This work studies the usage of well-known smoothed total variation regularization for solving an atmospheric tomography problem named as {\em GPS-tomography} in some quasi-Newton methods. That is we solve an unconstrained, convex, smooth minimization problem associated with a general type Tikhonov functional containing smoothed type total variation penalty term by quasi-Newton methods. As a result of the conducted experiments, it is concluded that limited memory BFGS algorithm with trust region is the effective algorithm in terms obtaining a reasonably optimum solution.
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 an…
As second-order methods, Gauss--Newton-type methods can be more effective than first-order methods for the solution of nonsmooth optimization problems with expensive-to-evaluate smooth components. Such methods, however, often do not converge. Motivated by nonlinear inverse problems with nonsmooth regularization, we propose a new Gauss--Newton-type method with inexact relaxed steps. We prove that the method converges to a set of disjoint critical points given that the linearisation of the forward operator for the inverse problem is sufficiently precise. We extensively evaluate the performance o…
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 significa…
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…
efficient matrix-free gauss-newton traveltime tomography on full-metric deformed grids
State Key Laboratory of Marine Geology, Tongji University（同济大学海洋地质国家重点实验室）；School of Mathematical Sciences, Tongji University（同济大学数学科学学院）；Earthquake Research Institute, The University of Tokyo（东京大学地震研究所）
An energetically balanced, implicit integrator for non-hydrostatic vertical atmospheric dynamics on the sphere is presented. The integrator allows for the exact balance of energy exchanges in space and time for vertical atmospheric motions by preserving the skew-symmetry of the non-canonical Hamiltonian formulation of the compressible Euler equations. The performance of the integrator is accelerated by a preconditioning strategy that reduces the dimensionality of the inner linear system. Here we reduce the four component velocity, density, density weighted potential temperature and Exner press…
Efficient TV regularization of large-scale linear inverse problems via the SCD semismooth* Newton method with applications in tomography
Johann Radon Institute for Computational and Applied Mathematics（约翰·拉登计算与应用数学研究所）；Institute of Information Theory and Automation, Czech Academy of Sciences（捷克科学院信息理论与自动化研究所）；Johannes Kepler University Linz（林茨约翰·开普勒大学）；University of Eastern Finland（芬兰东部大学）
In this paper, we formulate the reconstruction problem in diffuse optical tomography (DOT) in a statistical setting for determining the optical parameters, scattering and absorption, from boundary photon density measurements. A special kind of adaptive Metropolis algorithm for the reconstruction procedure using sparsity and total variation prior is presented. Finally, a simulation study of this technique with different regularization functions and its comparison to the deterministic Iteratively Regularized Gauss Newton method shows the effectiveness and stability of the method.
Nonlinearity parameter tomography leads to the problem of identifying a coefficient in a nonlinear wave equation (such as the Westervelt equation) modeling ultrasound propagation. In this paper we transfer this into frequency domain, where the Westervelt equation gets replaced by a coupled system of Helmholtz equations with quadratic nonlinearities. For the case of the to-be-determined nonlinearity coefficient being a characteristic function of an unknown, not necessarily connected domain $D$, we devise and test a reconstruction algorithm based on weighted point source approximations combined …
Spectral Computed Tomography (CT) is an emerging technology that enables to estimate the concentration of basis materials within a scanned object by exploiting different photon energy spectra. In this work, we aim at efficiently solving a model-based maximum-a-posterior problem to reconstruct multi-materials images with application to spectral CT. In particular, we propose to solve a regularized optimization problem based on a plug-in image-denoising function using a randomized second order method. By approximating the Newton step using a sketching of the Hessian of the likelihood function, it…
Non-smooth regularization is widely used in image reconstruction to eliminate the noise while preserving subtle image structures. In this work, we investigate the use of proximal Newton (PN) method to solve an optimization problem with a smooth data-fidelity term and total variation (TV) regularization arising from image reconstruction applications. Specifically, we consider a nonlinear Poisson-modeled single-energy X-ray computed tomography reconstruction problem with the data-fidelity term given by the I-divergence. The PN algorithm is compared to state-of-the-art first-order proximal algori…
University of Sheffield（谢菲尔德大学）；Purdue University（普渡大学）；University of Science and Technology of China（中国科学技术大学）；University of Oulu（奥卢大学）；University College London（伦敦大学学院）
Solving complex optimization problems in engineering and the physical sciences requires repetitive computation of multi-dimensional function derivatives. Commonly, this requires computationally-demanding numerical differentiation such as perturbation techniques, which ultimately limits the use for time-sensitive applications. In particular, in nonlinear inverse problems Gauss-Newton methods are used that require iterative updates to be computed from the Jacobian. Computationally more efficient alternatives are Quasi-Newton methods, where the repeated computation of the Jacobian is replaced by …
In this paper we the formulation of inverse problems as constrained minimization problems and their iterative solution by gradient or Newton type. We carry out a convergence analysis in the sense of regularization methods and discuss applicability to the problem of identifying the spatially varying diffusivity in an elliptic PDE from different sets of observations. Among these is a novel hybrid imaging techology known as impedance acoustic tomography, for which we provide numerical experiments.
For numerous parameter and state estimation problems, assimilating new data as they become available can help produce accurate and fast inference of unknown quantities. While most existing algorithms for solving those kind of ill-posed inverse problems can only be used with a single instance of the observed data, in this work we propose a new framework that enables existing algorithms to invert multiple instances of data in a sequential fashion. Specifically we will work with the well-known iteratively regularized Gauss-Newton method (IRGNM), a variational methodology for solving nonlinear inv…
