The Closest Point Method for solving partial differential equations (PDEs) posed on surfaces was recently introduced by Ruuth and Merriman [J. Comput. Phys. 2008] and successfully applied to a variety of surface PDEs. In this paper we study the theoretical foundations of this method. The main idea is that surface differentials of a surface function can be replaced with Cartesian differentials of its closest point extension, i.e., its composition with a closest point function. We introduce a general class of these closest point functions (a subset of differentiable retractions), show that these are exactly the functions necessary to satisfy the above idea, and give a geometric characterization of this class. Finally, we construct some closest point functions and demonstrate their effectiveness numerically on surface PDEs.Comment: 22 pages, 3 figures, 4 table

Abstract.We discuss an extension of our method image inpainting based on coherence transport. For the latter method the pixels of the inpainting domain have to be serialized into an ordered list. Until now, to induce the serialization we have used the distance to boundary map. But there are inpainting problems where the distance to boundary serialization causes unsatisfactory inpainting results. In the present work we demonstrate cases where we can resolve the difficulties by employing other distance functions which better suit the problem at hand. Key words. image processing, image inpainting, distance functions AMS subject classifications. 65D18, 51K99DOI. 10.1137/1008072961. Introduction. Nontexture image inpainting, also termed image interpolation, is the task of determining the values of a digital image for a destroyed, or consciously masked, subregion of the image domain.The simple idea of the generic single pass method-which forms the basis for our method image inpainting based on coherence transport published in [5]-is to fill the inpainting domain by traversing its pixels in an onion peeling fashion from the boundary inward and thereby setting new image values as weighted means of given or already calculated ones.Telea has been the first to use such an algorithm in [13]: the pixels are serialized according to their Euclidean distance to the boundary of the inpainting domain, and the weight is such that image values are propagated mainly along the gradient of the distance map. By his choices of the weight and the pixel serialization, the method of Telea is not adapted to the image.In [5], we addressed the adaption of the weight: our method uses an image dependent weight such that image values are propagated along the estimated tangents of color lines which have been interrupted by the inpainting domain. In that way we could improve the quality of the inpainting results compared to Telea (see Figure 1). Beyond that, we illustrated in [5] Figure 2 illustrates. The image in the middle shows the

We contribute to the mathematical modeling and analysis of magnetic particle imaging which is a promising new in-vivo imaging modality. Concerning modeling, we develop a structured decomposition of the imaging process and extract its core part which we reveal to be common to all previous contributions in this context. The central contribution of this paper is the development of reconstruction formulae for MPI in 2D and 3D. Until now, in the multivariate setup, only time consuming measurement approaches are available, whereas reconstruction formulae are only available in 1D. The 2D and the 3D (describing the real world) reconstruction formulae which we derive here are significantly different from the 1D situation -in particular there is no Dirac property in dimensions greater than one when the particle sizes approach zero. As a further result of our analysis, we conclude that the reconstruction problem in MPI is severely ill-posed. Finally, we obtain a model-based reconstruction algorithm.

Image inpainting is the process of touching-up damaged or unwanted portions of a picture and is an important task in image processing. For this purpose Bornemann and März [J. Math. Imaging Vis. , 28 (2007), pp. 259-278] introduced a very efficient method called Image Inpainting Based on Coherence Transport which fills the missing region by advecting the image information along integral curves of a coherence vector field from the boundary towards the interior of the hole. The mathematical model behind this method is a first-order functional advection PDE posed on a compact domain with all inflow boundary. We show that this problem is well-posed under certain conditions.

Magnetic particle imaging (MPI) is a promising new in vivo medical imaging modality in which distributions of super-paramagnetic nanoparticles are tracked based on their response in an applied magnetic field. In this paper we provide a mathematical analysis of the modeled MPI operator in the univariate situation. We provide a Hilbert space setup, in which the MPI operator is decomposed into simple building blocks and in which these building blocks are analyzed with respect to their mathematical properties. In turn, we obtain an analysis of the MPI forward operator and, in particular, of its ill-posedness properties. We further get that the singular values of the MPI core operator decrease exponentially. We complement our analytic results by some numerical studies which, in particular, suggest a rapid decay of the singular values of the MPI operator.