Deep Learning Methods for Solving Linear Inverse Problems: Research Directions and Paradigms2020-07-26 ${\displaystyle \cong }$ |

The linear inverse problem is fundamental to the development of various scientific areas. Innumerable attempts have been carried out to solve different variants of the linear inverse problem in different applications. Nowadays, the rapid development of deep learning provides a fresh perspective for solving the linear inverse problem, which has various well-designed network architectures results in state-of-the-art performance in many applications. In this paper, we present a comprehensive survey of the recent progress in the development of deep learning for solving various linear inverse problems. We review how deep learning methods are used in solving different linear inverse problems, and explore the structured neural network architectures that incorporate knowledge used in traditional methods. Furthermore, we identify open challenges and potential future directions along this research line. |

Regularization of Inverse Problems by Neural Networks2020-06-06 ${\displaystyle \cong }$ |

Inverse problems arise in a variety of imaging applications including computed tomography, non-destructive testing, and remote sensing. The characteristic features of inverse problems are the non-uniqueness and instability of their solutions. Therefore, any reasonable solution method requires the use of regularization tools that select specific solutions and at the same time stabilize the inversion process. Recently, data-driven methods using deep learning techniques and neural networks demonstrated to significantly outperform classical solution methods for inverse problems. In this chapter, we give an overview of inverse problems and demonstrate the necessity of regularization concepts for their solution. We show that neural networks can be used for the data-driven solution of inverse problems and review existing deep learning methods for inverse problems. In particular, we view these deep learning methods from the perspective of regularization theory, the mathematical foundation of stable solution methods for inverse problems. This chapter is more than just a review as many of the presented theoretical results extend existing ones. |

A General Framework Combining Generative Adversarial Networks and Mixture Density Networks for Inverse Modeling in Microstructural Materials Design2021-01-25 ${\displaystyle \cong }$ |

Microstructural materials design is one of the most important applications of inverse modeling in materials science. Generally speaking, there are two broad modeling paradigms in scientific applications: forward and inverse. While the forward modeling estimates the observations based on known parameters, the inverse modeling attempts to infer the parameters given the observations. Inverse problems are usually more critical as well as difficult in scientific applications as they seek to explore the parameters that cannot be directly observed. Inverse problems are used extensively in various scientific fields, such as geophysics, healthcare and materials science. However, it is challenging to solve inverse problems, because they usually need to learn a one-to-many non-linear mapping, and also require significant computing time, especially for high-dimensional parameter space. Further, inverse problems become even more difficult to solve when the dimension of input (i.e. observation) is much lower than that of output (i.e. parameters). In this work, we propose a framework consisting of generative adversarial networks and mixture density networks for inverse modeling, and it is evaluated on a materials science dataset for microstructural materials design. Compared with baseline methods, the results demonstrate that the proposed framework can overcome the above-mentioned challenges and produce multiple promising solutions in an efficient manner. |

Joint learning of variational representations and solvers for inverse problems with partially-observed data2020-06-05 ${\displaystyle \cong }$ |

Designing appropriate variational regularization schemes is a crucial part of solving inverse problems, making them better-posed and guaranteeing that the solution of the associated optimization problem satisfies desirable properties. Recently, learning-based strategies have appeared to be very efficient for solving inverse problems, by learning direct inversion schemes or plug-and-play regularizers from available pairs of true states and observations. In this paper, we go a step further and design an end-to-end framework allowing to learn actual variational frameworks for inverse problems in such a supervised setting. The variational cost and the gradient-based solver are both stated as neural networks using automatic differentiation for the latter. We can jointly learn both components to minimize the data reconstruction error on the true states. This leads to a data-driven discovery of variational models. We consider an application to inverse problems with incomplete datasets (image inpainting and multivariate time series interpolation). We experimentally illustrate that this framework can lead to a significant gain in terms of reconstruction performance, including w.r.t. the direct minimization of the variational formulation derived from the known generative model. |

Adversarial Regularizers in Inverse Problems2019-01-11 ${\displaystyle \cong }$ |

Inverse Problems in medical imaging and computer vision are traditionally solved using purely model-based methods. Among those variational regularization models are one of the most popular approaches. We propose a new framework for applying data-driven approaches to inverse problems, using a neural network as a regularization functional. The network learns to discriminate between the distribution of ground truth images and the distribution of unregularized reconstructions. Once trained, the network is applied to the inverse problem by solving the corresponding variational problem. Unlike other data-based approaches for inverse problems, the algorithm can be applied even if only unsupervised training data is available. Experiments demonstrate the potential of the framework for denoising on the BSDS dataset and for computed tomography reconstruction on the LIDC dataset. |

Numerical Solution of Inverse Problems by Weak Adversarial Networks2020-02-26 ${\displaystyle \cong }$ |

We consider a weak adversarial network approach to numerically solve a class of inverse problems, including electrical impedance tomography and dynamic electrical impedance tomography problems. We leverage the weak formulation of PDE in the given inverse problem, and parameterize the solution and the test function as deep neural networks. The weak formulation and the boundary conditions induce a minimax problem of a saddle function of the network parameters. As the parameters are alternatively updated, the network gradually approximates the solution of the inverse problem. We provide theoretical justifications on the convergence of the proposed algorithm. Our method is completely mesh-free without any spatial discretization, and is particularly suitable for problems with high dimensionality and low regularity on solutions. Numerical experiments on a variety of test inverse problems demonstrate the promising accuracy and efficiency of our approach. |

Provably Convergent Algorithms for Solving Inverse Problems Using Generative Models2021-05-13 ${\displaystyle \cong }$ |

The traditional approach of hand-crafting priors (such as sparsity) for solving inverse problems is slowly being replaced by the use of richer learned priors (such as those modeled by deep generative networks). In this work, we study the algorithmic aspects of such a learning-based approach from a theoretical perspective. For certain generative network architectures, we establish a simple non-convex algorithmic approach that (a) theoretically enjoys linear convergence guarantees for certain linear and nonlinear inverse problems, and (b) empirically improves upon conventional techniques such as back-propagation. We support our claims with the experimental results for solving various inverse problems. We also propose an extension of our approach that can handle model mismatch (i.e., situations where the generative network prior is not exactly applicable). Together, our contributions serve as building blocks towards a principled use of generative models in inverse problems with more complete algorithmic understanding. |

Deep Learning-Based Solvability of Underdetermined Inverse Problems in Medical Imaging2020-06-25 ${\displaystyle \cong }$ |

Recently, with the significant developments in deep learning techniques, solving underdetermined inverse problems has become one of the major concerns in the medical imaging domain. Typical examples include undersampled magnetic resonance imaging, interior tomography, and sparse-view computed tomography, where deep learning techniques have achieved excellent performances. Although deep learning methods appear to overcome the limitations of existing mathematical methods when handling various underdetermined problems, there is a lack of rigorous mathematical foundations that would allow us to elucidate the reasons for the remarkable performance of deep learning methods. This study focuses on learning the causal relationship regarding the structure of the training data suitable for deep learning, to solve highly underdetermined inverse problems. We observe that a majority of the problems of solving underdetermined linear systems in medical imaging are highly non-linear. Furthermore, we analyze if a desired reconstruction map can be learnable from the training data and underdetermined system. |

Solving Inverse Computational Imaging Problems using Deep Pixel-level Prior2018-04-23 ${\displaystyle \cong }$ |

Signal reconstruction is a challenging aspect of computational imaging as it often involves solving ill-posed inverse problems. Recently, deep feed-forward neural networks have led to state-of-the-art results in solving various inverse imaging problems. However, being task specific, these networks have to be learned for each inverse problem. On the other hand, a more flexible approach would be to learn a deep generative model once and then use it as a signal prior for solving various inverse problems. We show that among the various state of the art deep generative models, autoregressive models are especially suitable for our purpose for the following reasons. First, they explicitly model the pixel level dependencies and hence are capable of reconstructing low-level details such as texture patterns and edges better. Second, they provide an explicit expression for the image prior which can then be used for MAP based inference along with the forward model. Third, they can model long range dependencies in images which make them ideal for handling global multiplexing as encountered in various compressive imaging systems. We demonstrate the efficacy of our proposed approach in solving three computational imaging problems: Single Pixel Camera (SPC), LiSens and FlatCam. For both real and simulated cases, we obtain better reconstructions than the state-of-the-art methods in terms of perceptual and quantitative metrics. |

Applications of Deep Learning for Ill-Posed Inverse Problems Within Optical Tomography2020-03-21 ${\displaystyle \cong }$ |

Increasingly in medical imaging has emerged an issue surrounding the reconstruction of noisy images from raw measurement data. Where the forward problem is the generation of raw measurement data from a ground truth image, the inverse problem is the reconstruction of those images from the measurement data. In most cases with medical imaging, classical inverse Radon transforms, such as an inverse Fourier transform for MRI, work well for recovering clean images from the measured data. Unfortunately in the case of X-Ray CT, where undersampled data is very common, more than this is needed to resolve faithful and usable images. In this paper, we explore the history of classical methods for solving the inverse problem for X-Ray CT, followed by an analysis of the state of the art methods that utilize supervised deep learning. Finally, we will provide some possible avenues for research in the future. |

Efficient Decremental Learning Algorithms for Broad Learning System2019-12-30 ${\displaystyle \cong }$ |

The decremented learning algorithms are required in machine learning, to prune redundant nodes and remove obsolete inline training samples. In this paper, an efficient decremented learning algorithm to prune redundant nodes is deduced from the incremental learning algorithm 1 proposed in [9] for added nodes, and two decremented learning algorithms to remove training samples are deduced from the two incremental learning algorithms proposed in [10] for added inputs. The proposed decremented learning algorithm for reduced nodes utilizes the inverse Cholesterol factor of the Herminia matrix in the ridge inverse, to update the output weights recursively, as the incremental learning algorithm 1 for added nodes in [9], while that inverse Cholesterol factor is updated with an unitary transformation. The proposed decremented learning algorithm 1 for reduced inputs updates the output weights recursively with the inverse of the Herminia matrix in the ridge inverse, and updates that inverse recursively, as the incremental learning algorithm 1 for added inputs in [10]. |

Neumann Networks for Inverse Problems in Imaging2019-06-03 ${\displaystyle \cong }$ |

Many challenging image processing tasks can be described by an ill-posed linear inverse problem: deblurring, deconvolution, inpainting, compressed sensing, and superresolution all lie in this framework. Traditional inverse problem solvers minimize a cost function consisting of a data-fit term, which measures how well an image matches the observations, and a regularizer, which reflects prior knowledge and promotes images with desirable properties like smoothness. Recent advances in machine learning and image processing have illustrated that it is often possible to learn a regularizer from training data that can outperform more traditional regularizers. We present an end-to-end, data-driven method of solving inverse problems inspired by the Neumann series, which we call a Neumann network. Rather than unroll an iterative optimization algorithm, we truncate a Neumann series which directly solves the linear inverse problem with a data-driven nonlinear regularizer. The Neumann network architecture outperforms traditional inverse problem solution methods, model-free deep learning approaches, and state-of-the-art unrolled iterative methods on standard datasets. Finally, when the images belong to a union of subspaces and under appropriate assumptions on the forward model, we prove there exists a Neumann network configuration that well-approximates the optimal oracle estimator for the inverse problem and demonstrate empirically that the trained Neumann network has the form predicted by theory. |

Benchmarking deep inverse models over time, and the neural-adjoint method2020-09-27 ${\displaystyle \cong }$ |

We consider the task of solving generic inverse problems, where one wishes to determine the hidden parameters of a natural system that will give rise to a particular set of measurements. Recently many new approaches based upon deep learning have arisen generating impressive results. We conceptualize these models as different schemes for efficiently, but randomly, exploring the space of possible inverse solutions. As a result, the accuracy of each approach should be evaluated as a function of time rather than a single estimated solution, as is often done now. Using this metric, we compare several state-of-the-art inverse modeling approaches on four benchmark tasks: two existing tasks, one simple task for visualization and one new task from metamaterial design. Finally, inspired by our conception of the inverse problem, we explore a solution that uses a deep learning model to approximate the forward model, and then uses backpropagation to search for good inverse solutions. This approach, termed the neural-adjoint, achieves the best performance in many scenarios. |

Solving Inverse Wave Scattering with Deep Learning2019-11-27 ${\displaystyle \cong }$ |

This paper proposes a neural network approach for solving two classical problems in the two-dimensional inverse wave scattering: far field pattern problem and seismic imaging. The mathematical problem of inverse wave scattering is to recover the scatterer field of a medium based on the boundary measurement of the scattered wave from the medium, which is high-dimensional and nonlinear. For the far field pattern problem under the circular experimental setup, a perturbative analysis shows that the forward map can be approximated by a vectorized convolution operator in the angular direction. Motivated by this and filtered back-projection, we propose an effective neural network architecture for the inverse map using the recently introduced BCR-Net along with the standard convolution layers. Analogously for the seismic imaging problem, we propose a similar neural network architecture under the rectangular domain setup with a depth-dependent background velocity. Numerical results demonstrate the efficiency of the proposed neural networks. |

Task adapted reconstruction for inverse problems2018-08-27 ${\displaystyle \cong }$ |

The paper considers the problem of performing a task defined on a model parameter that is only observed indirectly through noisy data in an ill-posed inverse problem. A key aspect is to formalize the steps of reconstruction and task as appropriate estimators (non-randomized decision rules) in statistical estimation problems. The implementation makes use of (deep) neural networks to provide a differentiable parametrization of the family of estimators for both steps. These networks are combined and jointly trained against suitable supervised training data in order to minimize a joint differentiable loss function, resulting in an end-to-end task adapted reconstruction method. The suggested framework is generic, yet adaptable, with a plug-and-play structure for adjusting both the inverse problem and the task at hand. More precisely, the data model (forward operator and statistical model of the noise) associated with the inverse problem is exchangeable, e.g., by using neural network architecture given by a learned iterative method. Furthermore, any task that is encodable as a trainable neural network can be used. The approach is demonstrated on joint tomographic image reconstruction, classification and joint tomographic image reconstruction segmentation. |

Low Shot Learning with Untrained Neural Networks for Imaging Inverse Problems2019-10-23 ${\displaystyle \cong }$ |

Employing deep neural networks as natural image priors to solve inverse problems either requires large amounts of data to sufficiently train expressive generative models or can succeed with no data via untrained neural networks. However, very few works have considered how to interpolate between these no- to high-data regimes. In particular, how can one use the availability of a small amount of data (even $5-25$ examples) to one's advantage in solving these inverse problems and can a system's performance increase as the amount of data increases as well? In this work, we consider solving linear inverse problems when given a small number of examples of images that are drawn from the same distribution as the image of interest. Comparing to untrained neural networks that use no data, we show how one can pre-train a neural network with a few given examples to improve reconstruction results in compressed sensing and semantic image recovery problems such as colorization. Our approach leads to improved reconstruction as the amount of available data increases and is on par with fully trained generative models, while requiring less than $1 \%$ of the data needed to train a generative model. |

A machine learning approach to reconstruction of heart surface potentials from body surface potentials2018-01-19 ${\displaystyle \cong }$ |

Invasive cardiac catheterisation is a common procedure that is carried out before surgical intervention. Yet, invasive cardiac diagnostics are full of risks, especially for young children. Decades of research has been conducted on the so called inverse problem of electrocardiography, which can be used to reconstruct Heart Surface Potentials (HSPs) from Body Surface Potentials (BSPs), for non-invasive diagnostics. State of the art solutions to the inverse problem are unsatisfactory, since the inverse problem is known to be ill-posed. In this paper we propose a novel approach to reconstructing HSPs from BSPs using a Time-Delay Artificial Neural Network (TDANN). We first design the TDANN architecture, and then develop an iterative search space algorithm to find the parameters of the TDANN, which results in the best overall HSP prediction. We use real-world recorded BSPs and HSPs from individuals suffering from serious cardiac conditions to validate our TDANN. The results are encouraging, in that coefficients obtained by correlating the predicted HSP with the recorded patient' HSP approach ideal values. |

Denoising Score-Matching for Uncertainty Quantification in Inverse Problems2020-11-16 ${\displaystyle \cong }$ |

Deep neural networks have proven extremely efficient at solving a wide rangeof inverse problems, but most often the uncertainty on the solution they provideis hard to quantify. In this work, we propose a generic Bayesian framework forsolving inverse problems, in which we limit the use of deep neural networks tolearning a prior distribution on the signals to recover. We adopt recent denoisingscore matching techniques to learn this prior from data, and subsequently use it aspart of an annealed Hamiltonian Monte-Carlo scheme to sample the full posteriorof image inverse problems. We apply this framework to Magnetic ResonanceImage (MRI) reconstruction and illustrate how this approach not only yields highquality reconstructions but can also be used to assess the uncertainty on particularfeatures of a reconstructed image. |

NETT: Solving Inverse Problems with Deep Neural Networks2019-12-08 ${\displaystyle \cong }$ |

Recovering a function or high-dimensional parameter vector from indirect measurements is a central task in various scientific areas. Several methods for solving such inverse problems are well developed and well understood. Recently, novel algorithms using deep learning and neural networks for inverse problems appeared. While still in their infancy, these techniques show astonishing performance for applications like low-dose CT or various sparse data problems. However, there are few theoretical results for deep learning in inverse problems. In this paper, we establish a complete convergence analysis for the proposed NETT (Network Tikhonov) approach to inverse problems. NETT considers data consistent solutions having small value of a regularizer defined by a trained neural network. We derive well-posedness results and quantitative error estimates, and propose a possible strategy for training the regularizer. Our theoretical results and framework are different from any previous work using neural networks for solving inverse problems. A possible data driven regularizer is proposed. Numerical results are presented for a tomographic sparse data problem, which demonstrate good performance of NETT even for unknowns of different type from the training data. To derive the convergence and convergence rates results we introduce a new framework based on the absolute Bregman distance generalizing the standard Bregman distance from the convex to the non-convex case. |

Uncertainty Quantification by Ensemble Learning for Computational Optical Form Measurements2021-03-01 ${\displaystyle \cong }$ |

Uncertainty quantification by ensemble learning is explored in terms of an application from computational optical form measurements. The application requires to solve a large-scale, nonlinear inverse problem. Ensemble learning is used to extend a recently developed deep learning approach for this application in order to provide an uncertainty quantification of its predicted solution to the inverse problem. By systematically inserting out-of-distribution errors as well as noisy data the reliability of the developed uncertainty quantification is explored. Results are encouraging and the proposed application exemplifies the ability of ensemble methods to make trustworthy predictions on high dimensional data in a real-world application. |