Markus Haltmeier

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

Photoacoustic tomography using a Mach-Zehnder interferometer as an acoustic line detector

(f,0)

Thermoacoustic tomography with integrating area and line detectors

(or

Thermoacoustic computed tomography with large planar receivers

(0,g)

Sparse regularization with lq penalty term

) for the free space wave equation in even dimensions from the trace of the solution on the boundary of the ball, provided that the initial data has support in the ball.

Temporal back-projection algorithms for photoacoustic tomography with integrating line detectors

A three-dimensional photoacoustic imaging method is presented that uses a Mach-Zehnder interferometer for measurement of acoustic waves generated in an object by irradiation with short laser pulses. The signals acquired with the interferometer correspond to line integrals over the acoustic wave field. An algorithm for reconstruction of a three-dimensional image from such signals measured at multiple positions around the object is shown that is a combination of a frequency-domain technique and the inverse Radon transform. From images of a small source scanning across the interferometer beam it is estimated that the spatial resolution of the imaging system is in the range of 100 to about 300 mum, depending on the interferometer beam width and the size of the aperture formed by the scan length divided by the source-detector distance. By taking an image of a phantom it could be shown that the imaging system in its present configuration is capable of producing three-dimensional images of objects with an overall size in the range of several millimeters to centimeters. Strategies are proposed how the technique can be scaled for imaging of smaller objects with higher resolution.

THERMOACOUSTIC TOMOGRAPHY AND THE CIRCULAR RADON TRANSFORM: EXACT INVERSION FORMULA

Thermoacoustic (optoacoustic, photoacoustic) tomography is based on the generation of acoustic waves by illumination of a sample with a short electromagnetic pulse. The absorption density inside the sample is reconstructed from the acoustic pressure measured outside the illuminated sample. So far measurement data have been collected with small detectors as approximations of point detectors. Here, a novel measurement setup applying integrating detectors (e.g., lines or planes made of piezoelectric films) is presented. That way, the pressure is integrated along one or two dimensions, enabling the use of numerically efficient algorithms, such as algorithms for the inverse radon transformation, for thermoacoustic tomography. To reconstruct a three-dimensional sample, either an area detector has to be moved tangential around a sphere that encloses the sample or an array of line detectors is rotated around a single axis. The line detectors can be focused on cross sections perpendicular to the rotation axis using a synthetic aperture (SAFT) or by scanning with a cylindrical lens detector. Measurements were made with piezoelectric polyvinylidene fluoride film detectors and evaluated by comparison with numerical simulations. The resolution achieved in the resulting tomography images is demonstrated on the example of the reconstructed cross section of a grape.

KACZMARZ METHODS FOR REGULARIZING NONLINEAR ILL-POSED EQUATIONS I: CONVERGENCE ANALYSIS

Thermoacoustic imaging is a promising new modality for nondestructive evaluation. So far point measurement data for thermoacoustic imaging are used. In this paper we propose a novel measurement set-up with relatively large piezo foils (planar receivers) and an according real time imaging algorithm based on the Radon transform. We present numerical simulations for simulated and real world data.

On Steepest-Descent-Kaczmarz methods for regularizing systems of nonlinear ill-posed equations

We consider the stable approximation of sparse solutions to nonlinear operator equations by means of Tikhonov regularization with a subquadratic penalty term. Imposing certain assumptions, which for a linear operator are equivalent to the standard range condition, we derive the usual convergence rate of the regularized solutions in dependence of the noise level δ. Particular emphasis lies on the case, where the true solution is known to have a sparse representation in a given basis. In this case, if the differential of the operator satisfies a certain injectivity condition, we can show that the actual convergence rate improves up to O(δ).

A Reconstruction Algorithm for Photoacoustic Imaging Based on the Nonuniform FFT

Line detectors integrate the measured acoustic pressure over a straight line and can be realized by a thin line of a piezoelectric film or by a laser beam as part of an interferometer. Photoacoustic imaging with integrating line detectors is performed by rotating a sample or the detectors around an axis perpendicular to the line detectors. The subsequent reconstruction is a two-step procedure: first, two-dimensional (2D) projections parallel to the line detector are reconstructed, then the three-dimensional (3D) initial pressure distribution is obtained by applying the 2D inverse Radon transform. The first step involves an inverse problem for the 2D wave equation. Wave propagation in two dimensions is significantly different from 3D wave propagation and reconstruction algorithms from 3D photoacoustic imaging cannot be used directly. By integrating recently established 3D formulae in the direction parallel to the line detector we obtain novel back-projection formulae in two dimensions. Numerical simulations demonstrate the capability of the derived reconstruction algorithms, also for noisy measurement data, limited angle problems and 3D reconstruction with integrating line detectors.