Bloomberg L.P., New York, NY 10022, USA.
IEEE Trans Image Process. 2011 Apr;20(4):1094-111. doi: 10.1109/TIP.2010.2083677. Epub 2010 Oct 4.
Quantitatively accurate fluorescence diffuse optical tomographic (FDOT) image reconstruction is a computationally demanding problem that requires repeated numerical solutions of two coupled partial differential equations and an associated inverse problem. Recently, adaptive finite element methods have been explored to reduce the computation requirements of the FDOT image reconstruction. However, existing approaches ignore the ubiquitous presence of noise in boundary measurements. In this paper, we analyze the effect of finite element discretization on the FDOT forward and inverse problems in the presence of measurement noise and develop novel adaptive meshing algorithms for FDOT that take into account noise statistics. We formulate the FDOT inverse problem as an optimization problem in the maximum a posteriori framework to estimate the fluorophore concentration in a bounded domain. We use the mean-square-error (MSE) between the exact solution and the discretized solution as a figure of merit to evaluate the image reconstruction accuracy, and derive an upper bound on the MSE which depends upon the forward and inverse problem discretization parameters, noise statistics, a priori information of fluorophore concentration, source and detector geometry, as well as background optical properties. Next, we use this error bound to develop adaptive meshing algorithms for the FDOT forward and inverse problems to reduce the MSE due to discretization in the reconstructed images. Finally, we present a set of numerical simulations to illustrate the practical advantages of our adaptive meshing algorithms for FDOT image reconstruction.
定量准确的荧光漫射光学断层成像（FDOT）图像重建是一个计算密集型问题，需要反复求解两个耦合的偏微分方程和一个相关的反问题。最近，自适应有限元方法已经被探索用于降低 FDOT 图像重建的计算要求。然而，现有的方法忽略了边界测量中普遍存在的噪声。在本文中，我们分析了在存在测量噪声的情况下，有限元离散化对 FDOT 正问题和反问题的影响，并为 FDOT 开发了新的自适应网格算法，这些算法考虑了噪声统计。我们将 FDOT 反问题表述为最大后验概率框架中的优化问题，以估计有界域内的荧光团浓度。我们使用精确解和离散解之间的均方误差（MSE）作为评价图像重建精度的指标，并推导出一个依赖于正问题和反问题离散化参数、噪声统计、荧光团浓度的先验信息、源和探测器几何形状以及背景光学性质的 MSE 上限。接下来，我们使用这个误差界来开发 FDOT 正问题和反问题的自适应网格算法，以降低重建图像中离散化引起的 MSE。最后，我们提出了一组数值模拟，以说明我们的 FDOT 图像重建自适应网格算法的实际优势。
