Typical Inverse Problem Research Articles

Simultaneous optical image compression and encryption using error-reduction phase retrieval algorithm

We report a simultaneous image compression and encryption scheme based on solving a typical optical inverse problem. The secret images to be processed are multiplexed as the input intensities of a cascaded diffractive optical system. At the output plane, a compressed complex-valued data with a lot fewer measurements can be obtained by utilizing error-reduction phase retrieval algorithm. The magnitude of the output image can serve as the final ciphertext while its phase serves as the decryption key. Therefore the compression and encryption are simultaneously completed without additional encoding and filtering operations. The proposed strategy can be straightforwardly applied to the existing optical security systems that involve diffraction and interference. Numerical simulations are performed to demonstrate the validity and security of the proposal.

An agent-oriented hierarchic strategy for solving inverse problems

Abstract The paper discusses the complex, agent-oriented hierarchic memetic strategy (HMS) dedicated to solving inverse parametric problems. The strategy goes beyond the idea of two-phase global optimization algorithms. The global search performed by a tree of dependent demes is dynamically alternated with local, steepest descent searches. The strategy offers exceptionally low computational costs, mainly because the direct solver accuracy (performed by the hp-adaptive finite element method) is dynamically adjusted for each inverse search step. The computational cost is further decreased by the strategy employed for solution inter-processing and fitness deterioration. The HMS efficiency is compared with the results of a standard evolutionary technique, as well as with the multi-start strategy on benchmarks that exhibit typical inverse problems’ difficulties. Finally, an HMS application to a real-life engineering problem leading to the identification of oil deposits by inverting magnetotelluric measurements is presented. The HMS applicability to the inversion of magnetotelluric data is also mathematically verified.

High performance super-resolution reconstruction of multi-frame degraded images with local weighted anisotropy and successive regularization

Multi-frame image super-resolution (SR) is a procedure which takes several observed degraded low-resolution (LR) images, and processes them together to synthesize one or more high-quality high-resolution (HR) images. Due to an insufficient number of LR images and ill-conditioned blur operators, SR image reconstruction is a typical ill-posed inverse problem. However, the regularization approach can be used as an effective means to solve this kind of problems. In this paper, we propose an algorithm for multi-frame SR reconstruction with two new types of regularization terms, termed as local weighted anisotropy regularization and successive regularization toward iteration process, which are characterized by strong ability of suppressing noise and preserving edge information in HR image reconstruction. Furthermore, an iterative refinement procedure obtained superior result by employing Bregman iteration. Experiment results show that the effectiveness of the proposed regularization method and corresponding algorithm for SR reconstruction.

Box-constrained second-order total generalized variation minimization with a combined L 1 , 2 data-fidelity term for image reconstruction

Image reconstruction is a typical ill-posed inverse problem that has attracted increasing attention owing to its extensive use. To cope with the ill-posed nature of this problem, many regularizers have been presented to regularize the reconstruction process. One of the most popular regularizers in the literature is total variation (TV), known for its capability of preserving edges. However, TV-based reconstruction methods often tend to produce staircase-like artifacts since they favor piecewise constant solutions. To overcome this drawback, we propose to develop a second-order total generalized variation (TGVα2)-based image reconstruction model with a combined L1,2 data-fidelity term. The proposed model is applicable for restoration of blurred images with mixed Gaussian-impulse noise, and can be effectively used for undersampled magnetic resonance imaging. To further enhance the image reconstruction, a box constraint is incorporated into the proposed model by simply projecting all pixel values of the reconstructed image to lie in a certain interval (e.g., 0, 1 for normalized images and [0, 255] for 8-bit images). An optimization algorithm based on an alternating direction method of multipliers is developed to solve the proposed box-constrained image reconstruction model. Comprehensive numerical experiments have been conducted to compare our proposed method with some state-of-the-art reconstruction techniques. The experimental results have demonstrated its superior performance in terms of both quantitative evaluation and visual quality.

Reconstruction of Three Dimensional Convex Bodies from the Curvatures of Their Shadows

In this article, we study necessary and sufficient conditions for a function, defined on the space of flags to be the projection curvature radius function for a convex body. This type of inverse problems has been studied by Christoffel, Minkwoski for the case of mean and Gauss curvatures. We suggest an algorithm of reconstruction of a convex body from its projection curvature radius function by finding a representation for the support function of the body. We lead the problem to a system of differential equations of second order on the sphere and solve it applying a consistency method suggested by the author of the article.

The Method of Fundamental Solutions in Solving Coupled Boundary Value Problems for M/EEG

The estimation of neuronal activity in the human brain from electroencephalography (EEG) and magnetoencephalography (MEG) signals is a typical inverse problem whose solution process requires an accurate and fast forward solver. In this paper the method of fundamental solutions is, for the first time, proposed as a meshfree, boundary-type, and easy-to-implement alternative to the boundary element method (BEM) for solving the M/EEG forward problem. The solution of the forward problem is obtained by numerically solving a set of coupled boundary value problems for the three-dimensional Laplace equation. Numerical accuracy, convergence, and computational load are investigated. The proposed method is shown to be a competitive alternative to the state-of-the-art BEM for M/EEG forward solving.

Randomize-Then-Optimize for Sampling and Uncertainty Quantification in Electrical Impedance Tomography

In a typical inverse problem, a spatially distributed parameter in a physical model is estimated from measurements of model output. Since measurements are stochastic in nature, so is any parameter estimate. Moreover, in the Bayesian setting, the choice of regularization corresponds to the definition of the prior probability density function, which in turn is an uncertainty model for the unknown parameters. For both of these reasons, significant uncertainties exist in the solution of an inverse problem. Thus to fully understand the solution, quantifying these uncertainties is important. When the physical model is linear and the error model and prior are Gaussian, the posterior density function is Gaussian with a known mean and covariance matrix. However, the electrical impedance tomography inverse problem is nonlinear, and hence no closed form expression exists for the posterior density. The typical approach for such problems is to sample from the posterior and then use the samples to compute statistics (such as the mean and variance) of the unknown parameters. Sampling methods for electrical impedance tomography have been studied by various authors in the inverse problems community. However, up to this point the focus has been on the development of increasingly sophisticated implementations of the Gibbs sampler, whose samples are known to converge very slowly to the correct density for large-scale problems. In this paper, we implement a recently developed sampling method called randomize-then-optimize (RTO), which provides nearly independent samples for each application of an appropriate numerical optimization algorithm. The sample density for RTO is not the posterior density, but RTO can be used as a very effective proposal within a Metropolis--Hastings algorithm to obtain samples from the posterior. Here our focus is on implementing the method on synthetic examples from electrical impedance tomography, and we show that it is both computationally efficient and provides good results. We also compare RTO performance with the Metropolis adjusted Langevin algorithm and find RTO to be much more efficient.

Exact Reconstruction of Signals in Evolutionary Systems Via Spatiotemporal Trade-off

We consider the problem of spatiotemporal sampling in which an initial state $f$ of an evolution process $f_t=A_tf$ is to be recovered from a combined set of coarse samples from varying time levels $\{t_1,\dots,t_N\}$. This new way of sampling, which we call dynamical sampling, differs from standard sampling since at any fixed time $t_i$ there are not enough samples to recover the function $f$ or the state $f_{t_i}$. Although dynamical sampling is an inverse problem, it differs from the typical inverse problems in which $f$ is to be recovered from $A_Tf$ for a single time $T$. In this paper, we consider signals that are modeled by $\ell^2(\mathbb Z)$ or a shift invariant space $V\subset L^2(\mathbb R)$.

Estimation of the fracture diameter distributions using the maximum entropy principle

Fracture size is often estimated from trace length measurements. It is well recognized to be a typical non-unique inverse problem. This paper presents a distribution-free method for estimating the fracture diameter distributions using moments in conjunction with the maximum entropy principle. In this method, moments are used to characterize the statistical nature of the probability distributions. The method involves the inference of the moments of the true trace lengths from the sampled trace data, the explicit expression of moments of fracture diameters using general stereological relationship, and the estimation of fracture diameter distribution using the maximum entropy principle by a given set of moments of fracture diameters. The proposed method makes it possible to achieve a universal form for the fracture diameter distribution without any particular parametric form, and to match the moments of fracture diameters up to the fourth order. The overall behavior of the method is validated by an example, and the results show that it is capable of estimating the fracture diameter distributions.

Impedance imaging with first-order TV regularization.

EIT problem is a typical inverse problem with serious ill-posedness. In general, regularization techniques are necessary for such ill-posed inverse problems. To overcome ill-posedness, the total variation (TV) regularization is widely used and it is also successfully applied to EIT. For realtime monitoring, a fast and robust image reconstruction algorithm is required. By exploiting recent advances in optimization, we propose a first-order TV algorithm for EIT, which simply consists of matrix-vector multiplications and in which the sparse structure of the system can be easily exploited. Furthermore, a typical smoothing parameter to overcome nondifferentibility of the TV term is not needed and a closed form solution can be applied in part using soft thresholding. It shows a fast reconstruction in the beginning. Numerical experiments using simulated data and real experimental data support our claim.

Numerical solution for an inverse heat source problem by an iterative method

In this paper, we consider a typical inverse heat source problem, that is, we determine two separable source terms in a heat equation from the initial and boundary data along with two additional measurements. By a simple transformation, the original problem is changed into a nonlinear problem, and then we use an iterative method to solve it. After giving an algorithm, we prove some Hölder convergence rates for both the reconstructed heat source terms and the temperature distribution subject to certain bounds of the data. Numerical results show that our method is accurate and efficient.

Core axial power shape reconstruction based on radial basis function neural network

The core axial power shape reconstruction method based on radial basis function neural network (RBFNN) is proposed, and 18-node axial core power shape can be reconstructed from 6-segment in-core detector signals or 6-segment ex-core detector signals. Alternating conditional expectation algorithm is also used to validate the effectiveness of RBFNN algorithm. The results show that no matter what kind of detector is used, the RBFNN algorithm performs better than the ACE algorithm. The RBFNN algorithm has a good anti-noise ability when the in-core detector signals with noise are used, while the correct axial power shape can’t be reconstructed from the ex-core detector signals with noise. By analyzing the ex-core axial spatial weighting function’s condition number, the determination of the axial power shape from ex-core detector signals is found to be a typical ill-posed inverse problem. A regularized RBFNN algorithm is used to eliminate this ill-posedness and get the physically meaningful axial power shape.

Numerical aspects of grazing incidence XRF

In this article, we give a different mathematical approach for background aspects of grazing incidence x-ray fluorescence, GIXRF for short. Our contribution comes from an applied point of view, in order to have a computer program to simulate the fluorescence intensity from a stacking of thin layer films. A typical ill-posed inverse problem is formulated. Our aim is to reconstruct the fluorescence intensity for a variety of grazing angle measurements. We rederive some classical equations pointing out the numerical aspects of the inversion procedure and giving new directions for direct in inverse algorithms.

Inverse Bayesian Estimation of Gravitational Mass Density in Galaxies from Missing Kinematic Data

In this paper, we focus on a type of inverse problem in which the data are expressed as an unknown function of the sought and unknown model function (or its discretised representation as a model parameter vector). In particular, we deal with situations in which training data are not available. Then we cannot model the unknown functional relationship between data and the unknown model function (or parameter vector) with a Gaussian Process of appropriate dimensionality. A Bayesian method based on state space modelling is advanced instead. Within this framework, the likelihood is expressed in terms of the probability density function (pdf) of the state space variable and the sought model parameter vector is embedded within the domain of this pdf. As the measurable vector lives only inside an identified sub-volume of the system state space, the pdf of the state space variable is projected onto the space of the measurables, and it is in terms of the projected state space density that the likelihood is written; the final form of the likelihood is achieved after convolution with the distribution of measurement errors. Application motivated vague priors are invoked and the posterior probability density of the model parameter vectors, given the data are computed. Inference is performed by taking posterior samples with adaptive MCMC. The method is illustrated on synthetic as well as real galactic data.

Eddy Current Inversion Models for Estimating Dimensions of Defects in Multilayered Structures

In eddy current nondestructive evaluation, one of the principal challenges is to determine the dimensions of defects in multilayered structures from the measured signals. It is a typical inverse problem which is generally considered to be nonlinear and ill-posed. In the paper, two effective approaches have been proposed to estimate the defect dimensions. The first one is a partial least squares (PLS) regression method. The second one is a kernel partial least squares (KPLS) regression method. The experimental research is carried out. In experiments, the eddy current signals responding to magnetic field changes are detected by a giant magnetoresistive (GMR) sensor and preprocessed for noise elimination using a wavelet packet analysis (WPA) method. Then, the proposed two approaches are used to construct the inversion models of defect dimension estimation. Finally, the estimation results are analyzed. The performance comparison between the proposed two approaches and the artificial neural network (ANN) method is presented. The comparison results demonstrate the feasibility and validity of the proposed two methods. Between them, the KPLS regression method gives a better prediction performance than the PLS regression method at present.

The Adjoint Method for the Inverse Problem of Option Pricing

The estimation of implied volatility is a typical PDE inverse problem. In this paper, we propose theTV-L1model for identifying the implied volatility. The optimal volatility function is found by minimizing the cost functional measuring the discrepancy. The gradient is computed via the adjoint method which provides us with an exact value of the gradient needed for the minimization procedure. We use the limited memory quasi-Newton algorithm (L-BFGS) to find the optimal and numerical examples shows the effectiveness of the presented method.

Calibration of the Volatility in Option Pricing Using the Total Variation Regularization

In market transactions, volatility, which is a very important risk measurement in financial economics, has significantly intimate connection with the future risk of the underlying assets. Identifying the implied volatility is a typical PDE inverse problem. In this paper, based on the total variation regularization strategy, a bivariate total variation regularization model is proposed to estimate the implied volatility. We not only prove the existence of the solution, but also provide the necessary condition of the optimal control problem—Euler-Lagrange equation. The stability and convergence analyses for the proposed approach are also given. Finally, numerical experiments have been carried out to show the effectiveness of the method.

Expectation propagation for nonlinear inverse problems – with an application to electrical impedance tomography

In this paper, we study a fast approximate inference method based on expectation propagation for exploring the posterior probability distribution arising from the Bayesian formulation of nonlinear inverse problems. It is capable of efficiently delivering reliable estimates of the posterior mean and covariance, thereby providing an inverse solution together with quantified uncertainties. Some theoretical properties of the iterative algorithm are discussed, and the efficient implementation for an important class of problems of projection type is described. The method is illustrated with one typical nonlinear inverse problem, electrical impedance tomography with complete electrode model, under sparsity constraints. Numerical results for real experimental data are presented, and compared with that by Markov chain Monte Carlo. The results indicate that the method is accurate and computationally very efficient.

Sparse representation with morphologic regularizations for single image super-resolution

Due to the fact that natural images are inherently sparse in some domains, sparse representation has led to interesting results in image acquiring, representing, and compressing high-dimensional signals. Based on the experiences and learned priors in sparse domain from low and high resolution images, the typical ill-posed inverse problem of image super-resolution is effectively solved by the l1-norm optimization techniques. However, how to reasonably combine the sparse representation theory and the feature of natural images is still a critical issue for performances improvements of image super-resolution algorithms. Considering the fact that the different morphologic features in natural images should be regularized by different constrains in sparse domain, in this paper we present a novel sparse representation algorithm with reasonable morphologic regularization for single image super-resolution. Extensive experimental results on various natural images validate the superiority of the proposed algorithm in terms of qualitative and quantitative performance.

The combination of self-organizing feature maps and support vector regression for solving the inverse ECG problem

Noninvasive electrical imaging of the heart aims to quantitatively reconstruct transmembrane potentials (TMPs) from body surface potentials (BSPs), which is a typical inverse problem. Classically, electrocardiography (ECG) inverse problem is solved by regularization techniques. In this study, it is treated as a regression problem with multi-inputs (BSPs) and multi-outputs (TMPs). Then the resultant regression problem is solved by a hybrid method, which combines the support vector regression (SVR) method with self-organizing feature map (SOFM) techniques. The hybrid SOFM–SVR method conducts a two-step process: SOFM algorithm is used to cluster the training samples and the individual SVR method is employed to construct the regression model. For each testing sample, the cluster operation can effectively improve the efficiency of the regression algorithm, and also helps the setup of the corresponding SVR model for the TMPs reconstruction. The performance of the developed SOFM–SVR model is tested using our previously developed realistic heart-torso model. The experiment results show that, compared with traditional single SVR method in solving the inverse ECG problem, the proposed method can reduce the cost of training time and improve the reconstruction accuracy in solving the inverse ECG problem.