ALGORITHM: Scalable Algorithms for Regularized Tomography via Decoupling

X射线计算机断层扫描（CT）和相关成像模式（例如，PET）因其过度的计算需求而臭名昭著。 虽然早期的CT算法，如滤波反投影，现在在二维中是微不足道的，在三维中是可扩展的，但更抗噪声的概率方法，如正则化断层扫描仍然是禁止的。正则化的基本思想是计算一个光滑的图像，其模拟投影（线积分）近似于观察到的（但有噪声的）X射线投影。 在以前的方法中的计算费用源于显式地应用一个大的稀疏投影矩阵（以计算图像的线积分）和它的转置，以加强这些平滑度和数据近似约束在算法的许多迭代的每一个。 我们建议研究一种新的正规化层析成像公式，其中的平滑约束解析变换从图像的投影域，任何计算开始之前。 因此，迭代完全在投影域中进行，避免了重复的稀疏矩阵-向量乘积。 一个更令人惊讶的好处是将一个大的正则化方程系统解耦为许多小的简单方程系统。 计算因此变得“令人尴尬的并行”，因此延迟容忍和理想的可扩展并行计算是可能的，正如我们的初步结果在2-d中显示的那样。 我们建议将这种技术应用于CT以外的模式，在三维空间中实现它，并美化概率模型。此外，这种方法的网络友好性将使我们能够研究利用典型医院中日益浪费的桌面计算能力的可行性。 我们将解耦正则化视为断层扫描中令人兴奋的发展，通过向医生，患者和科学家提供更少伪影，更高分辨率和更强交互性的图像来造福社会。

X-ray computerized tomography (CT) and related imaging modalities (e.g., PET) are notorious for their excessive computational demands. While early CT algorithms such as filtered backprojection are now trivial in two-dimensions and scalable in three-dimensions, the more noise-resistant probabilistic methods such as regularized tomography are still prohibitive.The basic idea of regularization is to compute a smooth image whose simulated projections (line integrals) approximate the observed (but noisy) X-ray projections. The computational expense in previous methods stems from explicitly applying a large sparse projection matrix (to compute line integrals of the image) and its transpose to enforce these smoothness and data approximation constraints during each of many iterations of the algorithm. We propose to study a new formulation of regularized tomography in which the smoothness constraint is analytically transformed from the image to the projection domain, before any computations begin. As a result, iterations take place entirely in the projection domain, avoiding the repeated sparse matrix-vector products. A more surprising benefit is the decoupling of a large system of regularization equations into many small systems of simpler equations. The computation thus becomes ``embarassingly parallel'', so that latency tolerant and ideally scalable parallel computations are possible, as our preliminary results show in 2-d. We propose to apply this technique to modalities other than CT, to implement it in three-dimensions, and to embellish the probability models. Further, the network-friendly nature of this method will allow us to study the feasibility of harnessing the increasingly wasted desktop compute power in a typical hospital. We see decoupled regularization as an exciting development in tomography, benefiting society by providing images to doctors, patients, and scientists with fewer artifacts, at higher resolutions, and with greater interactivity.