Computerized tomography 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. Another difficulty arises from the fact that the projections are often contaminated by noise, practically limiting all current methods, including the recently proposed diffusion map approach. In this paper, we introduce two denoising steps that allow reconstructions at much lower signal-to-noise ratios (SNRs) when combined with the diffusion map framework. In the first denoising step we use principal component analysis (PCA) together with classical Wiener filtering to derive an asymptotically optimal linear filter. In the second step, we denoise the graph of similarities between the filtered projections using a network analysis measure such as the Jaccard index. Using this combination of PCA, Wiener filtering, graph denoising, and diffusion maps, we are able to reconstruct the two-dimensional (2-D) Shepp-Logan phantom from simulative noisy projections at SNRs well below their currently reported threshold values. We also report the results of a numerical experiment corresponding to an abdominal CT. Although the focus of this paper is the 2-D CT reconstruction problem, we believe that the combination of PCA, Wiener filtering, graph denoising, and diffusion maps is potentially useful in other signal processing and image analysis applications.

中文翻译：

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来自未知随机方向的噪声投影的二维断层扫描。

计算机断层扫描是从投影图像中获取物体内部结构的标准方法。虽然 CT 重建需要了解成像方向，但在某些情况下，成像方向是未知的，例如在对运动物体进行成像时。因此，需要根据在未知方向拍摄的投影图像设计一种重建方法。另一个困难源于投影经常被噪声污染，这实际上限制了所有当前的方法，包括最近提出的扩散图方法。在本文中，我们介绍了两个去噪步骤，当与扩散图框架结合时，它们允许以低得多的信噪比 (SNR) 进行重建。在第一个去噪步骤中，我们使用主成分分析 (PCA) 和经典维纳滤波来推导出渐近最优线性滤波器。在第二步中，我们使用网络分析度量（例如 Jaccard 指数）对过滤后的投影之间的相似性图进行去噪。使用 PCA、维纳滤波、图形去噪和扩散图的这种组合，我们能够从远低于其当前报告的阈值的 SNR 的模拟噪声投影重建二维 (2-D) Shepp-Logan 幻影。我们还报告了与腹部 CT 相对应的数值实验的结果。虽然本文的重点是二维 CT 重建问题，但我们认为结合 PCA、维纳滤波、图去噪、