We study an inverse problem where an unknown radiating source is observed with collimated detectors along a single line and the medium has a known attenuation. The research is motivated by applications in SPECT and beam hardening. If measurements are carried out with frequencies ranging in an open set, we show that the source density is uniquely determined by these measurements up to averaging over levelsets of the integrated attenuation. This leads to a generalized Laplace transform. We also discuss some numerical approaches and demonstrate the results with several examples.

中文翻译：

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沿一条线的多个频率的辐射源的最佳恢复

我们研究了一个反问题，其中用准直探测器沿着一条直线观察到未知的辐射源，并且介质具有已知的衰减。这项研究的动机是在SPECT和梁硬化中的应用。如果使用开放集合中的频率进行测量，则表明源密度是由这些测量唯一确定的，直到积分衰减的各个水平集的平均值为止。这导致了广义的拉普拉斯变换。我们还将讨论一些数值方法，并通过几个示例来证明结果。