In X-ray computed tomography (CT) it is generally acknowledged that reconstruction methods exploiting image sparsity allow reconstruction from a significantly reduced number of projections. The use of such reconstruction methods is inspired by recent progress in compressed sensing (CS). However, the CS framework provides neither guarantees of accurate CT reconstruction, nor any relation between sparsity and a sufficient number of measurements for recovery, i.e., perfect reconstruction from noise-free data. We consider reconstruction through 1-norm minimization, as proposed in CS, from data obtained using a standard CT fan-beam sampling pattern. In empirical simulation studies we establish quantitatively a relation between the image sparsity and the sufficient number of measurements for recovery within image classes motivated by tomographic applications. We show empirically that the specific relation depends on the image class and in many cases exhibits a sharp phase transition as seen in CS, i.e., same-sparsity images require the same number of projections for recovery. Finally we demonstrate that the relation holds independently of image size and is robust to small amounts of additive Gaussian white noise.

中文翻译：

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X射线CT的去采样和稀疏性之间的经验平均情况之间的关系。

在X射线计算机断层扫描（CT）中，通常公认的是，利用图像稀疏性的重建方法允许从数量显着减少的投影进行重建。压缩感测（CS）的最新进展启发了这种重建方法的使用。但是，CS框架既不能保证精确的CT重建，也不能保证稀疏性与足够数量的恢复测量之间的关系，即从无噪声的数据中完美重建。我们考虑通过CS中提出的1-范数最小化来重构，该重构是使用标准CT扇形光束采样模式获得的数据。在经验模拟研究中，我们定量地建立了图像稀疏度与足够数量的测量值之间的关系，以便在层析成像应用程序所激发的图像类别内进行恢复。我们凭经验表明，特定关系取决于图像类别，并且在许多情况下，如CS中所示，显示出急剧的相变，即，相同稀疏度的图像需要相同数量的投影才能恢复。最后，我们证明了该关系与图像大小无关，并且对少量加性高斯白噪声具有鲁棒性。