Solving Inverse Problems by Joint Posterior Maximization with Autoencoding Prior

Abstract

In this work we address the problem of solving ill-posed inverse problems in imaging where the prior is a variational autoencoder (VAE). Specifically we consider the decoupled case where the prior is trained once and can be reused for many different log-concave degradation models without retraining. Whereas previous MAP-based approaches to this problem lead to highly non-convex optimization algorithms, our approach computes the joint (space-latent) MAP that naturally leads to alternate optimization algorithms and to the use of a stochastic encoder to accelerate computations. The resulting technique (JPMAP) performs Joint Posterior Maximization using an Autoencoding Prior. We show theoretical and experimental evidence that the proposed objective function is quite close to bi-convex. Indeed it satisfies a weak bi-convexity property which is sufficient to guarantee that our optimization scheme converges to a stationary point. We also highlight the importance of correctly training the VAE using a denoising criterion, in order to ensure that the encoder generalizes well to out-of-distribution images, without affecting the quality of the generative model. This simple modification is key to providing robustness to the whole procedure. Finally we show how our joint MAP methodology relates to more common MAP approaches, and we propose a continuation scheme that makes use of our JPMAP algorithm to provide more robust MAP estimates. Experimental results also show the higher quality of the solutions obtained by our JPMAP approach with respect to other non-convex MAP approaches which more often get stuck in spurious local optima.

arXiv commentarXiv admin note: text overlap with arXiv:1911.06379

References in corpus (32)

• Auto-Encoding Variational Bayes
• Beyond a Gaussian Denoiser: Residual Learning of Deep CNN for Image Denoising
• Unsupervised Representation Learning with Deep Convolutional Generative Adversarial Networks
• FFDNet: Toward a Fast and Flexible Solution for CNN based Image Denoising
• Trainable Nonlinear Reaction Diffusion: A Flexible Framework for Fast and Effective Image Restoration
• Large Scale Adversarial Representation Learning
• CNN-Based Projected Gradient Descent for Consistent Image Reconstruction
• NVAE: A Deep Hierarchical Variational Autoencoder
• Normalizing Flows for Probabilistic Modeling and Inference
• Plug-and-Play Priors for Bright Field Electron Tomography and Sparse Interpolation
• Compressed Sensing using Generative Models
• DeepISP: Towards Learning an End-to-End Image Processing Pipeline
• SOPHIE velocimetry of Kepler transit candidates XVII. The physical properties of giant exoplanets within 400 days of period
• Learning Proximal Operators: Using Denoising Networks for Regularizing Inverse Imaging Problems
• Assessing Generative Models via Precision and Recall
• On the Spectral Bias of Neural Networks
• Unrolled Optimization with Deep Priors
• A primal-dual hybrid gradient method for non-linear operators with applications to MRI
• Plug-and-Play Methods Provably Converge with Properly Trained Denoisers
• Denoising Criterion for Variational Auto-Encoding Framework
• A Bayesian Hyperprior Approach for Joint Image Denoising and Interpolation, with an Application to HDR Imaging
• Adversarial Symmetric Variational Autoencoder
• Image Restoration using Autoencoding Priors
• Preconditioned ADMM with nonlinear operator constraint
• Solving Inverse Problems with Piecewise Linear Estimators: From Gaussian Mixture Models to Structured Sparsity
• Fast and Provable ADMM for Learning with Generative Priors
• Neumann Networks for Inverse Problems in Imaging
• A Provably Convergent Scheme for Compressive Sensing under Random Generative Priors
• Perceptual Generative Autoencoders
• Scene-Adapted Plug-and-Play Algorithm with Guaranteed Convergence: Applications to Data Fusion in Imaging
• Bayesian imaging using Plug & Play priors: when Langevin meets Tweedie
• Blind Image Restoration with Flow Based Priors

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