Department of Instrumentation and Applied Physics, Indian Institute of Science, Bangalore, India.
Med Phys. 2012 Feb;39(2):1092-101. doi: 10.1118/1.3679855.
The authors aim at developing a pseudo-time, sub-optimal stochastic filtering approach based on a derivative free variant of the ensemble Kalman filter (EnKF) for solving the inverse problem of diffuse optical tomography (DOT) while making use of a shape based reconstruction strategy that enables representing a cross section of an inhomogeneous tumor boundary by a general closed curve.
The optical parameter fields to be recovered are approximated via an expansion based on the circular harmonics (CH) (Fourier basis functions) and the EnKF is used to recover the coefficients in the expansion with both simulated and experimentally obtained photon fluence data on phantoms with inhomogeneous inclusions. The process and measurement equations in the pseudo-dynamic EnKF (PD-EnKF) presently yield a parsimonious representation of the filter variables, which consist of only the Fourier coefficients and the constant scalar parameter value within the inclusion. Using fictitious, low-intensity Wiener noise processes in suitably constructed "measurement" equations, the filter variables are treated as pseudo-stochastic processes so that their recovery within a stochastic filtering framework is made possible.
In our numerical simulations, we have considered both elliptical inclusions (two inhomogeneities) and those with more complex shapes (such as an annular ring and a dumbbell) in 2-D objects which are cross-sections of a cylinder with background absorption and (reduced) scattering coefficient chosen as μ(a) (b)=0.01mm(-1) and μ(s) ('b)=1.0mm(-1), respectively. We also assume μ(a) = 0.02 mm(-1) within the inhomogeneity (for the single inhomogeneity case) and μ(a) = 0.02 and 0.03 mm(-1) (for the two inhomogeneities case). The reconstruction results by the PD-EnKF are shown to be consistently superior to those through a deterministic and explicitly regularized Gauss-Newton algorithm. We have also estimated the unknown μ(a) from experimentally gathered fluence data and verified the reconstruction by matching the experimental data with the computed one.
The PD-EnKF, which exhibits little sensitivity against variations in the fictitiously introduced noise processes, is also proven to be accurate and robust in recovering a spatial map of the absorption coefficient from DOT data. With the help of shape based representation of the inhomogeneities and an appropriate scaling of the CH expansion coefficients representing the boundary, we have been able to recover inhomogeneities representative of the shape of malignancies in medical diagnostic imaging.
作者旨在开发一种基于无导数集合卡尔曼滤波（EnKF）的伪时、次最优随机滤波方法，用于解决漫射光学断层扫描（DOT）的反问题，同时利用基于形状的重建策略，该策略能够通过一般闭合曲线来表示不均匀肿瘤边界的横截面。
使用基于圆谐（CH）（傅里叶基函数）的展开来近似恢复的光学参数场，并使用 EnKF 来恢复展开的系数，同时使用具有不均匀内含物的体模的模拟和实验获得的光子荧光数据。伪动态 EnKF（PD-EnKF）中的过程和测量方程目前为滤波器变量提供了一种简洁的表示，该变量仅由内含物内的傅里叶系数和常数标量参数值组成。在适当构建的“测量”方程中使用虚构的低强度 Wiener 噪声过程，将滤波器变量视为伪随机过程，从而可以在随机滤波框架内恢复它们。
在我们的数值模拟中，我们考虑了 2-D 物体中的椭圆形内含物（两个不均匀性）和具有更复杂形状的内含物（如环形和哑铃形），这些物体是背景吸收和（降低）散射系数分别选为μ(a)（b）=0.01mm(-1)和μ(s)（'b）=1.0mm(-1)的圆柱体的横截面。我们还假设μ(a)在不均匀性内为 0.02mm(-1)（对于单个不均匀性情况），并且μ(a)为 0.02 和 0.03mm(-1)（对于两个不均匀性情况）。PD-EnKF 的重建结果被证明始终优于通过确定性和显式正则化高斯牛顿算法的结果。我们还从实验收集的荧光数据中估计了未知的μ(a)，并通过将实验数据与计算数据匹配来验证重建。
PD-EnKF 对虚假引入噪声过程的变化表现出很小的敏感性，并且在从 DOT 数据中恢复吸收系数的空间图方面也被证明是准确和鲁棒的。借助基于形状的不均匀性表示和代表边界的 CH 扩展系数的适当缩放，我们能够从医学诊断成像中恢复代表恶性肿瘤形状的不均匀性。
