Inhaltsverzeichnis - Mathematisches Institut der Universität zu Köln

DMV Tagung 2011 - Köln, 19. - 22. September
Florian Boßmann
Universität Göttingen
Model based image pattern recognition in ultrasonic non destructive testing
Ultrasonic non destructive testing methods have achieved rapidly gaining prominence as reli-able
techniques for tube inspection. Its use as inspection tool brings into light the challenge of developing
fast and reliable data processing methods in order to be able to characterize flaws in the material. In
particular, the determination of precise positions and dimensions of flaws is a complicated task due to
the huge amount of data. Moreover, the obtained data contains strong noise caused e.g. by (multiple)
reflections from mode converted signals.
In our group a new simplified physical model for the problem of tube inspection by ultrasound waves has
been developed. Using this model we are able to solve the inverse problem of defect reconstruction in
two steps. In particular, we use the sparsity of this problem: Normally the number and size of defects
will be small compared to the tube size. This means only a small number of reflections generates the
measured data. In a first step we compute the position and amplitude of those reflections by solving a
sparse representation problem. Taking this information we can solve the inverse problem of our model.
In this talk, the model and some basic ideas of its inversion will be summarized. First numerical results
can be shown. This is joined work with my supervisor Gerlind Plonka-Hoch (Göttingen).
Brigitte Forster
Technische Universität München
Interpolation with fundamental splines of fractional order
Fractional B-splines Bσ , σ ≥ 1, are piecewise polynomials of fractional degree that interpolate the
classical Schoenberg splines Bn, n ∈ N, with respect to the degree. As the Schoenberg splines of order
≥ 3, they in general do not satisfy the interpolation property Bσ(n − k) = δn,k, n,k ∈ Z. However, the
application of the interpolation filter 1/∑k∈Z �Bσ(ω − 2πk)—if well-defined—in the frequency domain
yields a fundamental spline of fractional order that does satisfy the interpolation property. We handle the
ambiguity of the complex exponential function appearing in the denominator of the interpolation filter, and
relate the filter to interesting properties of the Hurwitz zeta function. Finally, we show that the fundamental
splines of fractional order fits into the setting of Kramer’s Lemma and allows for a family of sampling,
resp., interpolation series. This is joint work with Peter Massopust, Helmholtz Zentrum München.
112