Recovering edges in ill-posed inverse problems: optimality of curvelet frames

Distribution of electrical potentials over the surface of the heart, which is called the epicardial potential distribution, is a valuable tool to understand whether there is a defect in the heart. Direct measurement of these potentials requires highly invasive procedures. An alternative is to reconstruct these epicardial potentials non-invasively from the body surface potentials, which constitutes one form of the ill-posed inverse problem of electrocardiography (ECG). The goal of this study is to solve the inverse problem of ECG using several regularization methods and compare their performances. We employed Tikhonov Regularization, Truncated Singular Value Decomposition (TSVD), Least Squares QR (LSQR) methods in this study. We compared the effectiveness of these regularization methods to solve the ill-posed inverse ECG problem. Some of the regularization methods require a regularization parameter to solve the inverse problem. We used the well-known L-Curve method to obtain the regularization parameter. The performance of the regularization methods for solving the inverse ECG problem was also evaluated based on a realistic heart-torso model simulation protocol. In this paper, we also investigated the usage of genetic algorithm (GA) for regularizing the ill-posed inverse ECG problem. The results showed that GA can be applied to regularize the ill-posed problem when combined with the results of conventional regularization methods or additional information about solutions.

When we use a discretization by finite differences, to solve differential equations we find problems at the border of the domain of the solution. If the solution is also immersed in a ill-posed inverse problem; we can find very bad solutions. In this paper we apply a discretization of two - sided difference quotients method to solve Ill-posed inverse self-adjoint elliptic problem [Kirsch(2011)]. Some numerical examples showing the effectiveness of this method and we will use mollification techniques to smooth the solutions.

The article discusses the problem of determining the learning outcomes for educational programs of the university, including at the program level and the level of individual disciplines included in the program. The goal is to substantiate the understanding of the definition of learning outcomes at a university as an inverse ill-posed problem within the framework of the theory of ill-posed problems, which will allow further reengineering of the educational process at a university based on models of inverse and ill-posed problems. The article identifies the types of uncertainty inherent in education, which affect the possibility of an incorrect decision. The publication substantiates a new understanding of the educational process focused on learning outcomes at a university, as a process of finding a solution to an inverse incorrect problem. In this case, the learning outcomes formulated for a specific educational program are considered as an available solution to the inverse problem. The research carried out by the authors made it possible to construct a mathematical model of the educational process at the university and identify its components. The authors demonstrate that the optimal way to design the educational process is to design it as an inverse problem, and the list of disciplines, their content and the corresponding learning outcomes are determined and developed at subsequent stages, focusing on the goals of the program.

Medical image reconstruction is typically an ill-posed inverse problem. In order to address such ill-posed problems, the prior distribution of the sought after object property is usually incorporated by means of some sparsity-promoting regularization. Recently, prior distributions for images estimated using generative adversarial networks (GANs) have shown great promise in regularizing some of these image reconstruction problems. In this work, we apply an image-adaptive GAN-based reconstruction method (IAGAN) to reconstruct high fidelity images from incomplete medical imaging data. It is observed that the IAGAN method can potentially recover fine structures in the object that are relevant for medical diagnosis but may be oversmoothed in reconstructions with traditional sparsity-promoting regularization.

Accelerometers have been widely used in almost every area of science and engineering. They are supposed to be physical instruments for measuring the acceleration of a moving object. Although huge technological advance is made in hardware of accelerometers over more than one century, accelerometers have been persistently designed and fabricated mechanically under the framework of forward problems of damped mass–spring systems. We show that accelerometers are essentially inverse ill-posed source problems from the mathematical point of view, implying that small measurement errors of equivalent displacements are inherently amplified significantly such that accelerations output from accelerometers can become extremely noisy, numerically incorrect and physically meaningless. The ill-posedness of accelerometers has been always implicitly circumvented approximately for more than one century. As a result, accelerometers theoretically can only produce approximate outputs of acceleration. Here we present the concept of computerized accelerometers, because inverse ill-posed problems, as in the case of accelerometers, cannot be rigorously solved mechanically. The acceleration can only rigorously be reconstructed computationally as a regularized solution to the inverse ill-posed source problem of acceleration from equivalent displacements. The concept of computerized accelerometers theoretically warrants precise measurement of acceleration without approximation, is valid for nonlinear time-dependent damping as well and provides a turning point for accelerometers to become a fully rigorous computerized physical instrument. Simulated examples have confirmed that least squares reconstruction of acceleration can be too extremely noisy to be physically meaningful and shown that elastic acceleration is incorrect by more than 100%.

We consider ill-posed inverse problems where the forward operator T is unknown, and instead we have access to training data consisting of functions fi and their noisy images Tfi. This is a practically relevant and challenging problem which current methods are able to solve only under strong assumptions on the training set. Here we propose a new method that requires minimal assumptions on the data, and prove reconstruction rates that depend on the number of training points and the noise level. We show that, in the regime of “many” training data, the method is minimax optimal. The proposed method employs a type of convolutional neural networks (U-nets) and empirical risk minimization in order to “fit” the unknown operator. In a nutshell, our approach is based on two ideas: the first is to relate U-nets to multiscale decompositions such as wavelets, thereby linking them to the existing theory, and the second is to use the hierarchical structure of U-nets and the low number of parameters of convolutional neural nets to prove entropy bounds that are practically useful. A significant difference with the existing works on neural networks in nonparametric statistics is that we use them to approximate operators and not functions, which we argue is mathematically more natural and technically more convenient.

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.

In this paper we study the relation between convergence rates of spectral regularization methods under Hölder-type source conditions resulting from the theory of ill-posed inverse problems, when the noise level $\delta$ goes to 0, and convergence rates resulting from statistical kernel learning, when the number of samples n goes to infinity. Toward this aim, we introduce a family of hybrid estimators in the statistical learning context whose convergence rates have the following properties: first, they are equal to those of spectral methods, and second, they are connected to the rates of spectral regularization in ill-posed inverse problems, provided that a suitable inverse proportionality relation between n and $\delta$ holds true. This family of estimators allows us to convert upper rates depending on $n$ to upper rates depending on $\delta$ and to convert lower rates vice versa, quantifying their deviation. The analysis is carried out under general source conditions in the case the rank of the forward operator is both finite and infinite, and, in the latter case, both by not making any assumptions on the eigenvalues and by assuming a polynomial eigenvalue decay.

Many works have shown strong connections between learning and regularization techniques for ill-posed inverse problems. A careful analysis shows that a rigorous connection between learning and regularization for inverse problem is not straightforward. In this study, pattern recognition will be viewed as an ill-posed inverse problem and applications of methods from the theory of inverse problems to pattern recognition are studied. A new learning algorithm derived from a well-known regularization model is generated and applied to the task of reconstruction of an inhomogeneous object as pattern recognition. Particularly, it is demonstrated that pattern recognition can be reformulated in terms of inverse problems defined by a Riesz-type kernel. This reformulation can be employed to design a learning algorithm based on a numerical solution of a system of linear equations. Finally, numerical experiments have been carried out with synthetic experimental data considering a reasonable level of noise. Good recoveries have been achieved with this methodology, and the results of these simulations are compatible with the existing methods. The comparison results show that the Regularization-based learning algorithm (RBA) obtains a promising performance on the majority of the test problems. In prospects, this method can be used for the creation of automated systems for diagnostics, testing, and control in various fields of scientific and applied research, as well as in industry.

Ultrasound tomography (UT) is a noninvasive procedure that can be used to detect breast cancer. Yet, to accomplish this, reconstruction algorithms must solve an inherent nonlinear, ill-posed inverse problem. One solution is to use the distorted Born iterative (DBI) method. However, in order for successful convergence, ill-posed inverse problems must also be solved for each individual iteration. We used the Tikhonov regularization with different algorithms for choosing the regularization parameter that provides optimal balance, a solution neither overregularized nor underregularized. In this article, we propose a novel algorithm for choosing a balanced parameter based on minimizing two inversely proportional components: signal loss and scaled noise errors (SNEs). This begins with an overestimation of the noise in the measured data, which is then appropriately adjusted within each iteration of the DBI method using the discrepancy between measured and calculated data. We compared our algorithm to the L-curve method, as well as generalized cross-validation (GCV) and projection-based regularized total least-squares (PB-RTLS) methods. Four numerical simulations with varying noise levels and aperture settings showed that our algorithm provided the lowest relative error (RE) for phantom reconstruction, signifying image quality compared to the other methods.

In this work, we unveil an analogy between the well-known lattice based learning with error problem and ill-posed inverse problems. We show that LWE problem is a structured inverse problem. Further, we propose a symmetric encryption scheme based on ill-posed problems and thoroughly discuss its security. Finally, we propose a public key encryption scheme based on our symmetric encryption scheme and CRYSTALS-Kyber KEM (key encapsulation mechanism) and discuss its security.

Limited-view Neutron CT Reconstruction with Sample Boundary

The task of Electrocardiographic Imaging is an ill-posed inverse problem, requiring regularization. However, it has special features, firstly because it is a "non-stationary" inverse problem, and secondly because the inherent dynamical variety (e.g., epicardial breakthroughs, arrhythmias, ischemic changes) may preclude a fruitful nontrivial process model. Importantly, its structure places it in the category of "partial inverse problems" - a theory that arises from this setting. Surprising features of the resulting regularization methodology include the ability to fashion nontrivial regularization matrices in part (and sometimes entirely) from the data. There is evidence that these theoretical results can have significant practical benefits.

The reconstruction of cone-beam x-ray luminescence computed tomography (CB-XLCT) is an ill-posed inversion problem because of incomplete data and lack of prior information. To improve the illness of the inversion problem, the data fidelity and regularization term are two key aspects for the reconstruction model. However, there is not much research considering the statistical characteristics of data in XLCT reconstruction, although many various regularizations are studied. To make full use of the data noise model, a strategy combing the maximum likelihood expectation estimation (MLEM) algorithm and the regularization-type algorithm is proposed. In the MLEM algorithm, the Poisson noise is considered for accurate data model. The result by the regularization-type algorithm is used as the specific initial image for the MLEM to improve the reconstruction quality and convergence speed of the MLEM. There are two main steps in the proposed strategy. Firstly, the fast iterative shrinkage-thresholding algorithm (FISTA) with a large regularization parameter is used to get the sparse solution quickly. Secondly, the sparse solution is used as the initial iteration value of the MLEM. The proposed algorithm is named as FISTA-MLEM. Through the stepwise strategy, the image sparsity is guaranteed and the accuracy of the reconstruction is maintained. Result of phantom experiment shows the FISTA-MLEM method presents better contrast to noise ratio and shape similarity compared with other traditional methods, such as ART, Tikhonov, FISTA and TSVD.

We investigate the ill-posedness of the inverse biosensor problem when the biosensor signals are corrupted by noise. To solve the problem, we employ feed-forward and convolutional neural networks. Computational experiments were performed with different levels of additive and multiplicative noises for the batch and flow injection analysis modes of the biosensor. Obtained results show that the largest errors of recovered concentrations are located on the edges of the training domain. We have found that the inverse problem is less ill-posed in the flow injection analysis mode and concentrations can be reliably recovered for higher levels of noise compared to the batch mode. This finding is confirmed by the application of the DIRECT global optimization method to the considered inverse biosensor problem.