arXiv subjects

Ronny Ramlau

Publications and source records attributed to Ronny Ramlau.

54 records · Page 3Linked to original sources

arXiv2020

MICADO PSF-Reconstruction work package description

The point spread function reconstruction (PSF-R) capability is a deliverable of the MICADO@ESO-ELT project. The PSF-R team works on the implementation of the instrument software devoted to reconstruct the point spread function (PSF), independently of the science data, using adaptive optics (AO) telemetry data, both for Single Conjugate (SCAO) and Multi-Conjugate Adaptive Optics (MCAO) mode of the MICADO camera and spectrograph. The PSF-R application will provide reconstructed PSFs through an archive querying system to restore the telemetry data synchronous to each science frame that MICADO will generate. Eventually, the PSF-R software will produce the output according to user specifications. The PSF-R service will support the state-of-the-art scientific analysis of the MICADO imaging and spectroscopic data.

astro-ph.IM↗

arXiv2020

An augmented wavelet reconstructor for atmospheric tomography

Atmospheric tomography, i.e. the reconstruction of the turbulence profile in the atmosphere, is a challenging task for adaptive optics (AO) systems of the next generation of extremely large telescopes. Within the community of AO the first choice solver is the so called Matrix Vector Multiplication (MVM), which directly applies the (regularized) generalized inverse of the system operator to the data. For small telescopes this approach is feasible, however, for larger systems such as the European Extremely Large Telescope (ELT), the atmospheric tomography problem is considerably more complex and the computational efficiency becomes an issue. Iterative methods, such as the Finite Element Wavelet Hybrid Algorithm (FEWHA), are a promising alternative. FEWHA is a wavelet based reconstructor that uses the well-known iterative preconditioned conjugate gradient (PCG) method as a solver. The number of floating point operations and memory usage are decreased significantly by using a matrix-free representation of the forward operator. A crucial indicator for the real-time performance are the number of PCG iterations. In this paper, we propose an augmented version of FEWHA, where the number of iterations is decreased by $50\%$ using a Krylov subspace recycling technique. We demonstrate that a parallel implementation of augmented FEWHA allows the fulfilment of the real-time requirements of the ELT.

astro-ph.IM↗

arXiv2020

A mathematical approach towards THz tomography for non-destructive imaging

In this paper, we consider the imaging problem of terahertz (THz) tomography, in particular as it appears in non-destructive testing. We derive a nonlinear mathematical model describing a full THz tomography experiment, and consider linear approximations connecting THz tomography with standard computerized tomography and the Radon transform. Based on the derived models we propose different reconstruction approaches for solving the THz tomography problem, which we then compare on experimental data obtained from THz measurements of a plastic sample.

math.NA↗

arXiv2020

Real-time implementation of an iterative solver for atmospheric tomography

The image quality of the new generation of earthbound Extremely Large Telescopes (ELTs) is heavily influenced by atmospheric turbulences. To compensate these optical distortions a technique called adaptive optics (AO) is used. Many AO systems require the reconstruction of the refractive index fluctuations in the atmosphere, called atmospheric tomography. The standard way of solving this problem is the Matrix Vector Multiplication, i.e., the direct application of a (regularized) generalized inverse of the system operator. However, over the last years the telescope sizes have increased significantly and the computational efficiency become an issue. Promising alternatives are iterative methods such as the Finite Element Wavelet Hybrid Algorithm (FEWHA), which is based on wavelets. Due to its efficient matrix-free representation of the underlying operators, the number of floating point operations and memory usage decreases significantly. In this paper, we focus on performance optimization techniques, such as parallel programming models, for the implementation of this iterative method on CPUs and GPUs. We evaluate the computational performance of our optimized, parallel version of FEWHA for ELT-sized test configurations.

math.NA↗

arXiv2020

A Frame Decomposition of the Atmospheric Tomography Operator

We consider the problem of atmospheric tomography, as it appears for example in adaptive optics systems for extremely large telescopes. We derive a frame decomposition, i.e., a decomposition in terms of a frame, of the underlying atmospheric tomography operator, extending the singular-value-type decomposition results of Neubauer and Ramlau (2017) by allowing a mixture of both natural and laser guide stars, as well as arbitrary aperture shapes. Based on both analytical considerations as well as numerical illustrations, we provide insight into the properties of the derived frame decomposition and its building blocks.

math.NA↗

arXiv2020

A note on the minimization of a Tikhonov functional with $\ell^1$-penalty

In this paper, we consider the minimization of a Tikhonov functional with an $\ell_1$ penalty for solving linear inverse problems with sparsity constraints. One of the many approaches used to solve this problem uses the Nemskii operator to transform the Tikhonov functional into one with an $\ell_2$ penalty term but a nonlinear operator. The transformed problem can then be analyzed and minimized using standard methods. However, by the nature of this transform, the resulting functional is only once continuously differentiable, which prohibits the use of second order methods. Hence, in this paper, we propose a different transformation, which leads to a twice differentiable functional that can now be minimized using efficient second order methods like Newton's method. We provide a convergence analysis of our proposed scheme, as well as a number of numerical results showing the usefulness of our proposed approach.

math.NA↗

arXiv2020

A Conjugate-Gradient Approach to the Parameter Estimation Problem of Magnetic Resonance Advection Imaging

We consider the inverse problem of estimating the spatially varying pulse wave velocity in blood vessels in the brain from dynamic MRI data, as it appears in the recently proposed imaging technique of Magnetic Resonance Advection Imaging (MRAI). The problem is formulated as a linear operator equation with a noisy operator and solved using a conjugate gradient type approach. Numerical examples on experimental data show the usefulness and advantages of the developed algorithm in comparison to previously proposed methods.

math.NA↗

arXiv2018

Error Estimates for Arnoldo-Tikhonov Regularization for Ill-Posed Operator Equations

Most of the literature on the solution of linear ill-posed operator equations, or their discretization, focuses only on the infinite-dimensional setting or only on the solution of the algebraic linear system of equations obtained by discretization. This paper discusses the influence of the discretization error on the computed solution. We consider the situation when the discretization used yields an algebraic linear system of equations with a large matrix. An approximate solution of this system is computed by first determining a reduced system of fairly small size by carrying out a few steps of the Arnoldi process. Tikhonov regularization is applied to the reduced problem and the regularization parameter is determined by the discrepancy principle. Errors incurred in each step of the solution process are discussed. Computed examples illustrate the error bounds derived.

math.NA↗

arXiv2018

On the singular value decomposition of n-fold integration operators

In theory and practice of inverse problems, linear operator equations $Tx=y$ with compact linear forward operators $T$ having a non-closed range $\mathcal{R}(T)$ and mapping between infinite dimensional Hilbert spaces plays some prominent role. As a consequence of the ill-posedness of such problems, regularization approaches are required, and due to its unlimited qualification spectral cut-off is an appropriate method for the stable approximate solution of corresponding inverse problems. For this method, however, the singular system $\{σ_i(T),u_i(T),v_i(T)\}_{i=1}^\infty$ of the compact operator $T$ is needed, at least for $i=1,2,...,N$, up to some stopping index $N$. In this note we consider $n$-fold integration operators $T=J^n\;(n=1,2,...)$ in $L^2([0,1])$ occurring in numerous applications, where the solution of the associated operator equation is characterized by the $n$-th generalized derivative $x=y^{(n)}$ of the Sobolev space function $y \in H^n([0,1])$. Almost all textbooks on linear inverse problems present the whole singular system $\{σ_i(J^1),u_i(J^1),v_i(J^1)\}_{i=1}^\infty$ in an explicit manner. However, they do not discuss the singular systems for $J^n,\;n \ge 2$. We will emphasize that this seems to be a consequence of the fact that for higher $n$ the eigenvalues $σ^2_i(J^n)$ of the associated ODE boundary value problems obey transcendental equations, the complexity of which is growing with $n$. We present the transcendental equations for $n=2,3,...$ and discuss and illustrate the associated eigenfunctions and some of their properties.

math.NA↗

arXiv2018

Real-time Adaptive Optics with pyramid wavefront sensors: A theoretical analysis of the pyramid sensor model

We consider the mathematical background of the wavefront sensor type that is widely used in Adaptive Optics systems for astronomy, microscopy, and ophthalmology. The theoretical analysis of the pyramid sensor forward operators presented in this paper is aimed at a subsequent development of fast and stable algorithms for wavefront reconstruction from data of this sensor type. In our analysis we allow the sensor to be utilized in both the modulated and non-modulated fashion. We derive detailed mathematical models for the pyramid sensor and the physically simpler roof wavefront sensor as well as their various approximations. Additionally, we calculate adjoint operators which build preliminaries for the application of several iterative mathematical approaches for solving inverse problems such as gradient based algorithms, Landweber iteration or Kaczmarz methods.

astro-ph.IM↗

arXiv2018

Real-time Adaptive Optics with pyramid wavefront sensors: Accurate wavefront reconstruction using iterative methods

In this paper, we address the inverse problem of fast, stable, and high-quality wavefront reconstruction from pyramid wavefront sensor data for Adaptive Optics systems on Extremely Large Telescopes. For solving the indicated problem we apply well-known iterative mathematical algorithms, namely conjugate gradient, steepest descent, Landweber, Landweber-Kaczmarz and steepest descent-Kaczmarz iteration based on theoretical studies of the pyramid wavefront sensor. We compare the performance (in terms of correction quality and speed) of these algorithms in end-to-end numerical simulations of a closed adaptive loop. The comparison is performed in the context of a high-order SCAO system for METIS, one of the first-light instruments currently under design for the Extremely Large Telescope. We show that, though being iterative, the analyzed algorithms, when applied in the studied context, can be implemented in a very efficient manner, which reduces the related computational effort significantly. We demonstrate that the suggested analytically developed approaches involving iterative algorithms provide comparable quality to standard matrix-vector-multiplication methods while being computationally cheaper.

astro-ph.IM↗

arXiv2018

Non-linear wavefront reconstruction methods for pyramid sensors using Landweber and Landweber-Kaczmarz iteration

Accurate and robust wavefront reconstruction methods for pyramid wavefront sensors are in high demand as these sensors are planned to be part of many instruments currently under development for ground based telescopes. The pyramid sensor relates the incoming wavefront and its measurements in a non-linear way. Nevertheless, almost all existing reconstruction algorithms are based on a linearization of the model. The assumption of a linear pyramid sensor response is justifiable in closed loop AO when the measured phase information is small but may not be reasonable in reality due to unpreventable errors depending on the system such as non common path aberrations. In order to solve the non-linear inverse problem of wavefront reconstruction from pyramid sensor data we introduce two new methods based on the non-linear Landweber and Landweber-Kaczmarz iteration. Using these algorithms we experience high-quality wavefront estimation especially for the non-modulated sensor by still keeping the numerical effort feasible for large-scale AO systems.

astro-ph.IM↗

arXiv2018

The MICADO first light imager for the ELT: overview, operation, simulation

MICADO will enable the ELT to perform diffraction limited near-infrared observations at first light. The instrument's capabilities focus on imaging (including astrometric and high contrast) as well as single object spectroscopy. This contribution looks at how requirements from the observing modes have driven the instrument design and functionality. Using examples from specific science cases, and making use of the data simulation tool, an outline is presented of what we can expect the instrument to achieve.

astro-ph.IM↗

arXiv2018

Nesterov's Accelerated Gradient Method for Nonlinear Ill-Posed Problems with a Locally Convex Residual Functional

In this paper, we consider Nesterov's Accelerated Gradient method for solving Nonlinear Inverse and Ill-Posed Problems. Known to be a fast gradient-based iterative method for solving well-posed convex optimization problems, this method also leads to promising results for ill-posed problems. Here, we provide a convergence analysis for ill-posed problems of this method based on the assumption of a locally convex residual functional. Furthermore, we demonstrate the usefulness of the method on a number of numerical examples based on a nonlinear diagonal operator and on an inverse problem in auto-convolution.

math.NA↗

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↗

arXiv2016

On the lifting of deterministic convergence rates for inverse problems with stochastic noise

Both for the theoretical and practical treatment of Inverse Problems, the modeling of the noise is a crucial part. One either models the measurement via a deterministic worst-case error assumption or assumes a certain stochastic behavior of the noise. Although some connections between both models are known, the communities develop rather independently. In this paper we seek to bridge the gap between the deterministic and the stochastic approach and show convergence and convergence rates for Inverse Problems with stochastic noise by lifting the theory established in the deterministic setting into the stochastic one. This opens the wide field of deterministic regularization methods for stochastic problems without having to do an individual stochastic analysis for each problem.

math.PR↗

arXiv2014

A global minimization algorithm for Tikhonov functionals with sparsity constraints

In this paper we present a globally convergent algorithm for the computation of a minimizer of the Tikhonov functional with sparsity promoting penalty term for nonlinear forward operators in Banach space. The dual TIGRA method uses a gradient descent iteration in the dual space at decreasing values of the regularization parameter $α_j$, where the approximation obtained with $α_j$ serves as the starting value for the dual iteration with parameter $α_{j+1}$. With the discrepancy principle as a global stopping rule the method further yields an automatic parameter choice. We prove convergence of the algorithm under suitable step-size selection and stopping rules and illustrate our theoretic results with numerical experiments for the nonlinear autoconvolution problem.

math.NA↗

arXiv2013

On the interplay of basis smoothness and specific range conditions occurring in sparsity regularization

The convergence rates results in $\ell^1$-regularization when the sparsity assumption is narrowly missed, presented by Burger et al. (2013 Inverse Problems 29 025013), are based on a crucial condition which requires that all basis elements belong to the range of the adjoint of the forward operator. Partly it was conjectured that such a condition is very restrictive. In this context, we study sparsity-promoting varieties of Tikhonov regularization for linear ill-posed problems with respect to an orthonormal basis in a separable Hilbert space using $\ell^1$ and sublinear penalty terms. In particular, we show that the corresponding range condition is always satisfied for all basis elements if the problems are well-posed in a certain weaker topology and the basis elements are chosen appropriately related to an associated Gelfand triple. The Radon transform, Symm's integral equation and linear integral operators of Volterra type are examples for such behaviour, which allows us to apply convergence rates results for non-sparse solutions, and we further extend these results also to the case of non-convex $\ell^q$-regularization with 0<q<1.

math.NA↗