Artifacts in incomplete data tomography - with applications to photoacoustic tomography and sonar

We develop a paradigm using microlocal analysis that allows one to characterize the visible and added singularities in a broad range of incomplete data tomography problems. We give precise characterizations for photo- and thermoacoustic tomography and Sonar, and provide artifact reduction strategies. In particular, our theorems show that it is better to arrange Sonar detectors so that the boundary of the set of detectors does not have corners and is smooth. To illustrate our results, we provide reconstructions from synthetic spherical mean data as well as from experimental photoacoustic data

Morera Theorems for Complex Manifolds

AbstractWe prove Morera theorems for the Radon transform integrating on geodesic spheres on complex analytic manifolds of arbitrary dimension. To avoid pathologies, we assume that the radius of each sphere of integration is less than the injectivity radius at its center. The proofs of the main results are local, and they involve the microlocal properties of associated Radon transforms and a theorem of Hörmander, Kawai, and Kashiwara on microlocal singularities. We consider Morera theorems for spheres of fixed radius and spheres of arbitrary radius

Microlocal Analysis of Bistatic SAR Imaging

A poster analyzing bistatic synthetic aperture radar imaging

Wavelet methods for a weighted sparsity penalty for region of interest tomography

We consider region of interest (ROI) tomography of piecewise constant functions. Additionally, an algorithm is developed for ROI tomography of piecewise constant functions using a Haar wavelet basis. A weighted ℓp–penalty is used with weights that depend on the relative location of wavelets to the region of interest. We prove that the proposed method is a regularization method, i.e., that the regularized solutions converge to the exact piecewise constant solution if the noise tends to zero. Tests on phantoms demonstrate the effectiveness of the method.FWF, T 529-N18, Mumford-Shah models for tomography IINSF, 1311558, Tomography and Microlocal AnalysisFWF, W 1214, Doktoratskolleg "Computational Mathematics