马尔可夫链蒙特卡罗和迭代法相结合的混合反演方法确定随机介质的光学性质
A hybrid inversion scheme combining Markov chain Monte Carlo and iterative methods for determining optical properties of random media
摘要：
近红外光谱（NIRS）包括漫射光学层析成像是一种成像模式，它利用漫射光在随机介质中传播。当从反射或透射光的边界测量研究随机介质的光学性质时，已知迭代反演方案（诸如Levenberg-Marquardt算法）在初始猜测不接近待重构系数的真实值时失败。在本文中，我们探讨如何克服这个弱点的迭代计划使用马尔可夫链蒙特卡罗。使用时间分辨的测量对基于多面体的幻影，我们提出了一种情况下，Levenberg-Marquardt算法无法工作，但建议的混合方法工作良好。然后用漫射光学层析成像的玩具模型说明了evenberg-Marquardt方法在陷入局部极小值时会失败，但混合方法可以通过使用Metropolis-Hastings马尔可夫链蒙特卡罗算法逃离局部极小值，直到到达全局极小值的谷值。所提出的混合格式可以应用于NIRS中不同的反问题，迭代求解。我们发现，对于数值和幻影实验的光学性能，如吸收和减少散射系数可以检索而不被困在一个局部最小值时，蒙特卡罗模拟运行只有约$100$步骤切换到迭代方法之前。将混合算法与模拟退火算法进行了比较。虽然Metropolis-Hastings MCMC在大约$10000$ Monte Carlo步骤处到达稳态，但是在混合方法中，蒙特卡罗模拟可以在老化时间之前停止。
英文摘要：
Near-infrared spectroscopy (NIRS) including diffuse optical tomography is an imaging modality which makes use of diffuse light propagation in random media. When optical properties of a random medium is investigated from boundary measurements of reflected or transmitted light, iterative inversion schemes such as the Levenberg-Marquardt algorithm are known to fail when initial guesses are not close to the true value of the coefficient to be reconstructed. In this paper, we investigate how this weakness of iterative schemes is overcome by the use of Markov chain Monte Carlo. Using time-resolved measurements performed against a polyurethane-based phantom, we present a case that the Levenberg-Marquardt algorithm fails to work but the proposed hybrid method works well. Then with a toy model of diffuse optical tomography we illustrate that the evenberg-Marquardt method fails when it is trapped by a local minimum but the hybrid method can escape from local minima by using the Metropolis-Hastings Markov chain Monte Carlo algorithm until it reaches the valley of the global minimum. The proposed hybrid scheme can be applied to different inverse problems in NIRS which are solved iteratively. We find that for both numerical and phantom experiments optical properties such as the absorption and reduced scattering coefficients can be retrieved without being trapped by a local minimum when Monte Carlo simulation is run only about $100$ steps before switching to an iterative method. The hybrid method is compared with simulated annealing. Although the Metropolis-Hastings MCMC arrives at the steady state at about $10000$ Monte Carlo steps, in the hybrid method the Monte Carlo simulation can be stopped way before the burn-in time.
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