2D and 3D Transmission Compton Scattering Imaging solved by Generalized Radon Transforms

康普顿散射成像作为用于诊断感兴趣对象的解剖结构的传统X射线成像的创新和补充方式而出现。康普顿散射，隐含在80%的衰减的一个旅行的光子通量，可以用来恢复电子密度的照明介质。类似于传统的层析成像，测量的几何形状产生的积分算子的重建技术必须开发。该建议是我们的论文[30]的后续，该论文通过研究一族Cormack型Radon变换，为康普顿散射层析成像（CST）中某些模态的图像重建建立了数学基础。然而，在文献中提出并在[30]中统一的积分建模过于简化，无法在实践中应用。康普顿散射光子的使用遇到了一些问题：衰减，能量分辨率和数据的不完整性。该项目旨在完成[30]中给出的数学框架，以克服这些未来成像概念中将出现的物理限制。这一完整的框架将为研究创新的国家技术服务小组模式的可行性提供有效的方法。这种模式，它合并了目前的CST的概念，使一个完美的利用散射辐射和出现感兴趣的未来imaging. Further，扩展到3D的研究模式可以构成一个新的概念康普顿散射成像的数学基础，在3D工作通过传输。在所设计的系统中，一个耦合源/探测器沿着整个球体的旋转使得能够直接在3D中恢复电子密度。这一开创性的概念将在具有3D对象的兴趣，因为只需要一个检测器，由于康普顿散射模态的高灵敏度数据。此外，其在非破坏性测试中的应用，其中数据采集时间不受限制，显得特别相关。总之，拟议项目的目的是提供一个完整的数学框架，在图像重建的方式在2D和3D康普顿成像，将出现在未来十年。

Compton scattering imaging is emerging as an innovative and complementary way to conventional X-ray imaging for diagnosing the anatomic structure of an object of interest. The Compton scattering, implied in 80 percent of the attenuation of a travelling photon flux, can be used to recover the electron density of the illuminated medium. Similarly to conventional tomography, the geometry of measurement gives rise to integral operators for which reconstruction techniques have to be developed. This proposal follows on from our paper [30] which established the mathematical foundations for image reconstruction of some modalities in Compton scattering tomography (CST) via the study of a family of Cormack-type Radon transforms. Nevertheless, the integral modelings proposed in the literature and unified in [30] are too much simplified to be applied in practice. The use of Compton scattered photons encounters some issues : attenuation, energy resolution and incompleteness in the data. This project intends to complete the mathematical framework given in [30] in order to overcome the physical limitations which will occur in these future imaging concepts. This complete framework will provide efficient methods to study the feasibility of an innovative CST modality. This modality, which merges the concept of the current CST ones, makes a perfect use of the scattered radiation and appears of interest for the future of imaging.Furthermore, an extension to 3D of the studied modalities could constitute the mathematical foundations for a new concept of Compton scattering imaging in 3D working by transmission. In the designed system, the revolution of a couple source/detector along the whole sphere enables the recovery of the electron density directly in 3D. This pioneering concept would be of interest in featuring a 3D object since only one detector is required and due to the high sensitive data of Compton scattering modalities. Furthermore, its application in non-destructive testing, where data acquisition time is not limited, appears especially relevant. In conclusion, the aim of the proposed project is to provide a complete mathematical framework in image reconstruction for modalities in 2D and 3D Compton imaging which will emerge in the next decade.