基于全变差的加速梯度法CT图像重建
Accelerated gradient methods for total-variation-based CT image reconstruction
摘要：
基于全变差（TV）的计算机断层扫描（CT）图像重建实验表明，能够从稀疏视图数据产生准确的重建。特别是基于TV的重建非常适合于具有分段近似恒定区域的图像。然而，在计算上，基于TV的重建要求更高，特别是对于3D成像，并且从临床数据集的重建远未接近实时。从临床角度来看，这是不期望的，因此存在加速基本最优化问题的解决的动机。原则上可以通过任何最优化方法找到TV重建，但是实际上，在CT图像重建中出现的大规模系统排除了使用诸如牛顿法的需要存储器的方法。简单的梯度法具有低得多的存储器需求，但表现出缓慢的收敛。在目前的工作中，我们考虑使用两个加速的基于梯度的方法，GPBB和UPN，为减少梯度法迭代的数量，需要实现高精度的TV解决方案在CT图像重建。前者采用了Barzilai-Borwein（BB）步长选择和非单调线搜索等优化文献中的几种算法。后者使用一个巧妙选择的辅助点序列，以实现更好的收敛速度。该方法是高效的存储器，并配备了停止准则，以确保确实找到了TV重建。http://www2.imm.dtu.dk/~pch/TVReg/我们比较所提出的方法与标准的梯度法，应用到3D测试问题与合成的少视图数据。实验结果表明，对于实际参数，本文提出的方法明显优于梯度法。
英文摘要：
Total-variation (TV)-based Computed Tomography (CT) image reconstruction has shown experimentally to be capable of producing accurate reconstructions from sparse-view data. In particular TV-based reconstruction is very well suited for images with piecewise nearly constant regions. Computationally, however, TV-based reconstruction is much more demanding, especially for 3D imaging, and the reconstruction from clinical data sets is far from being close to real-time. This is undesirable from a clinical perspective, and thus there is an incentive to accelerate the solution of the underlying optimization problem. The TV reconstruction can in principle be found by any optimization method, but in practice the large-scale systems arising in CT image reconstruction preclude the use of memory-demanding methods such as Newton's method. The simple gradient method has much lower memory requirements, but exhibits slow convergence. In the present work we consider the use of two accelerated gradient-based methods, GPBB and UPN, for reducing the number of gradient method iterations needed to achieve a high-accuracy TV solution in CT image reconstruction. The former incorporates several heuristics from the optimization literature such as Barzilai-Borwein (BB) step size selection and nonmonotone line search. The latter uses a cleverly chosen sequence of auxiliary points to achieve a better convergence rate. The methods are memory efficient and equipped with a stopping criterion to ensure that the TV reconstruction has indeed been found. An implementation of the methods (in C with interface to Matlab) is available for download from http://www2.imm.dtu.dk/~pch/TVReg/. We compare the proposed methods with the standard gradient method, applied to a 3D test problem with synthetic few-view data. We find experimentally that for realistic parameters the proposed methods significantly outperform the gradient method.
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