arXiv subjects

M. Haltmeier

Publications and source records attributed to M. Haltmeier.

8 recordsLinked to original sources

arXiv2020

Regularization of systems of nonlinear ill-posed equations: I. Convergence Analysis

In this article we develop and analyze novel iterative regularization techniques for the solution of systems of nonlinear ill--posed operator equations. The basic idea consists in considering separately each equation of this system and incorporating a loping strategy. The first technique is a Kaczmarz-type method, equipped with a novel stopping criteria. The second method is obtained using an embedding strategy, and again a Kaczmarz-type approach. We prove well-posedness, stability and convergence of both methods.

math.NA↗

arXiv2020

Regularization of systems of nonlinear ill-posed equations: II. Applications

In part I we introduced modified Landweber-Kaczmarz methods and have established a convergence analysis. In the present work we investigate three applications: an inverse problem related to thermoacoustic tomography, a nonlinear inverse problem for semiconductor equations, and a nonlinear problem in Schlieren tomography. Each application is considered in the framework established in the previous part. The novel algorithms show robustness, stability, computational efficiency and high accuracy.

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arXiv2014

Exact Reconstruction Formula for the Spherical Mean Radon Transform on Ellipsoids

Many modern imaging and remote sensing applications require reconstructing a function from spherical averages (mean values). Examples include photoacoustic tomography, ultrasound imaging or SONAR. Several formulas of the back-projection type for recovering a function in $n$ spatial dimensions from mean values over spheres centered on a sphere have been derived in [D. Finch, S. K. Patch, and Rakesh, SIAM J. Math. Anal. 35(5), pp. 1213--1240, 2004] for odd spatial dimension and in [D. Finch, M. Haltmeier, and Rakesh, SIAM J. Appl. Math. 68(2), pp. 392--412, 2007] for even spatial dimension. In this paper we generalize some of these formulas to the case where the centers of integration lie on the boundary of an arbitrary ellipsoid. For the special cases $n=2$ and $n=3$ our results have recently been established in [Y. Salman, J. Math. Anal. Appl., 2014, in press]. For the higher dimensional case $n > 3$ we establish proof techniques extending the ones in the above references. Back-projection type inversion formulas for recovering a function from spherical means with centers on an ellipsoid have first been derived in [F. Natterer, Inverse Probl. Imaging 6(2), pp. 315--320, 2012] for $n=3$ and in [V. Palamodov, Inverse Probl. 28(6), 065014, 2012] for arbitrary dimension. The results of Natterer have later been generalized to arbitrary dimension in [M. Haltmeier, SIAM J. Math. Anal. 46(1), pp. 214--232, 2014]. Note that these formulas are different from the ones derived in the present paper.

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arXiv2008

Exact Series Reconstruction in Photoacoustic Tomography with Circular Integrating Detectors

A method for photoacoustic tomography is presented that uses circular integrals of the acoustic wave for the reconstruction of a three-dimensional image. Image reconstruction is a two-step process: In the first step data from a stack of circular integrating are used to reconstruct the circular projection of the source distribution. In the second step the inverse circular Radon transform is applied. In this article we establish inversion formulas for the first step, which involves an inverse problem for the axially symmetric wave equation. Numerical results are presented that show the validity and robustness of the resulting algorithm.

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arXiv2008

A Reconstruction Algorithm for Photoacoustic Imaging based on the Nonuniform FFT

Fourier reconstruction algorithms significantly outperform conventional back-projection algorithms in terms of computation time. In photoacoustic imaging, these methods require interpolation in the Fourier space domain, which creates artifacts in reconstructed images. We propose a novel reconstruction algorithm that applies the one-dimensional nonuniform fast Fourier transform to photoacoustic imaging. It is shown theoretically and numerically that our algorithm avoids artifacts while preserving the computational effectiveness of Fourier reconstruction.

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arXiv2008

On Steepest-Descent-Kaczmarz Methods for Regularizing Systems of Nonlinear Ill-posed Equations

We investigate modified steepest descent methods coupled with a loping Kaczmarz strategy for obtaining stable solutions of nonlinear systems of ill-posed operator equations. We show that the proposed method is a convergent regularization method. Numerical tests are presented for a linear problem related to photoacoustic tomography and a non-linear problem related to the testing of semiconductor devices.

math.NA↗

arXiv2007

Inversion of spherical means and the wave equation in even dimensions

We establish inversion formulas of the so called filtered back-projection type to recover a function supported in the ball in even dimensions from its spherical means over spheres centered on the boundary of the ball. We also find several formulas to recover initial data of the form (f,0) (or (0,g)) for the free space wave equation in even dimensions from the trace of the solution on the boundary of the ball, provided the initial data has support in the ball.

math.AP↗

arXiv2006

Mathematical Challenges Arising in Thermoacoustic Tomography with Line Detectors

Thermoacoustic computed tomography (thermoacoustic CT) has the potential to become a mayor non-invasive medical imaging method. In this paper we derive a general mathematical framework of a novel measuring setup introduced in [P. Burgholzer, C. Hofer, G. Paltauf, M. Haltmeier, and O. Scherzer, "Thermoacoustic tomography with integrating area and line detectors", IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control, 52 (2005)], that uses line shaped detectors instead of the usual point like ones. We show that the three dimensional thermoacoustic imaging problem reduces to the mathematical problem of reconstructing the initial data of the two dimensional wave equation from boundary measurements of its solution. We derive and analyze an analytic reconstruction formula which allows for fast numerical implementation.

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