Inhaltsverzeichnis - Mathematisches Institut der Universität zu Köln

DMV Tagung 2011 - Köln, 19. - 22. September
Jürgen Frikel
Institut für Biomathematik und Biometrie, Helmholtz Zentrum München
A new framework for tomographic reconstruction at a limited angular range
We investigate the reconstruction problem for limited angle tomography. Such problems arise naturally
in applications like digital breast tomosynthesis, dental tomography, etc. Since the acquired tomographic
data is highly incomplete, the reconstruction problem is severely ill-posed and the traditional reconstruction
methods, as filtered backprojection (FBP), do not perform well in such situations. Our approach
is based on the observation that for a given acquisition geometry only a few (visible) structures of the
unknown object can be reconstructed using a limited angle data set. By formulating the problem in the
curvelet domain, we can characterize those curvelet coefficients, which correspond to visible structures
in the image domain. The integration of this a-priori information into the reconstruction problem leads to a
considerable dimensionality reduction in the transform domain and, thus, accelerates the corresponding
reconstruction algorithms.
Jan Hamaekers
Fraunhofer SCAI, Sankt Augustin
HCFFT: A fast Fourier transformation software library for general hyperbolic cross/sparse grid
spaces
In this talk, we will present our software library HCFFT for fast Fourier transformations on general sparse
grid approximation spaces. The curse of dimension limits the application of standard full grid spaces to
low dimensional approximation problems and thus limits also the application of the conventional multidimensional
fast Fourier transformation method. For functions which fulfill certain additional regularity
assumptions, sparse grid spaces allows us to circumvent the curse of dimension at least to some extend.
Our library HCFFT enables us to perform a fast Fourier transformation on these spaces. In particular, this
includes optimized sparse grid approximation spaces, e.g. energy-norm sparse grid like spaces, and also
dimension-adaptive sparse grid approximation spaces. We will discuss costs, accuracy, convergence
rates, and some implementational details and applications.
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