A class of Landweber-type iterative methods based on the Radon transform for incomplete view tomography.

Background: We study the reconstruction problem for incomplete view tomography, including sparse view tomography and limited angle tomography, by the Landweber iteration and its accelerated version. Traditional implementations of these Landweber-type iterative methods necessitate multiple large-scale matrix-vector multiplications, which in turn require substantial time and storage resources.

Objective: This paper aims to develop and test a novel and efficient discretization approach for a class of Landweber-type methods that minimizes storage requirements by incorporating the specific structure of the incomplete view Radon transform.

Methods: We prove that the normal operator of incomplete view Radon transform in these methods is a compact convolution operator, and derive the explicit representation of its convolution kernel. Discretized by the pixel basis, these Landweber-type iterative methods can be implemented quickly and accurately by introducing a discretized convolution operation between two small-scale matrices with minimal storage requirements.

Results: For the simulated complete and limited angle data, the reconstruction results using various Landweber-type methods with our proposed discretization scheme achieve a 1-5dB improvement in PSNR and require one-third of computation time compared to the traditional approach. For the simulated sparse view data, our discretization scheme yields a valid image with the highest PSNR.

Conclusions: The Landweber-type iterative methods, when combined with our proposed discretization approach based on the Radon transform, are effective for addressing the incomplete view tomography problem.