The two dimensional spherical Radon transform assigns to a function its integrals over circles and is one of the mathematical models in sonar. Thus, stable inversion algorithms for that transform are of great interest. In contrast to the Radon
transform, where we integrate over lines, the spherical Radon transform
can not be formulated as bounded operator between Hilbert spaces,
rather between distribution spaces. To establish a stable
inversion scheme we have to think about how to approximate a distribution
in an appropriate manner. That is why we expand the method of approximate
inverse, which led to efficient reconstruction algorithms in several
areas of tomography, such that it may be applied to this distribution space
setting. The approximate inverse consists of evaluations of the dual pairing
of the given Radon data with precomputed reconstruction kernels. These
kernels are smooth and rapidly decreasing functions whose images under
the Radon transform are so called mollifiers.
In the talk we first define what we mean by a mollifier. We design a mollifier
and outline how the corresponding reconstruction kernel can be approximated by
numerical integration. We show that the Radon transform as well as the
constructed mollifier satisfy a certain operator invariance which accelerates
the reconstruction process significantly. Plots of the mollifier and the reconstruction kernel are presented.
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