arXiv subjects

Tapio Helin

Publications and source records attributed to Tapio Helin.

At least 55 records · Page 3Linked to original sources

arXiv2019

Simultaneous Reconstruction of Emission and Attenuation in Passive Gamma Emission Tomography of Spent Nuclear Fuel

The International Atomic Energy Agency (IAEA) has recently approved passive gamma emission tomography (PGET) as a method for inspecting spent nuclear fuel assemblies (SFAs), an important aspect of international nuclear safeguards which aim at preventing the proliferation of nuclear weapons. The PGET instrument is essentially a single photon emission computed tomography (SPECT) system that allows the reconstruction of axial cross-sections of the emission map of the SFA. The fuel material heavily self-attenuates its gamma-ray emissions, so that correctly accounting for the attenuation is a critical factor in producing accurate images. Due to the nature of the inspections, it is desirable to use as little a priori information as possible about the fuel, including the attenuation map, in the reconstruction process. Current reconstruction methods either do not correct for attenuation, assume a uniform attenuation throughout the fuel assembly, or assume an attenuation map based on an initial filtered back projection (FBP) reconstruction. Here, we propose a method to simultaneously reconstruct the emission and attenuation maps by formulating the reconstruction as a constrained minimization problem with a least squares data fidelity term and regularization terms. Using simulated data, we compare the proposed method to FBP, showing that our approach produces significantly better reconstructions by various numerical metrics and a much better classification of spent, missing, and fresh fuel rods.

physics.ins-det↗

arXiv2019

Inverse problems for heat equation and space-time fractional diffusion equation with one measurement

Given a connected compact Riemannian manifold $(M,g)$ without boundary, $\dim M\ge 2$, we consider a space--time fractional diffusion equation with an interior source that is supported on an open subset $V$ of the manifold. The time-fractional part of the equation is given by the Caputo derivative of order $α\in(0,1]$, and the space fractional part by $(-Δ_g)^β$, where $β\in(0,1]$ and $Δ_g$ is the Laplace--Beltrami operator on the manifold. The case $α=β=1$, which corresponds to the standard heat equation on the manifold, is an important special case. We construct a specific source such that measuring the evolution of the corresponding solution on $V$ determines the manifold up to a Riemannian isometry.

math.AP↗

arXiv2019

Generalized modes in Bayesian inverse problems

Uncertainty quantification requires efficient summarization of high- or even infinite-dimensional (i.e., non-parametric) distributions based on, e.g., suitable point estimates (modes) for posterior distributions arising from model-specific prior distributions. In this work, we consider non-parametric modes and MAP estimates for priors that do not admit continuous densities, for which previous approaches based on small ball probabilities fail. We propose a novel definition of generalized modes based on the concept of approximating sequences, which reduce to the classical mode in certain situations that include Gaussian priors but also exist for a more general class of priors. The latter includes the case of priors that impose strict bounds on the admissible parameters and in particular of uniform priors. For uniform priors defined by random series with uniformly distributed coefficients, we show that generalized MAP estimates -- but not classical MAP estimates -- can be characterized as minimizers of a suitable functional that plays the role of a generalized Onsager--Machlup functional. This is then used to show consistency of nonlinear Bayesian inverse problems with uniform priors and Gaussian noise.

math.ST↗

arXiv2018

Inverse random source problems for time-harmonic acoustic and elastic waves

This paper concerns the random source problems for the time-harmonic acoustic and elastic wave equations in two and three dimensions. The goal is to determine the compactly supported external force from the radiated wave field measured in a domain away from the source region. The source is assumed to be a microlocally isotropic generalized Gaussian random function such that its covariance operator is a classical pseudo-differential operator. Given such a distributional source, the direct problem is shown to have a unique solution by using an integral equation approach and the Sobolev embedding theorem. For the inverse problem, we demonstrate that the amplitude of the scattering field averaged over the frequency band, obtained from a single realization of the random source, determines uniquely the principle symbol of the covariance operator. The analysis employs asymptotic expansions of the Green functions and microlocal analysis of the Fourier integral operators associated with the Helmholtz and Navier equations.

math.AP↗

arXiv2018

An inverse random source problem in a stochastic fractional diffusion equation

In this work the authors consider an inverse source problem in the following stochastic fractional diffusion equation $$\partial_t^αu(x,t)+\mathcal{A} u(x,t)=f(x)h(t)+g(x) \dot{\mathbb{W}}(t).$$ The interested inverse problem is to reconstruct $f(x)$ and $g(x)$ by the statistics of the final time data $u(x,T).$ Some direct problem results are proved at first, such as the existence, uniqueness, representation and regularity of the solution. Then the reconstruction scheme for $f$ and $g$ is given. To tackle the ill-posedness, the Tikhonov regularization is adopted. Finally we give a regularized reconstruction algorithm and some numerical results are displayed.

math.AP↗

arXiv2018

Correlation imaging in inverse scattering is tomography on probability distributions

Scattering from a non-smooth random field on the time domain is studied for plane waves that propagate simultaneously through the potential in variable angles. We first derive sufficient conditions for stochastic moments of the field to be recovered from correlations between amplitude measurements of the leading singularities, detected in the exterior of a region where the potential is almost surely supported. The result is then applied to show that if two sufficiently regular random fields yield the same data, they have identical laws as function-valued random variables.

math.AP↗

arXiv2018

Atmospheric turbulence profiling with unknown power spectral density

Adaptive optics (AO) is a technology in modern ground-based optical telescopes to compensate the wavefront distortions caused by atmospheric turbulence. One method that allows to retrieve information about the atmosphere from telescope data is so-called SLODAR, where the atmospheric turbulence profile is estimated based on correlation data of Shack--Hartmann wavefront measurements. This approach relies on a layered Kolmogorov turbulence model. In this article, we propose a novel extension of the SLODAR concept by including a general non-Kolmogorov turbulence layer close to the ground with an unknown power spectral density. We prove that the joint estimation problem of the turbulence profile above ground simultaneously with the unknown power spectral density at the ground is ill-posed and propose three numerical reconstruction methods. We demonstrate by numerical simulations that our methods lead to substantial improvements in the turbulence profile reconstruction, compared to standard SLODAR-type approach. Also, our methods can accurately locate local perturbations in non-Kolmogorov power spectral densities.

astro-ph.IM↗

arXiv2018

Large Noise in Variational Regularization

In this paper we consider variational regularization methods for inverse problems with large noise that is in general unbounded in the image space of the forward operator. We introduce a Banach space setting that allows to define a reasonable notion of solutions for more general noise in a larger space provided one has sufficient mapping properties of the forward operators. A key observation, which guides us through the subsequent analysis, is that such a general noise model can be understood with the same setting as approximate source conditions (while a standard model of bounded noise is related directly to classical source conditions). Based on this insight we obtain a quite general existence result for regularized variational problems and derive error estimates in terms of Bregman distances. The latter are specialized for the particularly important cases of one- and p-homogeneous regularization functionals. As a natural further step we study stochastic noise models and in particular white noise, for which we derive error estimates in terms of the expectation of the Bregman distance. The finiteness of certain expectations leads to a novel class of abstract smoothness conditions on the forward operator, which can be easily interpreted in the Hilbert space case. We finally exemplify the approach and in particular the conditions for popular examples of regularization functionals given by squared norm, Besov norm and total variation, respectively.

math.NA↗

arXiv2017

Sparsity-promoting and edge-preserving maximum a posteriori estimators in non-parametric Bayesian inverse problems

We consider the inverse problem of recovering an unknown functional parameter $u$ in a separable Banach space, from a noisy observation $y$ of its image through a known possibly non-linear ill-posed map ${\mathcal G}$. The data $y$ is finite-dimensional and the noise is Gaussian. We adopt a Bayesian approach to the problem and consider Besov space priors (see Lassas et al. 2009), which are well-known for their edge-preserving and sparsity-promoting properties and have recently attracted wide attention especially in the medical imaging community. Our key result is to show that in this non-parametric setup the maximum a posteriori (MAP) estimates are characterized by the minimizers of a generalized Onsager--Machlup functional of the posterior. This is done independently for the so-called weak and strong MAP estimates, which as we show coincide in our context. In addition, we prove a form of weak consistency for the MAP estimators in the infinitely informative data limit. Our results are remarkable for two reasons: first, the prior distribution is non-Gaussian and does not meet the smoothness conditions required in previous research on non-parametric MAP estimates. Second, the result analytically justifies existing uses of the MAP estimate in finite but high dimensional discretizations of Bayesian inverse problems with the considered Besov priors.

math.ST↗

arXiv2017

A Variational Reconstruction Method for Undersampled Dynamic X-ray Tomography based on Physical Motion Models

In this paper we study the reconstruction of moving object densities from undersampled dynamic X-ray tomography in two dimensions. A particular motivation of this study is to use realistic measurement protocols for practical applications, i.e. we do not assume to have a full Radon transform in each time step, but only projections in few angular directions. This restriction enforces a space-time reconstruction, which we perform by incorporating physical motion models and regularization of motion vectors in a variational framework. The methodology of optical flow, which is one of the most common methods to estimate motion between two images, is utilized to formulate a joint variational model for reconstruction and motion estimation. We provide a basic mathematical analysis of the forward model and the variational model for the image reconstruction. Moreover, we discuss the efficient numerical minimization based on alternating minimizations between images and motion vectors. A variety of results are presented for simulated and real measurement data with different sampling strategy. A key observation is that random sampling combined with our model allows reconstructions of similar amount of measurements and quality as a single static reconstruction.

math.NA↗

arXiv2016

Correlation based passive imaging with a white noise source

Passive imaging refers to problems where waves generated by unknown sources are recorded and used to image the medium through which they travel. The sources are typically modelled as a random variable and it is assumed that some statistical information is available. In this paper we study the stochastic wave equation $\partial_t^2 u - Δ_g u = χW$, where $W$ is a random variable with the white noise statistics on ${\mathbb R}^{1+n}$, $n \ge 3$, $χ$ is a smooth function vanishing for negative times and outside a compact set in space, and $Δ_g$ is the Laplace-Beltrami operator associated to a smooth non-trapping Riemannian metric tensor $g$ on ${\mathbb R}^n$. The metric tensor $g$ models the medium to be imaged, and we assume that it coincides with the Euclidean metric outside a compact set. We consider the empirical correlations on an open set $\mathcal X \subset {\mathbb R}^n$, $$ C_T(t_1, x_1, t_2, x_2) = \frac 1 T \int_0^T u(t_1+s,x_1) u(t_2+s,x_2) ds, \quad t_1,t_2>0,\ x_1,x_2\in \mathcal X, $$ for $T>0$. Supposing that $χ$ is non-zero on $\mathcal X$ and constant in time after $t > 1$, we show that in the limit $T \to \infty$, the data $C_T$ becomes statistically stable, that is, independent of the realization of $W$. Our main result is that, with probability one, this limit determines the Riemannian manifold $({\mathbb R}^n,g)$ up to an isometry. To our knowledge, this is the first result showing that a medium can be determined in a passive imaging setting, without assuming a separation of scales.

math.AP↗

arXiv2016

Inverse scattering for a random potential

In this paper we consider an inverse problem for the $n$-dimensional random Schrödinger equation $(Δ-q+k^2)u = 0$. We study the scattering of plane waves in the presence of a potential $q$ which is assumed to be a Gaussian random function such that its covariance is described by a pseudodifferential operator. Our main result is as follows: given the backscattered far field, obtained from a single realization of the random potential $q$, we uniquely determine the principal symbol of the covariance operator of $q$. Especially, for $n=3$ this result is obtained for the full non-linear inverse backscattering problem. Finally, we present a physical scaling regime where the method is of practical importance.

math.AP↗

arXiv2015

Towards analytical model optimization in atmospheric tomography

Modern ground-based telescopes rely on a technology called adaptive optics (AO) in order to compensate for the loss of image quality caused by atmospheric turbulence. Next-generation AO systems designed for a wide field of view require a stable and high-resolution reconstruction of the refractive index fluctuations in the atmosphere. By introducing a novel Bayesian method, we address the problem of estimating an atmospheric turbulence strength profile and reconstructing the refractive index fluctuations simultaneously, where we only use wavefront measurements of incoming light from guide stars. Most importantly, we demonstrate how this method can be used for model optimization as well. We propose two different algorithms for solving the maximum a posteriori estimate: the first approach is based on alternating minimization and has the advantage of integrability into existing atmospheric tomography methods. In the second approach, we formulate a convex non-differentiable optimization problem, which is solved by an iterative thresholding method. This approach clearly illustrates the underlying sparsity-enforcing mechanism for the strength profile. By introducing a tuning/regularization parameter, an automated model reduction of the layer structure of the atmosphere is achieved. Using numerical simulations, we demonstrate the performance of our method in practice.

math.NA↗

arXiv2015

Maximum a posteriori probability estimates in infinite-dimensional Bayesian inverse problems

A demanding challenge in Bayesian inversion is to efficiently characterize the posterior distribution. This task is problematic especially in high-dimensional non-Gaussian problems, where the structure of the posterior can be very chaotic and difficult to analyse. Current inverse problem literature often approaches the problem by considering suitable point estimators for the task. Typically the choice is made between the maximum a posteriori (MAP) or the conditional mean (CM) estimate. The benefits of either choice are not well-understood from the perspective of infinite-dimensional theory. Most importantly, there exists no general scheme regarding how to connect the topological description of a MAP estimate to a variational problem. The results by Dashti et. al. (2013) resolve this issue for non-linear inverse problems in Gaussian framework. In this work we improve the current understanding by introducing a novel concept called the weak MAP (wMAP) estimate. We show that any MAP estimate in the sense of Dashti et. al. (2013) is a wMAP estimate and, moreover, how in general infinite-dimensional non-Gaussian problems the wMAP estimate connects to a variational formulation. Such a formulation yields many properties of the estimate that were earlier impossible to study. In a recent work by Burger and Lucka (2014) the MAP estimator was studied in the context of Bayes cost method. Using Bregman distances, proper convex Bayes cost functions were introduced for which the MAP estimator is the Bayes estimator. Here, we generalize these results to the infinite-dimensional setting. Moreover, we discuss the implications of our results for some examples of prior models such as the Besov prior and hierarchical prior.

math.ST↗

arXiv2014

Inverse acoustic scattering problem in half-space with anisotropic random impedance

We study an inverse acoustic scattering problem in half-space with a probabilistic impedance boundary value condition. The Robin coefficient (surface impedance) is assumed to be a Gaussian random function $λ= λ(x)$ with a pseudodifferential operator describing the covariance. We measure the amplitude of the backscattered field averaged over the frequency band and assume that the data is generated by a single realization of $λ$. Our main result is to show that under certain conditions the principal symbol of the covariance operator of $λ$ is uniquely determined. Most importantly, no approximations are needed and we can solve the full non-linear inverse problem. We concentrate on anisotropic models for the principal symbol, which leads to the analysis of a novel anisotropic spherical Radon transform and its invertibility.

math.AP↗

arXiv2013

Inverse problem for the wave equation with a white noise source

We consider a smooth Riemannian metric tensor $g$ on $\R^n$ and study the stochastic wave equation for the Laplace-Beltrami operator $\p_t^2 u - Δ_g u = F$. Here, $F=F(t,x,ω)$ is a random source that has white noise distribution supported on the boundary of some smooth compact domain $M \subset \R^n$. We study the following formally posed inverse problem with only one measurement. Suppose that $g$ is known only outside of a compact subset of $M^{int}$ and that a solution $u(t,x,ω_0)$ is produced by a single realization of the source $F(t,x,ω_0)$. We ask what information regarding $g$ can be recovered by measuring $u(t,x,ω_0)$ on $\R_+ \times \p M$? We prove that such measurement together with the realization of the source determine the scattering relation of the Riemannian manifold $(M, g)$ with probability one. That is, for all geodesics passing through $M$, the travel times together with the entering and exit points and directions are determined. In particular, if $(M,g)$ is a simple Riemannian manifold and $g$ is conformally Euclidian in $M$, the measurement determines the metric $g$ in $M$.

math.AP↗

arXiv2013

Wavelet methods in multi-conjugate adaptive optics

The next generation ground-based telescopes rely heavily on adaptive optics for overcoming the limitation of atmospheric turbulence. In the future adaptive optics modalities, like multi-conjugate adaptive optics (MCAO), atmospheric tomography is the major mathematical and computational challenge. In this severely ill-posed problem a fast and stable reconstruction algorithm is needed that can take into account many real-life phenomena of telescope imaging. We introduce a novel reconstruction method for the atmospheric tomography problem and demonstrate its performance and flexibility in the context of MCAO. Our method is based on using locality properties of compactly supported wavelets, both in the spatial and frequency domain. The reconstruction in the atmospheric tomography problem is obtained by solving the Bayesian MAP estimator with a conjugate gradient based algorithm. An accelerated algorithm with preconditioning is also introduced. Numerical performance is demonstrated on the official end-to-end simulation tool OCTOPUS of European Southern Observatory.

math.NA↗

arXiv2010

An inverse problem for the wave equation with one measurement and the pseudorandom noise

We consider the wave equation $(\p_t^2-Δ_g)u(t,x)=f(t,x)$, in $\R^n$, $u|_{\R_-\times \R^n}=0$, where the metric $g=(g_{jk}(x))_{j,k=1}^n$ is known outside an open and bounded set $M\subset \R^n$ with smooth boundary $\p M$. We define a deterministic source $f(t,x)$ called the pseudorandom noise as a sum of point sources, $f(t,x)=\sum_{j=1}^\infty a_jδ_{x_j}(x)δ(t)$, where the points $x_j,\ j\in\Z_+$, form a dense set on $\p M$. We show that when the weights $a_j$ are chosen appropriately, $u|_{\R\times \p M}$ determines the scattering relation on $\p M$, that is, it determines for all geodesics which pass through $M$ the travel times together with the entering and exit points and directions. The wave $u(t,x)$ contains the singularities produced by all point sources, but when $a_j=λ^{-λ^{j}}$ for some $λ>1$, we can trace back the point source that produced a given singularity in the data. This gives us the distance in $(\R^n, g)$ between a source point $x_j$ and an arbitrary point $y \in \p M$. In particular, if $(\bar M,g)$ is a simple Riemannian manifold and $g$ is conformally Euclidian in $\bar M$, these distances are known to determine the metric $g$ in $M$. In the case when $(\bar M,g)$ is non-simple we present a more detailed analysis of the wave fronts yielding the scattering relation on $\p M$.

math.AP↗