In many areas of two-dimensional imaging science, data acquisition arises as an integral process. The inverse process—or image reconstruction—means the solving of a Radon problem mathematically. It may happen that there exists classes of integral data which are mirror symmetric with respect to a line. Common sense suggests that the occurrence of a symmetry usually provides significant help in the search of the problem solution. Here, we showed an example of the contrary to this popular belief. In fact, to solve such a Radon problem with inherent reflection symmetry, there is a need to split it into two new Radon problems on half-spaces, which do not have solutions at hand. In this paper, a full solution is obtained via geometric inversion mapping of the two original half-spaces Radon problems to the disk interior/exterior Radon problem arising in recent modalities of Compton scattering tomography, which fortunately has explicit worked out inverse formulas.

中文翻译：

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反射对称氡数据的函数重建

在二维成像科学的许多领域，数据采集是作为一个整体过程出现的。逆过程——或图像重建——意味着在数学上解决氡问题。可能会发生存在关于一条线镜像对称的积分数据类。常识表明，对称性的出现通常为寻找问题解决方案提供重要帮助。在这里，我们展示了一个与这种流行观念相反的例子。事实上，要解决这样一个具有固有反射对称性的氡问题，需要将其拆分为两个新的半空间上的氡问题，而这两个问题手头没有解。在本文中，