We consider reconstructing multi-channel images from measurements performed by photon-counting and energy-discriminating detectors in the setting of multi-spectral x-ray computed tomography (CT). Our aim is to exploit the strong structural correlation that is known to exist between the channels of multi-spectral CT images. To that end, we adopt the multi-channel Potts prior to jointly reconstruct all channels. This nonconvex prior produces piecewise constant solutions with strongly correlated channels. In particular, edges are strictly enforced to have the same spatial position across channels which is a benefit over TV-based methods whose channel-couplings are typically less strict. We consider the Potts prior in two frameworks: (a) in the context of a variational Potts model, and (b) in a Potts-superiorization approach that perturbs the iterates of a basic iterative least squares solver. We identify an alternating direction method of multipliers approach as well as a Potts-superiorized conjugate gradient method as particularly suitable. In numerical experiments, we compare the Potts prior based approaches to existing TV-type approaches on realistically simulated multi-spectral CT data and obtain improved reconstruction for compound solid bodies.

我们考虑在多光谱X射线计算机断层扫描（CT）的设置中，根据光子计数和能量分配检测器的测量结果来重建多通道图像。我们的目标是利用已知在多光谱CT图像通道之间存在的强结构相关性。为此，在联合重建所有通道之前，我们采用多通道Potts。该非凸先验产生具有强相关通道的分段常数解。特别地，边缘被严格地强制为在通道之间具有相同的空间位置，这相对于基于电视的方法（其信道耦合通常不太严格）是有好处的。我们在以下两个框架中考虑了Potts先验：（a）在变型Potts模型的背景下，（b）以Potts-superiorization方法，该方法扰动基本迭代最小二乘法求解器的迭代。我们确定乘数方法的交替方向方法以及适用于Potts的高级共轭梯度方法。在数值实验中，我们在实际模拟的多光谱CT数据上将基于Potts先验的方法与现有的TV型方法进行了比较，并获得了复合固体的改进重建方法。