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?

TL;DR

This paper introduces a universal PDE for 2D Radon transform revealing its redundancy property, enabling sparse-view CT reconstruction without image priors, potentially transforming CT scanning and reconstruction methods.

Contribution

It proposes the local correlation equation (LCE) as a universal PDE for Radon transform, and develops a new sparse-view CT reconstruction framework leveraging this redundancy.

Findings

LCE reveals the universal correlation property of Radon transform.

Sparse-view CT can be effectively reconstructed without image priors.

The proposed methods are validated through experiments, showing promising results.

Abstract

In this study, we proposed a universal n-th order partial differential equation (PDE) of 2-D Radon transform to disclose the relationship of Radon transform over a neighborhood of the integral line, named as local correlation equation (LCE). It is independent to the imaging object while in present CT theory, the relationship of Radon transform over neighboring integral line had been described depended on the imaging objection. Hence, the LCE is the first PDE to reveal the universal correlation property of Radon transform. The LCE can be applied to either of 2D CT projections or any 2-D profile of 3-D CT projections. The correlation also provides the redundancy property of Radon transform. In this regard, we carried out a preliminary study on sparse-view CT reconstruction by using a discrete first order LCE to interpolate missing projections in sparse-view sampling without knowing image prior. Meanwhile, we also proposed a unified reconstruction framework that combines a regularized iterative reconstruction with the LCE based interpolation method to handle the sparse-view CT problem with higher sparsity level. The conducted experiments have credibly validated the proposed LCE, projection interpolation method, and the unified reconstruction scheme. The result of this study suggests an attractive possibility that a sparse-view projection may contain enough information of the complete projection, by which projection completeness in CT scanning may not be necessity. This possibility would bring profound changes in CT geometry designs and reconstruction algorithms. Moreover, this study initiates an appealing research topic of exploring the redundancy property of Radon transform and investigating new CT theories based on the redundancy property, which will boost the further development of CT reconstructions.