity in a transform domain : wavelets, TV, etc
    Low-rankness : nuclear norm
    Structured sparsity : 12
    (group-lasso)
    Bayesian priors −→ MMSE, MAP
    Data-driven
    Dictionary learning
    Neural networks
    T. Oberlin Regularization via deep generative models 4 / 28

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  5. Example : fast acquisition of EELS spectrum-images
    T. Oberlin Regularization via deep generative models 5 / 28

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  6. Example : Fusion imagery/spectroscopy for the JWST
    [Guilloteau et al 2020]
    Problem similar to super-resolution
    Need for spatial and spectral regularizations
    T. Oberlin Regularization via deep generative models 6 / 28

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  7. Outline
    1. Introduction : inverse problems
    2. Deep neural networks for inverse problems
    Regression DNNs
    Unrolling
    Plug and play
    Generative models
    3. Analysis vs synthesis
    4. Experiments
    5. Conclusion
    T. Oberlin Regularization via deep generative models 7 / 28

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  8. Regression DNNs
    Examples in single-image super-resolution :
    SRCNN [Dong et al., TPAMI 2015]
    VDSR [Kim et al, CVPR 2016]
    T. Oberlin Regularization via deep generative models 8 / 28

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  9. Unrolling/unfolding
    Learns some parameters or building blocks of an iterative algorithm
    (ADMM, proximal gradient, etc) −→ can include some information about
    model or prior
    [Yang et al. NeurIPS 2016]
    T. Oberlin Regularization via deep generative models 9 / 28

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  10. Plug and play
    Decouple the regularization and the degradation model −→ the
    same network can help to solve different inverse problems
    Early works : standard denoising algorithms such as BM3D
    [Venkatakrishnan et al., 2013]
    More recently : denoising neural networks
    Limitation : hard to tune the strength of regularization (related to
    the noise level used for training the network)
    T. Oberlin Regularization via deep generative models 10 / 28

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  11. Generative models
    Generative neural networks as priors
    1. Learn a generative NN
    2. Use it as a prior in a model-based formulation [Bora et al., ICML
    2017]
    Pros and cons
    (+) Well-posed formulation (variational/MAP)
    (+) Tunable regularization
    (+) Generic : same network for any inverse problem
    (-) Performance can be lower than with regression
    (-) Do not generalize well for ood examples
    T. Oberlin Regularization via deep generative models 11 / 28

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  12. Outline
    1. Introduction : inverse problems
    2. Deep neural networks for inverse problems
    3. Analysis vs synthesis
    Generative NNs
    Invertible Neural Networks
    Analysis vs Synthesis
    Related works
    4. Experiments
    5. Conclusion
    T. Oberlin Regularization via deep generative models 12 / 28

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  13. Generative neural networks : Variational auto-encoder
    (VAE)
    Generative model
    z ∼ p(z) “prior”
    x ∼ pθ
    (x|z) = N(Dθ
    (z), ηI), avec Dθ
    decoder/generator network
    Maximum likelihood :

    (x) = pθ
    (x|z)p(z) dz.
    Intractable −→ variational inference
    log pθ
    (x) = Ez∼qφ
    (z|x)
    log pθ
    (x)
    ≥ Ez∼qφ
    (z|x)
    log

    (x, z)

    (z|x)
    termed ELBO or VLB
    T. Oberlin Regularization via deep generative models 13 / 28

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  14. Learning a VAE
    Loss = ELBO :
    Lθ,φ
    (x) = Ez∼qφ
    (z|x)
    log pθ
    (x|z)
    Reconstruction error
    + Ez∼qφ
    (z|x)
    log
    p(z)

    (z|x)
    Regularization
    Reconstruction error :
    log pθ
    (x|z) = −
    1

    x − Dθ
    (z) 2
    2
    Regularization :
    log
    p(z)

    (z|x)
    = −DKL
    (qφ
    (z|x)||p(z))
    Reparametrization of qφ
    :

    (z|x) = N(µ(x), σ(x)I),
    where µ and σ are parameterized by encoder E.
    T. Oberlin Regularization via deep generative models 14 / 28

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  15. Illustration of a VAE
    [credit : Lilian Weng]
    T. Oberlin Regularization via deep generative models 15 / 28

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  16. Invertible Neural Networks (INNs)
    Same generative model
    z ∼ p(z) prior
    x ∼ pθ
    (x|z) = N(Dθ
    (z), ηI) decoder / generator
    INN : design Dθ
    so as to allow for exact inference.
    Main ideas :
    Invertible (bijective) layers −→ no convolution
    Triangular Jacobian for tractable gradient backpropagation
    Simple layers and deep networks
    T. Oberlin Regularization via deep generative models 16 / 28

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  17. Glow [Kingma & Dhariwal, NeurIPS 2018]
    Building blocks : realNVP [Dinh et al 2016] and 1 × 1 convolution
    Actnorm : channel-wise affine transformation
    Invertible 1 × 1 convolution (i.e., invertible linear transformation in
    the channel dimension)
    Affine coupling layer : split + affine transformation (computed from
    an auxiliary NN)
    [Kingma & Dhariwal 2018]
    T. Oberlin Regularization via deep generative models 17 / 28

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  18. Synthesis formulation
    1. Learn a generative NN
    2. Use it as a prior in a synthesis formulation [Bora et al., ICML 2017]
    [Asim et al, 2020]
    ˆ
    x = D arg min
    z
    1
    2
    AD(z) − y 2
    2
    + λ z 2
    2
    . (3)
    Limitations
    Hard to initialize
    Sensisitive to any bias in the prior
    Bad generalization ability
    T. Oberlin Regularization via deep generative models 18 / 28

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  19. Analysis regularization
    Our proposal :
    ˆ
    x = arg min
    x
    1
    2
    Ax − y 2
    2
    + λ E(x) 2
    2
    . (4)
    Similar to analysis vs synthesis for sparsity in a dictionary
    Not a MAP (no Jacobian)
    Intractable for VAEs
    Well suited to INNs
    Generalizes well
    T. Oberlin Regularization via deep generative models 19 / 28

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  20. Related works
    Joint Posterior MAP [González, Almanda and Tan 2021]
    Focused on MAPs with VAEs
    Joint MAP in x and z with alternate optimization
    Add noise during VAE training to improve generalization
    Bayesian estimation of z [Holden, Pereyra and Zygalakis 2021]
    Bayesian computation in the manifold given by a VAE or GAN
    (“synthesis”)
    Sample the posterior with MCMC
    T. Oberlin Regularization via deep generative models 20 / 28

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  21. Outline
    1. Introduction : inverse problems
    2. Deep neural networks for inverse problems
    3. Analysis vs synthesis
    4. Experiments
    5. Conclusion
    T. Oberlin Regularization via deep generative models 21 / 28

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  22. Experimental setting
    Experimental setting similar to [Asim et al 2020]
    Dataset : CelebA, face images of size 64 x 64
    Network : Glow with 32 steps of flow, trained with Adam
    Parameter λ : tuned manually
    Initialization : z0 = 0 and x0 = D(0)
    Inverse problems :
    Inpainting with random mask (60% of missing pixels) or structured
    (squared of 10 x 10)
    Deblurring with 7 x 7 uniform filter
    Super-resolution with factor 2 and 4 and uniform filters
    T. Oberlin Regularization via deep generative models 22 / 28

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  23. Deblurring and super-resolution
    True
    Observed
    Synthesis
    Analysis
    Deblurring 2x super-resolution
    T. Oberlin Regularization via deep generative models 23 / 28

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  24. Inpainting
    True
    Observed
    Synthesis
    Analysis
    Random mask Structured mask
    T. Oberlin Regularization via deep generative models 24 / 28

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  25. Scores
    Task PSNR (synthesis) PSNR (analysis) SSIM (synthesis) SSIM (analysis)
    Deblurring 23.38 ±2.04 32.16 ±1.56 0.74 ±0.09 0.94 ±0.01
    Super-res. (x2) 22.26 ±4.21 31.19 ±1.33 0.76 ±0.12 0.93 ±0.01
    Super-res. (x4) 18.94 ±2.81 24.12 ±1.21 0.61 ±0.11 0.76 ±0.03
    Inp. (random mask) 21.84 ±3.57 27.89 ±2.24 0.71 ±0.14 0.87 ±0.05
    Inp. (struct. mask) 30.40 ±2.53 27.50 ±3.26 0.94 ±0.02 0.91 ±0.03
    Average performance over 50 images ± standard deviation. Best score
    between analysis and synthesis highlighted in bold, for both metrics.
    T. Oberlin Regularization via deep generative models 25 / 28

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  26. Out of distribution examples
    Synthesis
    Analysis
    Deblurring Inpainting (r) Inpainting (s)
    T. Oberlin Regularization via deep generative models 26 / 28

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  27. Out of distribution examples
    Synthesis
    Analysis
    Deblurring Inpainting (r) Inpainting (s)
    T. Oberlin Regularization via deep generative models 27 / 28

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  28. Conclusion & perspectives
    Summary
    Regularization with DNNs : use data-driven priors within a
    model-based inversion −→ generic and well grounded
    Analysis regularization : generalizes better, less sensitive to bias or
    mismatch in the prior
    Perspectives
    Short-term perspectives : MAP with Glow ; Analysis with VAE or
    other non-invertible DNNs
    Long-term perspective : close the gap between toy datasets and real
    applications
    T. Oberlin Regularization via deep generative models 28 / 28

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