2-d Tomography from Noisy Projections Taken at Unknown Random Directions

Computerized Tomography (CT) is a standard method for obtaining internal structure of objects from their projection images. While CT reconstruction requires the knowledge of the imaging directions, there are some situations in which the imaging directions are unknown, for example, when imaging a moving object. It is therefore desirable to design a reconstruction method from projection images taken at unknown directions. Recently, it was shown that the imaging directions can be obtained by the diffusion map framework. Another difficulty arises from the fact that projections are often contaminated by noise, practically limiting all current methods, including the diffusion map approach. In this paper, we introduce two denoising steps that allow reconstructions at much lower signal-to-noise ratios (SNR) when combined with the diffusion map framework. The first denoising step consists of using the singular value decomposition (SVD) in order to find an adaptive basis for the projection data set, leading to improved similarities between different projections. In the second step, we denoise the graph of similarities using the Jaccard index, which is a widely used measure in network analysis. Using this combination of SVD, Jaccard index and diffusion map, we are able to reconstruct the 2-D Shepp-Logan phantom from simulative noisy projections at SNRs well below their currently reported threshold values. Although the focus of this paper is the 2-D CT reconstruction problem, we believe that the combination of SVD, Jaccard index graph denoising and diffusion maps is potentially useful in other signal processing and image analysis applications.