Zheng J, Saquib S S, Sauer K, Bouman C A
Delphi Delco Electron. Syst., Kokomo, IN 46904-9005, USA.
IEEE Trans Image Process. 2000;9(10):1745-59. doi: 10.1109/83.869186.
Bayesian tomographic reconstruction algorithms generally require the efficient optimization of a functional of many variables. In this setting, as well as in many other optimization tasks, functional substitution (FS) has been widely applied to simplify each step of the iterative process. The function to be minimized is replaced locally by an approximation having a more easily manipulated form, e.g., quadratic, but which maintains sufficient similarity to descend the true functional while computing only the substitute. We provide two new applications of FS methods in iterative coordinate descent for Bayesian tomography. The first is a modification of our coordinate descent algorithm with one-dimensional (1-D) Newton-Raphson approximations to an alternative quadratic which allows convergence to be proven easily. In simulations, we find essentially no difference in convergence speed between the two techniques. We also present a new algorithm which exploits the FS method to allow parallel updates of arbitrary sets of pixels using computations similar to iterative coordinate descent. The theoretical potential speed up of parallel implementations is nearly linear with the number of processors if communication costs are neglected.
贝叶斯断层扫描重建算法通常需要对多个变量的泛函进行高效优化。在这种情况下，以及在许多其他优化任务中，函数替换（FS）已被广泛应用于简化迭代过程的每一步。待最小化的函数在局部被一个具有更易于处理形式（例如二次型）的近似函数所取代，但该近似函数要与原函数保持足够的相似性，以便在仅计算替代函数时仍能使真实泛函下降。我们给出了FS方法在贝叶斯断层扫描的迭代坐标下降法中的两个新应用。第一个是对我们的坐标下降算法的改进，将一维（1 - D）牛顿 - 拉夫森近似替换为另一种二次型近似，从而使得收敛性易于证明。在模拟中，我们发现这两种技术在收敛速度上基本没有差异。我们还提出了一种新算法，该算法利用FS方法，通过类似于迭代坐标下降的计算来允许对任意像素集进行并行更新。如果忽略通信成本，并行实现的理论潜在加速比与处理器数量几乎呈线性关系。
