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In this paper we investigate a variety of deep learning strategies for solving inverse problems. We classify existing deep learning solutions for inverse problems into three categories of Direct Mapping, Data Consistency Optimizer, and Deep Regularizer. We choose a sample of each inverse problem type, so as to compare the robustness of the three categories, and report a statistical analysis of their differences. We perform extensive experiments on the classic problem of linear regression and three well-known inverse problems in computer vision, namely image denoising, 3D human face inverse rendering, and object tracking, in presence of noise and outliers, are selected as representative prototypes for each class of inverse problems. The overall results and the statistical analyses show that the solution categories have a robustness behaviour dependent on the type of inverse problem domain, and specifically dependent on whether or not the problem includes measurement outliers. Based on our experimental results, we conclude by proposing the most robust solution category for each inverse problem class.

•We provide a tutorial and evaluation of MNE-type and beamforming methods.•We highlight the importance of resolution matrix, point-spread and cross-talk.•We present intuitive resolution metrics to evaluate and compare methods.•We applied these tools to five MNE-type methods and two beamformers.•Point-spread localization error can be low but cross-talk is fundamentally limited. The spatial resolution of EEG/MEG source estimates, often described in terms of source leakage in the context of the inverse problem, poses constraints on the inferences that can be drawn from EEG/MEG source estimation results. Software packages for EEG/MEG data analysis offer a large choice of source estimation methods but few tools to experimental researchers for methods evaluation and comparison. Here, we describe a framework and tools for objective and intuitive resolution analysis of EEG/MEG source estimation based on linear systems analysis, and apply those to the most widely used distributed source estimation methods such as L2-minimum-norm estimation (L2-MNE) and linearly constrained minimum variance (LCMV) beamformers. Within this framework it is possible to define resolution metrics that define meaningful aspects of source estimation results (such as localization accuracy in terms of peak localization error, PLE, and spatial extent in terms of spatial deviation, SD) that are relevant to the task at hand and can easily be visualized. At the core of this framework is the resolution matrix, which describes the potential leakage from and into point sources (point-spread and cross-talk functions, or PSFs and CTFs, respectively). Importantly, for linear methods these functions allow generalizations to multiple sources or complex source distributions. This paper provides a tutorial-style introduction into linear EEG/MEG source estimation and resolution analysis aimed at experimental (rather than methods-oriented) researchers. We used this framework to demonstrate how L2-MNE-type as well as LCMV beamforming methods can be evaluated in practice using software tools that have only recently become available for routine use. Our novel methods comparison includes PLE and SD for a larger number of methods than in similar previous studies, such as unweighted, depth-weighted and normalized L2-MNE methods (including dSPM, sLORETA, eLORETA) and two LCMV beamformers. The results demonstrate that some methods can achieve low and even zero PLE for PSFs. However, their SD as well as both PLE and SD for CTFs are far less optimal for all methods, in particular for deep cortical areas. We hope that our paper will encourage EEG/MEG researchers to apply this approach to their own tasks at hand.

Deep convolutional networks have become a popular tool for image generation and restoration. Generally, their excellent performance is imputed to their ability to learn realistic image priors from a large number of example images. In this paper, we show that, on the contrary, the structure of a generator network is sufficient to capture a great deal of low-level image statistics prior to any learning. In order to do so, we show that a randomly-initialized neural network can be used as a handcrafted prior with excellent results in standard inverse problems such as denoising, super-resolution, and inpainting. Furthermore, the same prior can be used to invert deep neural representations to diagnose them, and to restore images based on flash-no flash input pairs. Apart from its diverse applications, our approach highlights the inductive bias captured by standard generator network architectures. It also bridges the gap between two very popular families of image restoration methods: learning-based methods using deep convolutional networks and learning-free methods based on handcrafted image priors such as self-similarity (Code and supplementary material are available at https://dmitryulyanov.github.io/deep_image_prior).

The need for global damage detection methods that can be applied in complex structures has led to the development of methods that examine the structural dynamic behavior. The damage detection problem can be considered as a inverse problem with minimization of a objective function. For those reasons, a new nature-inspired optimization method based on sunflowers’ motion is introduced. The proposed sunflower optimization algorithm (SFO) technique is a population-based iterative heuristic global optimization algorithm for multi-modal problems. Compared to traditional algorithms, SFO employs terms as root velocity and pollination providing robustness. The new method is then applied in an inverse problem of structural damage detection in composite laminated plates.

Kirigami tessellations, regular planar patterns formed by partially cutting flat, thin sheets, allow compact shapes to morph into open structures with rich geometries and unusual material properties. However, geometric and topological constraints make the design of such structures challenging. Here we pose and solve the inverse problem of determining the number, size and orientation of cuts that enables the deployment of a closed, compact regular kirigami tessellation to conform approximately to any prescribed target shape in two or three dimensions. We first identify the constraints on the lengths and angles of generalized kirigami tessellations that guarantee that their reconfigured face geometries can be contracted from a non-trivial deployed shape to a compact, non-overlapping planar cut pattern. We then encode these conditions into a flexible constrained optimization framework to obtain generalized kirigami patterns derived from various periodic tesselations of the plane that can be deployed into a wide variety of prescribed shapes. A simple mechanical analysis of the resulting structure allows us to determine and control the stability of the deployed state and control the deployment path. Finally, we fabricate physical models that deploy in two and three dimensions to validate this inverse design approach. Altogether, our approach, combining geometry, topology and optimization, highlights the potential for generalized kirigami tessellations as building blocks for shape-morphing mechanical metamaterials.

We propose a prototypical Split Inverse Problem (SIP) and a new variational problem, called the Split Variational Inequality Problem (SVIP), which is a SIP. It entails finding a solution of one inverse problem (e.g., a Variational Inequality Problem (VIP)), the image of which under a given bounded linear transformation is a solution of another inverse problem such as a VIP. We construct iterative algorithms that solve such problems, under reasonable conditions, in Hilbert space and then discuss special cases, some of which are new even in Euclidean space.

Computer-aided engineering (CAE) is now an essential instrument that aids in engineering decision-making. Statistical model calibration and validation has recently drawn great attention in the engineering community for its applications in practical CAE models. The objective of this paper is to review the state-of-the-art and trends in statistical model calibration and validation, based on the available extensive literature, from the perspective of uncertainty structures. After a brief discussion about uncertainties, this paper examines three problem categories—the forward problem, the inverse problem, and the validation problem—in the context of techniques and applications for statistical model calibration and validation.

We consider nonparametric Bayesian inference in a reflected diffusion model dXt = b(Xt)dt + σ(Xt)dWt, with discretely sampled observations X0, XΔ,...,XnΔ. We analyse the nonlinear inverse problem corresponding to the \"low frequency sampling\" regime where Δ > 0 is fixed and n → ∞. A general theorem is proved that gives conditions for prior distributions Π on the diffusion coefficient σ and the drift function b that ensure minimax optimal contraction rates of the posterior distribution over Hölder–Sobolev smoothness classes. These conditions are verified for natural examples of nonparametric random wavelet series priors. For the proofs, we derive new concentration inequalities for empirical processes arising from discretely observed diffusions that are of independent interest.

We consider the statistical inverse problem of recovering a function f : M → ℝ, where M is a smooth compact Riemannian manifold with boundary, from measurements of general X-ray transforms Iₐ(f) of f, corrupted by additive Gaussian noise. For M equal to the unit disk with “flat” geometry and a = 0 this reduces to the standard Radon transform, but our general setting allows for anisotropic media M and can further model local “attenuation” effects—both highly relevant in practical imaging problems such as SPECT tomography. We study a nonparametric Bayesian inference method based on standard Gaussian process priors for f. The posterior reconstruction of f corresponds to a Tikhonov regulariser with a reproducing kernel Hilbert space norm penalty that does not require the calculation of the singular value decomposition of the forward operator Iₐ. We prove Bernstein–von Mises theorems for a large family of one-dimensional linear functionals of f, and they entail that posterior-based inferences such as credible sets are valid and optimal from a frequentist point of view. In particular we derive the asymptotic distribution of smooth linear functionals of the Tikhonov regulariser, which attains the semiparametric information lower bound. The proofs rely on an invertibility result for the “Fisher information” operator I a * I a between suitable function spaces, a result of independent interest that relies on techniques from microlocal analysis. We illustrate the performance of the proposed method via simulations in various settings.