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This paper proposes a new method for the exact reconstruction of gray-scale images from projections. The image projections construct an accumulator array, which is used afterwards to reconstruct the original grayscale image by applying the proposed decomposition algorithm. The proposed method determines the number of projections and the number of rays in each projection that are required in order to achieve the reconstruction. These two parameters also define the dimensions of the accumulator array. Using an accumulator array with proper dimensions ensures that there is always a unique characteristic sample for each pixel, which is used during the reconstruction process to extract the pixel's grayscale value. During the reconstruction phase, the sinusoidal contribution of each pixel is removed from the accumulator array. At the end of the decomposition process the accumulator array becomes empty and the original image is exactly reconstructed. The experimental results confirm the robustness and efficiency of the proposed method.

This paper deals with the scattering of time-harmonic acoustic waves by inhomogeneous medium. We study the problem to recover the near and the far field using a priori information about the refractive index and the support of inhomogeneity. The incident spherical wave is modified in such a way as to recover the plane wave incidence when the source point approaches infinity. Applying the low-frequency expansions, the scattering medium problem is reduced to a sequence of potential problems for the approximation coefficients in the presence of a monopole singularity located at the source of incidence. Complete expansions for the integral representation formula in the near field as well as for the scattering amplitude in the far field are provided. The method is applied to the case of a spherical region of inhomogeneity and a radial dependent refractive index. As the point singularity tends to infinity, the relative results recover the scattering medium problem for plane wave incidence.

We consider the reconstruction of singular surfaces from the over-determined boundary conditions of an elliptic problem. The problem arises in optical and impedance tomography, where void-like structure or cracks may be modeled as diffusion processes supported on co-dimension one surfaces. The reconstruction of such surfaces is obtained theoretically and numerically by combining a shape sensitivity analysis with a level set method. The shape sensitivity analysis is used to define a velocity field, which allows us to update the surface while decreasing a given cost function, which quantifies the error between the prediction of the forward model and the measured data. The velocity field depends on the geometry of the surface and the tangential diffusion process supported on it. The latter process is assumed to be known in this paper. The level set method is next applied to evolve the surface in the direction of the velocity field. Numerical simulations show how the surface may be reconstructed from noisy estimates of the full, or local, Neumann-to-Dirichlet map.

Using compressive sensing and sparse regularization, one can nearly completely reconstruct the input (sparse) signal using limited numbers of observations. At the same time, the reconstruction methods by compressing sensing and optimizing techniques overcome the obstacle of the number of sampling requirement of the Shannon/Nyquist sampling theorem. It is well known that seismic reflection signal may be sparse, sometimes and the number of sampling is insufficient for seismic surveys. So, the seismic signal reconstruction problem is ill-posed. Considering the ill-posed nature and the sparsity of seismic inverse problems, we study reconstruction of the wavefield and the reflection seismic signal by Tikhonov regularization and the compressive sensing. The l₀, l₁ and l₂ regularization models are studied. Relationship between Tikhonov regularization and the compressive sensing is established. In particular, we introduce a general l_p - l_q (p, q ≥ 0) regularization model, which overcome the limitation on the assumption of convexity of the objective function. Interior point methods and projected gradient methods are studied. To show the potential for application of the regularized compressive sensing method, we perform both synthetic seismic signal and field data compression and restoration simulations using a proposed piecewise random sub-sampling. Numerical performance indicates that regularized compressive sensing is applicable for practical seismic imaging.

A tumor growth model based on a parametric system of partial differential equations is considered. The system corresponds to a phenomenological description of a multi-species population evolution. A velocity field taking into account the volume increase due to cellular division is introduced and the mechanical closure is provided by a Darcy-type law. The complexity of the biological phenomenon is taken into account through a set of parameters included in the model that need to be calibrated. To this end, a system identification method based on a low-dimensional representation of the solution space is introduced. We solve several idealized identification cases corresponding to typical situations where the information is scarce in time and in terms of observable fields. Finally, applications to actual clinical data are presented.

In this paper we consider inverse problems for resistor networks and for models obtained via the finite element method (FEM) for the conductivity equation. These correspond to discrete versions of the inverse conductivity problem of Calderón. We characterize FEM models corresponding to a given triangulation of the domain that are equivalent to certain resistor networks, and apply the results to study nonuniqueness of the discrete inverse problem. It turns out that the degree of nonuniqueness for the discrete problem is larger than the one for the partial differential equation. We also study invisibility cloaking for FEM models, and show how an arbitrary body can be surrounded with a layer so that the cloaked body has the same boundary measurements as a given background medium.

A method for estimating parameters in dynamic stochastic (Markov Chain) models based on Kurtz's limit theory coupled with inverse problem methods developed for deterministic dynamical systems is proposed and illustrated in the context of disease dynamics. This methodology relies on finding an approximate large-population behavior of an appropriate scaled stochastic system. The approach leads to a deterministic approximation obtained as solutions of rate equations (ordinary differential equations) in terms of the large sample size average over sample paths or trajectories (limits of pure jump Markov processes). Using the resulting deterministic model, we select parameter subset combinations that can be estimated using an ordinary-least-squares (OLS) or generalized-least-squares (GLS) inverse problem formulation with a given data set. The selection is based on two criteria of the sensitivity matrix: the degree of sensitivity measured in the form of its condition number and the degree of uncertainty measured in the form of its parameter selection score. We illustrate the ideas with a stochastic model for the transmission of vancomycin-resistant enterococcus (VRE) in hospitals and VRE surveillance data from an oncology unit.

One of the most recent results in empirical finance is the achievement of an accuracy gain for the volatility estimates from the use of high-frequency samples. In many applications, for instance, the observed intra-day data become the best informative source of estimation for the daily volatilites. This information set, or filtration in probability terms, can in theory be optimally exploited while adopting an increasingly finer sampling rate, a fact which finds justification when assuming a stochastic characterization of the problem in terms of so-called semimartingales. These instruments are very useful for representing asset price dynamics, and in recent proposals supply realized and integrated volatility measures. The former volatility is the empirical approximation of the latter, an average of high frequency returns observed within a certain time frame. Semimartingales become suitable model tools by allowing for the quadratic variation principle to hold. This in turn means that the convergence of the realized to the integrated volatility can be verified; conversely, from both theoretical and experimental standpoints, the cumulative squared high frequency returns represent consistent estimators of the integrated volatility measure. The goal of this work is twofold: first, to show with simulations the quality of the convergence for time-based estimators, compared to that obtained when time-scale coordinate wavelet tranforms are considered. Second, to verify that special families of wavelet decompose returns and allow for multi-scaling features to be revealed, together with the possible presence of underlying nonlinear dynamics of stock index return volatility.

We show that under certain hypotheses, an iterated function system on mappings (IFSM) is a contraction on the complete space of functions of bounded variation (BV). It then possesses a unique attractor of BV. Some BV-based inverse problems based on the Collage Theorem for contraction maps are considered.

In this paper, a new technique for solving the two-dimensional inverse scattering problem for ultrasound inverse imaging is presented. Reconstruction of a two-dimensional object is accomplished using an iterative algorithm which combines the conjugate gradient (CG) method and a neural network (NN) approach. The neural network technique is used to exploit knowledge of the statistical characteristics of the object to enhance the performance of the conjugate gradient method. The results for simulations show that the CGNN algorithm is more accurate than the CG method and, in addition, convergence occurs more rapidly. For the CGNN algorithm, approximately 50% fewer iterations are needed to obtain the inverse solution for a signal-to-noise ratio (SNR) of 50 dB. For a smaller SNR of 35 dB, the CGNN method is not as accurate, but it still gives reasonable results.

In ultrasound inverse problems, the integral equation can be nonlinear, ill-posed, and computationally expensive. One approach to solving such problems is the conjugate gradient (CG) method. A key parameter in the CG method is the conjugate gradient direction. In this paper, we investigate the CG directions proposed by Polyak et al. (PPR), Hestenes and Stiefel (HS), Fletcher and Reeves (FR), Dai and Yuan (YD), and the two-parameter family generalization proposed by Nazareth (TPF). Each direction is applied to three test cases with different contrasts and phase shifts. Test case 1 has low contrast with a phase shift of 0.2π. Reconstruction of the object is obtained for all directions. The performances of the PPR, HS, YD, and TPF directions are comparable, while the FR direction gives the poorest performance. Test case 2 has medium contrast with a phase shift of 0.75π. Reconstruction is obtained for all but the FR direction. The PPR, HS, YD, and TPF directions have similar mean square error; the YD direction takes the least amount of CPU time. Test case 3 has the highest contrast with a phase shift of 1.003π. Only the YD direction gives reasonably accurate results.

In this paper we investigate the unknown body problem in a waveguide. The Rayleigh conjecture states that every point on an illuminated body radiates sound from that point as if the point lies on its tangent sphere. This conjecture is the cornerstone of the intersecting canonical body approximation ICBA for solving the unknown body inverse problem. Therefore, the use of the ICBA requires that an analytical solution be known exterior to the sphere in the waveguide, which leads us to analytically compute the exterior solution for a sphere between two parallel plates. A least-squares matching of theoretical acoustic fields against the measured, scattered field permits a reconstruction of the unknown object.

In this review paper, the use of the Time Reversal (TR) method as a computational tool for solving some classes of inverse problems is surveyed. The basics of computational TR are explained, using the scalar wave equation as a simple model. The application of TR to various problems in acoustics and elastodynamics is reviewed, in a selective and biased way as it leans on the author's personal view, referring to representative articles published on the subject.

Time reversal is a powerful procedure in application fields involving wave propagation. It is based on the invariance of the wave equations, in the absence of dissipation, in the time direction. This allows going backward in time to recover past events. We use time reversal to recover the location of a source applied at the initial time based on measurements at a later time. We generalize the procedure previously developed for the scalar wave equation¹ to elastodynamics. We show that the technique is quite robust, sometimes even in the presence of very high noise levels. Also it is not very sensitive to the medium characterizations, when a sufficient amount of measurement data is available. We extend previous work to get good refocusing for multiple sources. We introduce a new score to assess the quality of the numerical solution for the refocusing problem which produces good results. Furthermore, we use the refocusing technique as a basis for scatterer location recovery. By adding noise in a controlled manner we improve the scheme of finding the location of the scatterer.

This paper proposes a technique for the design of reflector shapes from prescribed optical properties (far field radiance distribution) and geometrical constraints, which is of high importance in the field of Lighting Engineering, more specifically for Luminaire Design. The reflector shape to be found is just a part of a set of pieces of what is known in lighting engineering as an optical set, and is composed of a lamp (light source), a reflector, a holding case and a glass that protects the system from dust and other environmental phenomena. Thus, we aim at the design and development of a system capable of generating a reflector shape in a way such that the optical set emits a given, user defined, far field radiance distribution. This problem can be put in the mathematical context of inverse problems, which refer to all the problems where, contrary to what happens with traditional direct problems, several aspects of the scene are unknown. Then, the algorithm is allowed to work backwards to establish the missing parameters. In order to do so, light propagation inside and outside the optical set must be computed and the resulting radiance distribution compared to the desired one. Finally, constraints on the shape imposed by industry needs must be taken into account, bounding the set of possible shape definitions. The general approach taken is based on a minimization procedure on the space of possible reflector shapes. The algorithm moves towards minimizing the distance, in the l² metric, between the resulting illumination far from the reflector and a prescribed, ideal optical radiance distribution specified at the far field by the user.

We present an analytical procedure for the exact, explicit construction of Euler–Bernoulli beams with given values of the first $N$ buckling loads. The result is valid for pinned–pinned (P–P) end conditions and for beams with regular bending stiffness. The analysis is based on a reduction of the buckling problem to an eigenvalue problem for a vibrating string, and uses recent results on the exact construction of Sturm–Liouville operators with prescribed natural frequencies.

In this paper, we discuss the following inverse problem: how to reconstruct the mass distribution function of a two-span beam with an overhang via its polynomial fundamental mode and polynomial stiffness function. This leads to a basic equations group constituted by the coefficients of the mass distribution and the stiffness distribution function. To make the basic equations group match, one way is to divide it into two sub- equations, and solve them. We specify the method for solving this inverse problem, and research the existence and rationality of the positive solutions.

We study the discretization of inverse problems defined by a Carleman operator. In particular, we develop a discretization strategy for this class of inverse problems and we give a convergence analysis. Learning from examples, as well as the discretization of integral equations, can be analyzed in our setting.

This survey treats a number of interconnected topics related in one way or another to the famous paper of Mark Kac, "Can one hear the shape of a drum?": wave motion, classical and quantum inverse problems, integrable systems, and the relations between spectra and geometry. We sketch the history and some of the principal developments from the vibrating string to quantum inverse problems, the KdV equation and integrable systems, spectral geometry and the index problem.

We study the problem of reconstructing solutions of inverse problems when only noisy measurements are available. We assume that the problem can be modeled with an infinite-dimensional forward operator that is not continuously invertible. Then, we restrict this forward operator to finite-dimensional spaces so that the inverse is Lipschitz continuous. For the inverse operator, we demonstrate that there exists a neural network which is a robust-to-noise approximation of the operator. In addition, we show that these neural networks can be learned from appropriately perturbed training data. We demonstrate the admissibility of this approach to a wide range of inverse problems of practical interest. Numerical examples are given that support the theoretical findings.