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Semiclassical sampling and discretization of certain linear inverse problems

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arxiv 1811.01240 v2 pith:YQQT6BBV submitted 2018-11-03 math.AP

classification math.AP
keywords samplingsemiclassicalanalysisratesaliasingapplyartifactsaveraging
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abstract

We study sampling of Fourier Integral Operators $A$ at rates $sh$ with $s$ fixed and $h$ a small parameter. We show that the Nyquist sampling limit of $Af$ and $f$ are related by the canonical relation of $A$ using semiclassical analysis. We apply this analysis to the Radon transform in the parallel and the fan-beam coordinates. We explain and illustrate the optimal sampling rates for $Af$, the aliasing artifacts, and the effect of averaging (blurring) the data $Af$. We prove a Weyl type of estimate on the minimal number of sampling points to recover $f$ stably in terms of the volume of its semiclassical wave front set.

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  1. Resolution analysis of inverting the generalized Radon transform from discrete data in $\mathbb R^3$

    math.NA 2019-08 accept novelty 6.0 of 10

    For a general class of 3D generalized Radon transforms, the resolution of reconstructed jump discontinuities from discrete data is an explicit edge profile determined by the interpolation kernel, with no non-local art...

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