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Register a new userDeep Unfolding with Normalizing Flow Priors for Inverse Problems
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Added by
Xinyi Wei
Publication date
2021
fields
Electronic Engineering
and research's language is
English
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Many application domains, spanning from computational photography to medical imaging, require recovery of high-fidelity images from noisy, incomplete or partial/compressed measurements. State of the art methods for solving these inverse problems combine deep learning with iterative model-based solvers, a concept known as deep algorithm unfolding. By combining a-priori knowledge of the forward measurement model with learned (proximal) mappings based on deep networks, these methods yield solutions that are both physically feasible (data-consistent) and perceptually plausible. However, current proximal mappings only implicitly learn such image priors. In this paper, we propose to make these image priors fully explicit by embedding deep generative models in the form of normalizing flows within the unfolded proximal gradient algorithm. We demonstrate that the proposed method outperforms competitive baselines on various image recovery tasks, spanning from image denoising to inpainting and deblurring.
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Recent work in machine learning shows that deep neural networks can be used to solve a wide variety of inverse problems arising in computational imaging. We explore the central prevailing themes of this emerging area and present a taxonomy that can be used to categorize different problems and reconstruction methods. Our taxonomy is organized along two central axes: (1) whether or not a forward model is known and to what extent it is used in training and testing, and (2) whether or not the learning is supervised or unsupervised, i.e., whether or not the training relies on access to matched ground truth image and measurement pairs. We also discuss the trade-offs associated with these different reconstruction approaches, caveats and common failure modes, plus open problems and avenues for future work.
Recent efforts on solving inverse problems in imaging via deep neural networks use architectures inspired by a fixed number of iterations of an optimization method. The number of iterations is typically quite small due to difficulties in training networks corresponding to more iterations; the resulting solvers cannot be run for more iterations at test time without incurring significant errors. This paper describes an alternative approach corresponding to an infinite number of iterations, yielding a consistent improvement in reconstruction accuracy above state-of-the-art alternatives and where the computational budget can be selected at test time to optimize context-dependent trade-offs between accuracy and computation. The proposed approach leverages ideas from Deep Equilibrium Models, where the fixed-point iteration is constructed to incorporate a known forward model and insights from classical optimization-based reconstruction methods.
To enhance low-light images to normally-exposed ones is highly ill-posed, namely that the mapping relationship between them is one-to-many. Previous works based on the pixel-wise reconstruction losses and deterministic processes fail to capture the complex conditional distribution of normally exposed images, which results in improper brightness, residual noise, and artifacts. In this paper, we investigate to model this one-to-many relationship via a proposed normalizing flow model. An invertible network that takes the low-light images/features as the condition and learns to map the distribution of normally exposed images into a Gaussian distribution. In this way, the conditional distribution of the normally exposed images can be well modeled, and the enhancement process, i.e., the other inference direction of the invertible network, is equivalent to being constrained by a loss function that better describes the manifold structure of natural images during the training. The experimental results on the existing benchmark datasets show our method achieves better quantitative and qualitative results, obtaining better-exposed illumination, less noise and artifact, and richer colors.
In inverse problems, we often have access to data consisting of paired samples $(x,y)sim p_{X,Y}(x,y)$ where $y$ are partial observations of a physical system, and $x$ represents the unknowns of the problem. Under these circumstances, we can employ supervised training to learn a solution $x$ and its uncertainty from the observations $y$. We refer to this problem as the supervised case. However, the data $ysim p_{Y}(y)$ collected at one point could be distributed differently than observations $ysim p_{Y}(y)$, relevant for a current set of problems. In the context of Bayesian inference, we propose a two-step scheme, which makes use of normalizing flows and joint data to train a conditional generator $q_{theta}(x|y)$ to approximate the target posterior density $p_{X|Y}(x|y)$. Additionally, this preliminary phase provides a density function $q_{theta}(x|y)$, which can be recast as a prior for the unsupervised problem, e.g.~when only the observations $ysim p_{Y}(y)$, a likelihood model $y|x$, and a prior on $x$ are known. We then train another invertible generator with output density $q_{phi}(x|y)$ specifically for $y$, allowing us to sample from the posterior $p_{X|Y}(x|y)$. We present some synthetic results that demonstrate considerable training speedup when reusing the pretrained network $q_{theta}(x|y)$ as a warm start or preconditioning for approximating $p_{X|Y}(x|y)$, instead of learning from scratch. This training modality can be interpreted as an instance of transfer learning. This result is particularly relevant for large-scale inverse problems that employ expensive numerical simulations.
Image restoration has seen great progress in the last years thanks to the advances in deep neural networks. Most of these existing techniques are trained using full supervision with suitable image pairs to tackle a specific degradation. However, in a blind setting with unknown degradations this is not possible and a good prior remains crucial. Recently, neural network based approaches have been proposed to model such priors by leveraging either denoising autoencoders or the implicit regularization captured by the neural network structure itself. In contrast to this, we propose using normalizing flows to model the distribution of the target content and to use this as a prior in a maximum a posteriori (MAP) formulation. By expressing the MAP optimization process in the latent space through the learned bijective mapping, we are able to obtain solutions through gradient descent. To the best of our knowledge, this is the first work that explores normalizing flows as prior in image enhancement problems. Furthermore, we present experimental results for a number of different degradations on data sets varying in complexity and show competitive results when comparing with the deep image prior approach.
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