Exploring the redundancy of Radon transform using a set of partial derivative equations: could we precisely reconstruct the image from a sparse-view projection without any image prior?

In this study, we propose a universalnth order partial differential equation (PDE) of 2D Radon transform to disclose the correlation of Radon transform among neighboring integration line. Specifically, a CT geometry of dual centers of rotation is introduced to formulate an object independent PDE that presents the local correlation of Radon transform on the variables of distance and angle, named LCE (local correlation equation). The LCE is directly available to divergent beam CT geometries, e.g. fan beam CT or cone beam CT. In this case, one rotation center is set at the focal spot, so that the LCE becomes a general PDE for actually used CT systems with single rotation center (origin). Thus, we deduce two equivalent LCE forms for two widely used CT geometries, i.e. cLCE for circular scanning trajectory and sLCE for stationary linear array scanning trajectory, respectively. The LCE also explores the redundancy property existed in Radon transform. One usage of the LCE is that it supports a sparse-view projection could contain enough information of complete projection, and hence projection completeness in CT scanning would be no longer needed. In this regard, based on the circular scanning trajectory, we explore whether the cLCE is able to solve sparse-view problem without the help of image prior. We propose a discrete cLCE based interpolation scheme that can be solved by a matrix inversion based on Lagrange multiplier method. The analysis on the matrix inversion shows that the interpolation matrix is full rank although the condition number of the matrix is larger when the sparsity increases. The fact suggests that sparse-view CT projection indeed contains enough information of complete projection, which is independent of the scanned object. Moreover, a unified reconstruction framework combining a regularized iterative reconstruction with the cLCE based interpolation is also proposed to cope with higher sparsity level. In experimental validation, we chose 1/4 and 1/8 sparsity to verify the discrete cLCE interpolation method and the unified reconstruction scheme, respectively. The results confirm that the sparse-view projection is feasible to realize a comparable reconstruction as from complete projection based on the LCE. It would be expected that combining the LCE property will boost various researches on CT reconstructions in the future.