Revisit FBP: Analyze the tensor data after view-by-view backprojection

For a very long time, low-dose computed tomography (CT) imaging techniques have been performed by either preprocessing the projection data or regularizing the iterative reconstruction. The conventional filtered backprojection (FBP) algorithm is rarely studied. In this work, we show that the intermediate data during FBP possess some fascinating properties and can be readily processed to reduce the noise and artifacts. The FBP algorithm can be technically decomposed into three steps: filtering, view-by-view backprojection and summing. The data after view-by-view backprojection is naturally a tensor, which is supposed to contain useful information for processing in higher dimensionality. We here introduce a sorting operation to the tensor along the angular direction based on the pixel intensity. The sorting for each point in the image plane is independent. Through the sorting operation, the structures of the object can be explicitly encoded into the tensor data and the artifacts can be automatically driven into the top and bottom slices of the tensor. The sorted tensor also provides high dimensional information and good low-rank properties. Therefore, any advanced processing methods can be applied. In the experiments, we demonstrate that under the proposed scheme, even the Gaussian smoothing can be used to remove the streaking artifacts in the ultra-low dose case, with nearly no compromising of the image resolution. It is noted that the scheme presented in this paper is a heuristic idea for developing new algorithms of low-dose CT imaging.