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In recent years, deep learning has emerged as a promising technique for medical image analysis. However, this application domain is likely to suffer from a limited availability of large public datasets and annotations. A common solution to these challenges in deep learning is the usage of a transfer learning framework, typically with a fine-tuning protocol, where a large-scale source dataset is used to pre-train a model, further fine-tuned on the target dataset. In this paper, we present a systematic study analyzing whether the usage of small-scale in-domain x-ray image datasets may provide any improvement for landmark detection over models pre-trained on large natural image datasets only. We focus on the multi-landmark localization task for three datasets, including chest, head, and hand x-ray images. Our results show that using in-domain source datasets brings marginal or no benefit with respect to an ImageNet out-of-domain pre-training. Our findings can provide an indication for the development of robust landmark detection systems in medical images when no large annotated dataset is available., Comment: Accepted to ISBI 2024

We consider the inverse scattering problem for time-harmonic acoustic waves in a medium with pointwise inhomogeneities. In the Foldy-Lax model, the estimation of the scatterers' locations and intensities from far field measurements can be recast as the recovery of a discrete measure from nonlinear observations. We propose a "linearize and locally optimize" approach to perform this reconstruction. We first solve a convex program in the space of measures (known as the Beurling LASSO), which involves a linearization of the forward operator (the far field pattern in the Born approximation). Then, we locally minimize a second functional involving the nonlinear forward map, using the output of the first step as initialization. We provide guarantees that the output of the first step is close to the sought-after measure when the scatterers have small intensities and are sufficiently separated. We also provide numerical evidence that the second step still allows for accurate recovery in settings that are more involved.

In inverse problems, it is widely recognized that the incorporation of a sparsity prior yields a regularization effect on the solution. This approach is grounded on the a priori assumption that the unknown can be appropriately represented in a basis with a limited number of significant components, while most coefficients are close to zero. This occurrence is frequently observed in real-world scenarios, such as with piecewise smooth signals. In this study, we propose a probabilistic sparsity prior formulated as a mixture of degenerate Gaussians, capable of modeling sparsity with respect to a generic basis. Under this premise, we design a neural network that can be interpreted as the Bayes estimator for linear inverse problems. Additionally, we put forth both a supervised and an unsupervised training strategy to estimate the parameters of this network. To evaluate the effectiveness of our approach, we conduct a numerical comparison with commonly employed sparsity-promoting regularization techniques, namely LASSO, group LASSO, iterative hard thresholding, and sparse coding/dictionary learning. Notably, our reconstructions consistently exhibit lower mean square error values across all $1$D datasets utilized for the comparisons, even in cases where the datasets significantly deviate from a Gaussian mixture model.

Representing a manifold of very high-dimensional data with generative models has been shown to be computationally efficient in practice. However, this requires that the data manifold admits a global parameterization. In order to represent manifolds of arbitrary topology, we propose to learn a mixture model of variational autoencoders. Here, every encoder-decoder pair represents one chart of a manifold. We propose a loss function for maximum likelihood estimation of the model weights and choose an architecture that provides us the analytical expression of the charts and of their inverses. Once the manifold is learned, we use it for solving inverse problems by minimizing a data fidelity term restricted to the learned manifold. To solve the arising minimization problem we propose a Riemannian gradient descent algorithm on the learned manifold. We demonstrate the performance of our method for low-dimensional toy examples as well as for deblurring and electrical impedance tomography on certain image manifolds.

Compressed sensing allows for the recovery of sparse signals from few measurements, whose number is proportional to the sparsity of the unknown signal, up to logarithmic factors. The classical theory typically considers either random linear measurements or subsampled isometries and has found many applications, including accelerated magnetic resonance imaging, which is modeled by the subsampled Fourier transform. In this work, we develop a general theory of infinite-dimensional compressed sensing for abstract inverse problems, possibly ill-posed, involving an arbitrary forward operator. This is achieved by considering a generalized restricted isometry property, and a quasi-diagonalization property of the forward map. As a notable application, for the first time, we obtain rigorous recovery estimates for the sparse Radon transform (i.e., with a finite number of angles $\theta_1,\dots,\theta_m$), which models computed tomography, in both the parallel-beam and the fan-beam settings. In the case when the unknown signal is $s$-sparse with respect to an orthonormal basis of compactly supported wavelets, we prove stable recovery under the condition \[ m\gtrsim s, \] up to logarithmic factors., Comment: 57 pages

In this work, we present and study Continuous Generative Neural Networks (CGNNs), namely, generative models in the continuous setting: the output of a CGNN belongs to an infinite-dimensional function space. The architecture is inspired by DCGAN, with one fully connected layer, several convolutional layers and nonlinear activation functions. In the continuous $L^2$ setting, the dimensions of the spaces of each layer are replaced by the scales of a multiresolution analysis of a compactly supported wavelet. We present conditions on the convolutional filters and on the nonlinearity that guarantee that a CGNN is injective. This theory finds applications to inverse problems, and allows for deriving Lipschitz stability estimates for (possibly nonlinear) infinite-dimensional inverse problems with unknowns belonging to the manifold generated by a CGNN. Several numerical simulations, including signal deblurring, illustrate and validate this approach., Comment: 40 pages, 8 figures

This paper proposes a spatial model with a realistic geography where a continuous distribution of agents (e.g., farmers) engages in economic interactions with one location from a finite set (e.g., cities). The spatial structure of the equilibrium consists of a tessellation, i.e., a partition of space into a collection of mutually exclusive market areas. After proving the existence of a unique equilibrium, we characterize how the location of borders and, in the case with mobile labor, the set of inhabited cities change in response to economic shocks. To deal with a two-dimensional space, we draw on tools from computational geometry and from the theory of shape optimization. Finally, we provide an empirical application to illustrate the usefulness of the framework for applied work., Comment: revised version, 64 pages

In this work, we consider the linear inverse problem $y=Ax+\epsilon$, where $A\colon X\to Y$ is a known linear operator between the separable Hilbert spaces $X$ and $Y$, $x$ is a random variable in $X$ and $\epsilon$ is a zero-mean random process in $Y$. This setting covers several inverse problems in imaging including denoising, deblurring, and X-ray tomography. Within the classical framework of regularization, we focus on the case where the regularization functional is not given a priori but learned from data. Our first result is a characterization of the optimal generalized Tikhonov regularizer, with respect to the mean squared error. We find that it is completely independent of the forward operator $A$ and depends only on the mean and covariance of $x$. Then, we consider the problem of learning the regularizer from a finite training set in two different frameworks: one supervised, based on samples of both $x$ and $y$, and one unsupervised, based only on samples of $x$. In both cases, we prove generalization bounds, under some weak assumptions on the distribution of $x$ and $\epsilon$, including the case of sub-Gaussian variables. Our bounds hold in infinite-dimensional spaces, thereby showing that finer and finer discretizations do not make this learning problem harder. The results are validated through numerical simulations.

We consider the problem of the detection of brain hemorrhages from three dimensional (3D) electrical impedance tomography (EIT) measurements. This is a condition requiring urgent treatment for which EIT might provide a portable and quick diagnosis. We employ two neural network architectures -- a fully connected and a convolutional one -- for the classification of hemorrhagic and ischemic strokes. The networks are trained on a dataset with $40\,000$ samples of synthetic electrode measurements generated with the complete electrode model on realistic heads with a 3-layer structure. We consider changes in head anatomy and layers, electrode position, measurement noise and conductivity values. We then test the networks on several datasets of unseen EIT data, with more complex stroke modeling (different shapes and volumes), higher levels of noise and different amounts of electrode misplacement. On most test datasets we achieve $\geq 90\%$ average accuracy with fully connected neural networks, while the convolutional ones display an average accuracy $\geq 80\%$. Despite the use of simple neural network architectures, the results obtained are very promising and motivate the applications of EIT-based classification methods on real phantoms and ultimately on human patients., Comment: 17 pages, 11 figures

Photoacoustic tomography (PAT) is an emerging imaging modality that aims at measuring the high-contrast optical properties of tissues by means of high-resolution ultrasonic measurements. The interaction between these two types of waves is based on the thermoacoustic effect. In recent years, many works have investigated the applicability of compressed sensing to PAT, in order to reduce measuring times while maintaining a high reconstruction quality. However, in most cases, theoretical guarantees are missing. In this work, we show that in many measurement setups of practical interest, compressed sensing PAT reduces to compressed sensing for undersampled Fourier measurements. This is achieved by applying known reconstruction formulae in the case of the free-space model for wave propagation, and by applying the theories of Riesz bases and nonuniform Fourier series in the case of the bounded domain model. Extensive numerical simulations illustrate and validate the approach., Comment: 38 pages, 8 figures

We consider abstract inverse problems between infinite-dimensional Banach spaces. These inverse problems are typically nonlinear and ill-posed, making the inversion with limited and noisy measurements a delicate process. In this work, we assume that the unknown belongs to a finite-dimensional manifold: this assumption arises in many real-world scenarios where natural objects have a low intrinsic dimension and belong to a certain submanifold of a much larger ambient space. We prove uniqueness and H\"older and Lipschitz stability results in this general setting, also in the case when only a finite discretization of the measurements is available. Then, a Landweber-type reconstruction algorithm from a finite number of measurements is proposed, for which we prove global convergence, thanks to a new criterion for finding a suitable initial guess. These general results are then applied to several examples, including two classical nonlinear ill-posed inverse boundary value problems. The first is Calder\'on's inverse conductivity problem, for which we prove a Lipschitz stability estimate from a finite number of measurements for piece-wise constant conductivities with discontinuities on an unknown triangle. A similar stability result is then obtained for Gel'fand-Calder\'on's problem for the Schr\"odinger equation, in the case of piece-wise constant potentials with discontinuities on a finite number of non-intersecting balls., Comment: 70 pages, 5 figures. This revised version contains the application of the abstract results to the Calder\'on problem with a triangular inclusion and to the Gel'fand-Calder\'on problem with piecewise constant potentials on non-intersecting balls

Electrical Impedance Tomography (EIT) is an emerging non-invasive medical imaging modality. It is based on feeding electrical currents into the patient, measuring the resulting voltages at the skin, and recovering the internal conductivity distribution. The mathematical task of EIT image reconstruction is a nonlinear and ill-posed inverse problem. Therefore any EIT image reconstruction method needs to be regularized, typically resulting in blurred images. One promising application is stroke-EIT, or classification of stroke into either ischemic or hemorrhagic. Ischemic stroke involves a blood clot, preventing blood flow to a part of the brain causing a low-conductivity region. Hemorrhagic stroke means bleeding in the brain causing a high-conductivity region. In both cases the symptoms are identical, so a cost-effective and portable classification device is needed. Typical EIT are not optimal for stroke-EIT because of blurriness. This paper explores the possibilities of machine learning in improving the classification results. Two paradigms are compared: (a) learning from the EIT data, that is Dirichlet-to-Neumann (DN) maps and (b) extracting robust features from data and learning from them. The features of choice are Virtual Hybrid Edge Detection (VHED) functions [Greenleaf {\it et al.}, Analysis \& PDE 11, 2018] that have a geometric interpretation and whose computation from EIT data does not involve calculating a full image of the conductivity. We report the measures of accuracy, sensitivity and specificity of the networks trained with EIT data and VHED functions separately. Computational evidence based on simulated noisy EIT data suggests that the regularized grey-box paradigm (b) leads to significantly better classification results than the black-box paradigm (a).

We prove a local Lipschitz stability estimate for Gel'fand-Calder\'on's inverse problem for the Schr\"odinger equation. The main novelty is that only a finite number of boundary input data is available, and those are independent of the unknown potential, provided it belongs to a known finite-dimensional subspace of $L^\infty$. A similar result for Calder\'on's problem is obtained as a corollary. This improves upon two previous results of the authors on several aspects, namely the number of measurements and the stability with respect to mismodeling errors. A new iterative reconstruction scheme based on the stability result is also presented, for which we prove exponential convergence in the number of iterations and stability with respect to noise in the data and to mismodeling errors., Comment: 19 pages

We consider the problem of the recovery of a Robin coefficient on a part $\gamma \subset \partial \Omega$ of the boundary of a bounded domain $\Omega$ from the principal eigenvalue and the boundary values of the normal derivative of the principal eigenfunction of the Laplace operator with Dirichlet boundary condition on $\partial \Omega \setminus \gamma$. We prove uniqueness, as well as local Lipschitz stability of the inverse problem. Moreover, we present an iterative reconstruction algorithm with numerical computations in two dimensions showing the accuracy of the method., Comment: 18 pages, 4 figures

We present a general framework to study uniqueness, stability and reconstruction for infinite-dimensional inverse problems when only a finite-dimensional approximation of the measurements is available. For a large class of inverse problems satisfying Lipschitz stability we show that the same estimate holds even with a finite number of measurements. We also derive a globally convergent reconstruction algorithm based on the Landweber iteration. This theory applies to nonlinear ill-posed problems such as electrical impedance tomography, inverse scattering and quantitative photoacoustic tomography, under the assumption that the unknown belongs to a finite-dimensional subspace., Comment: 26 pages

The unique determination of a measurable conductivity from the Dirichlet-to-Neumann map of the equation $\mathrm{div} (\sigma \nabla u) = 0$ is the subject of this note. A new strategy, based on Clifford algebras and a higher dimensional analogue of the Beltrami equation, is here proposed. This represents a possible first step for a proof of uniqueness for the Calder\'on problem in three and higher dimensions in the $L^\infty$ case., Comment: 11 pages

We prove that an $L^\infty$ potential in the Schr\"odinger equation in three and higher dimensions can be uniquely determined from a finite number of boundary measurements, provided it belongs to a known finite dimensional subspace $\mathcal W$. As a corollary, we obtain a similar result for Calder\'on's inverse conductivity problem. Lipschitz stability estimates and a globally convergent nonlinear reconstruction algorithm for both inverse problems are also presented. These are the first results on global uniqueness, stability and reconstruction for nonlinear inverse boundary value problems with finitely many measurements. We also discuss a few relevant examples of finite dimensional subspaces $\mathcal W$, including bandlimited and piecewise constant potentials, and explicitly compute the number of required measurements as a function of $\dim \mathcal W$., Comment: 15 pages

We consider a compressed sensing problem in which both the measurement and the sparsifying systems are assumed to be frames (not necessarily tight) of the underlying Hilbert space of signals, which may be finite or infinite dimensional. The main result gives explicit bounds on the number of measurements in order to achieve stable recovery, which depends on the mutual coherence of the two systems. As a simple corollary, we prove the efficiency of nonuniform sampling strategies in cases when the two systems are not incoherent, but only asymptotically incoherent, as with the recovery of wavelet coefficients from Fourier samples. This general framework finds applications to inverse problems in partial differential equations, where the standard assumptions of compressed sensing are often not satisfied. Several examples are discussed, with a special focus on electrical impedance tomography., Comment: 42 pages

In this paper, we develop a shape optimization-based algorithm for the electrical impedance tomography (EIT) problem of determining a piecewise constant conductivity on a polygonal partition from boundary measurements. The key tool is to use a distributed shape derivative of a suitable cost functional with respect to movements of the partition. Numerical simulations showing the robustness and accuracy of the method are presented for simulated test cases in two dimensions.

The ill-posedness of Calder\'on's inverse conductivity problem, responsible for the poor spatial resolution of Electrical Impedance Tomography (EIT), has been an impetus for the development of hybrid imaging techniques, which compensate for this lack of resolution by coupling with a second type of physical wave, typically modeled by a hyperbolic PDE. Here we show how, using EIT data alone, to efficiently detect interior jumps and other singularities of the conductivity. Analysis of the complex geometrical optics solutions of Astala and P\"aiv\"arinta [\emph{Ann. Math.}, {\bf 163} (2006)] in 2D makes it possible to exploit an underlying complex principal type structure of the problem. We show that the leading term in a Neumann series is an invertible nonlinear generalized Radon transform of the conductivity. The wave front set of all higher-order terms can be characterized, and, under a prior, some are smoother than the leading term. Numerics indicate that this approach effectively detects inclusions within inclusions via EIT., Comment: Revised with additional commentary and references. 54 pages, 10 figures

In Electrical Impedance Tomography (EIT) one wants to image the conductivity distribution of a body from current and voltage measurements carried out on its boundary. In this paper we consider the underlying mathematical model, the inverse conductivity problem, in two dimensions and under the realistic assumption that only a part of the boundary is accessible to measurements. In this framework our data are modeled as a partial Neumann-to-Dirichlet map (ND map). We compare this data to the full-boundary ND map and prove that the error depends linearly on the size of the missing part of the boundary. The same linear dependence is further proved for the difference of the reconstructed conductivities -- from partial and full boundary data. The reconstruction is based on a truncated and linearized D-bar method. Auxiliary results include an extrapolation method to obtain the full-boundary data from the measured one, an approximation of the complex geometrical optics solutions computed directly from the ND map as well as an approximate scattering transform for reconstructing the conductivity. Numerical verification of the convergence results and reconstructions are presented for simulated test cases.

A new computational method for reconstructing a potential from the Dirichlet-to-Neumann map at positive energy is developed. The method is based on D-bar techniques and it works in absence of exceptional points -- in particular, if the potential is small enough compared to the energy. Numerical tests reveal exceptional points for perturbed, radial potentials. Reconstructions for several potentials are computed using simulated Dirichlet-to-Neumann maps with and without added noise. The new reconstruction method is shown to work well for energy values between $10^{-5}$ and $5$, smaller values giving better results.

The problem of the recovery of a real-valued potential in the two-dimensional Schrodinger equation at positive energy from the Dirichlet-to-Neumann map is considered. It is know that this problem is severely ill-posed and the reconstruction of the potential is only logarithmic stable in general. In this paper a new stability estimate is proved, which is explicitly dependent on the regularity of the potentials and on the energy. Its main feature is an efficient increasing stability phenomenon at sufficiently high energies: in some sense, the stability rapidly changes from logarithmic type to Holder type. The paper develops also several estimates for a non-local Riemann-Hilbert problem which could be of independent interest., Comment: 34 pages. arXiv admin note: text overlap with arXiv:1204.4610

We study the inverse problem of determining a real-valued potential in the two-dimensional Schr\"odinger equation at negative energy from the Dirichlet-to-Neumann map. It is known that the problem is ill-posed and a stability estimate of logarithmic type holds. In this paper we prove three new stability estimates. The main feature of the first one is that the stability increases exponentially with respect to the smoothness of the potential, in a sense to be made precise. The others show how the first estimate depends on the energy, for low and high energies (in modulus). In particular it is found that for high energies the stability estimate changes, in some sense, from logarithmic type to Lipschitz type: in this sense the ill-posedness of the problem decreases when increasing the energy (in modulus)., Comment: 17 pages

We prove a new global stability estimate for the Gel'fand-Calder\'on inverse problem on a two-dimensional bounded domain or, more precisely, the inverse boundary value problem for the equation $-\Delta \psi + v\, \psi = 0$ on $D$, where $v$ is a smooth real-valued potential of conductivity type defined on a bounded planar domain $D$. The principal feature of this estimate is that it shows that the more a potential is smooth, the more its reconstruction is stable, and the stability varies exponentially with respect to the smoothness (in a sense to be made precise). As a corollary we obtain a similar estimate for the Calder\'on problem for the electrical impedance tomography., Comment: 18 pages

We consider two inverse problems for the multi-channel two-dimensional Schr\"odinger equation at fixed positive energy, i.e. the equation $-\Delta \psi + V(x)\psi = E \psi$ at fixed positive $E$, where $V$ is a matrix-valued potential. The first is the Gel'fand inverse problem on a bounded domain $D$ at fixed energy and the second is the inverse fixed-energy scattering problem on the whole plane $\R^2$. We present in this paper two algorithms which give efficient approximate solutions to these problems: in particular, in both cases we show that the potential $V$ is reconstructed with Lipschitz stability by these algorithms up to $O(E^{-(m-2)/2})$ in the uniform norm as $E \to +\infty$, under the assumptions that $V$ is $m$-times differentiable in $L^1$, for $m \geq 3$, and has sufficient boundary decay.

We prove a global logarithmic stability estimate for the multi-channel Gel'fand-Calder\'on inverse problem on a two-dimensional bounded domain, i.e. the inverse boundary value problem for the equation $-\Delta \psi + v\, \psi = 0$ on $D$, where $v$ is a smooth matrix-valued potential defined on a bounded planar domain $D$.

We study the multi-channel Gel'fand-Calder\'on inverse problem in two dimensions, i.e. the inverse boundary value problem for the equation $-\Delta \psi + v(x) \psi = 0$, $x\in D$, where $v$ is a smooth matrix-valued potential defined on a bounded planar domain $D$. We give an exact global reconstruction method for finding $v$ from the associated Dirichlet-to-Neumann operator. This also yields a global uniqueness results: if two smooth matrix-valued potentials defined on a bounded planar domain have the same Dirichlet-to-Neumann operator then they coincide.

We prove a global logarithmic stability estimate for the Gel'fand-Calderon inverse problem on a two-dimensional domain.

Let $X$ be a smooth bordered surface in $\real^3$ with smooth boundary and $\hat \sigma$ a smooth anisotropic conductivity on $X$. If the genus of $X$ is given, then starting from the Dirichlet-to-Neumann operator $\Lambda_{\hat \sigma}$ on $\partial X$, we give an explicit procedure to find a unique Riemann surface $Y$ (up to a biholomorphism), an isotropic conductivity $\sigma$ on $Y$ and the boundary values of a quasiconformal diffeomorphism $F: X \to Y$ which transforms $\hat \sigma$ into $\sigma$. As a corollary we obtain the following uniqueness result: if $\sigma_1, \sigma_2$ are two smooth anisotropic conductivities on $X$ with $\Lambda_{\sigma_1}= \Lambda_{\sigma_2}$, then there exists a smooth diffeomorphism $\Phi: \bar X \to \bar X$ which transforms $\sigma_1$ into $\sigma_2$., Comment: 21 pages, no figures; added corrections in Theorem 5.1

Let $\hat \Omega \subset \mathbb R^2$ be a bounded domain with smooth boundary and $\hat \sigma$ a smooth anisotropic conductivity on $\hat \Omega$. Starting from the Dirichlet-to-Neumann operator $\Lambda_{\hat \sigma}$ on $\partial \hat \Omega$, we give an explicit procedure to find a unique domain $\Omega$, an isotropic conductivity $\sigma$ on $\Omega$ and the boundary values of a quasiconformal diffeomorphism $F:\hat \Omega \to \Omega$ which transforms $\hat \sigma$ into $\sigma$., Comment: 9 pages, no figure