环面计算机断层扫描
Torus computed tomography
摘要：
提出了一种新的计算机层析成像（CT）方法，用于反演二维Radon变换。这个想法依赖于平面环面的几何形状，因此我们称之为环面CT的新方法。我们证明了新的反演公式的可积函数，解决了最小化问题，在索伯列夫空间的Tikhonov正则化，并证明了该解决方案提供了一个可接受的正则化策略与定量的稳定性估计。该正则化是用于体模的傅里叶级数的简单后处理低通滤波器。我们还研究了平面环面上X射线变换的伴随算子和正规算子。X射线变换在平坦环面上是么正的。我们已经实现了环面CT方法，使用Matlab和模拟数据进行了测试，有希望的结果。反演方法是无网格的，因为它给出了一个封闭的形式函数，可以在任何感兴趣的点进行评估。
英文摘要：
We present a new computed tomography (CT) method for inverting the Radon transform in 2D. The idea relies on the geometry of the flat torus, hence we call the new method Torus CT. We prove new inversion formulas for integrable functions, solve a minimization problem associated to Tikhonov regularization in Sobolev spaces and prove that the solution operator provides an admissible regularization strategy with a quantitative stability estimate. This regularization is a simple post-processing low-pass filter for the Fourier series of a phantom. We also study the adjoint and the normal operator of the X-ray transform on the flat torus. The X-ray transform is unitary on the flat torus. We have implemented the Torus CT method using Matlab and tested it with simulated data with promising results. The inversion method is meshless in the sense that it gives out a closed form function that can be evaluated at any point of interest.
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