In this work we introduce a new Radon transform which arises from a new modality of Compton Scattering Tomography (CST). This new system is made of a single detector rotating around a fixed source. Unlike some previous CST, no collimator is used at the detector. Such a system allows us to collect scattered photons coming from two opposite sides of the source-detector segment, hence the manifold of the associated Radon transform is a family of double circular arcs. As first main theoretical result, an analytic inversion formula is established for this new Radon transform. This is achieved through the formulation of the transform in terms of circular harmonic expansion satisfying the consistency conditions in Cormack's sense. Moreover, a fast and efficient numerical implementation via an alternative formulation based on Hilbert transform is carried out. Simulation results illustrate the theoretical feasibility of the new system. From a practical point of view, an uncollimated detector system considerably increases the amount of collected data, which is particularly significant in a scatter imaging system.

中文翻译：

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双圆弧上Radon变换的解析反演在康普顿散射断层扫描中的应用

在这项工作中，我们引入了一种新的氡变换，它源于康普顿散射断层扫描 (CST) 的新模式。这个新系统由围绕固定源旋转的单个探测器组成。与之前的一些 CST 不同，检测器不使用准直器。这样的系统允许我们收集来自源探测器段的两个相对侧的散射光子，因此相关的氡变换的流形是双圆弧族。作为第一个主要的理论结果，建立了这个新的 Radon 变换的解析反演公式。这是通过根据满足 Cormack 意义上的一致性条件的圆谐展开来制定变换来实现的。而且，通过基于希尔伯特变换的替代公式进行快速有效的数值实现。仿真结果说明了新系统的理论可行性。从实用的角度来看，非准直探测器系统大大增加了收集的数据量，这在散射成像系统中尤为重要。