Biswas Samir Kumar, Kanhirodan Rajan, Vasu Ram Mohan, Roy Debasish
Department of Physics, Indian Institute of Science, Bangalore, India.
J Opt Soc Am A Opt Image Sci Vis. 2012 Jun 1;29(6):1017-26. doi: 10.1364/JOSAA.29.001017.
We have developed an efficient fully three-dimensional (3D) reconstruction algorithm for diffuse optical tomography (DOT). The 3D DOT, a severely ill-posed problem, is tackled through a pseudodynamic (PD) approach wherein an ordinary differential equation representing the evolution of the solution on pseudotime is integrated that bypasses an explicit inversion of the associated, ill-conditioned system matrix. One of the most computationally expensive parts of the iterative DOT algorithm, the reevaluation of the Jacobian in each of the iterations, is avoided by using the adjoint-Broyden update formula to provide low rank updates to the Jacobian. In addition, wherever feasible, we have also made the algorithm efficient by integrating along the quadratic path provided by the perturbation equation containing the Hessian. These algorithms are then proven by reconstruction, using simulated and experimental data and verifying the PD results with those from the popular Gauss-Newton scheme. The major findings of this work are as follows: (i) the PD reconstructions are comparatively artifact free, providing superior absorption coefficient maps in terms of quantitative accuracy and contrast recovery; (ii) the scaling of computation time with the dimension of the measurement set is much less steep with the Jacobian update formula in place than without it; and (iii) an increase in the data dimension, even though it renders the reconstruction problem less ill conditioned and thus provides relatively artifact-free reconstructions, does not necessarily provide better contrast property recovery. For the latter, one should also take care to uniformly distribute the measurement points, avoiding regions close to the source so that the relative strength of the derivatives for measurements away from the source does not become insignificant.
我们已经为扩散光学断层扫描（DOT）开发了一种高效的全三维（3D）重建算法。3D DOT是一个严重不适定问题，通过伪动态（PD）方法解决，其中对表示伪时间上解的演化的常微分方程进行积分，绕过了对相关病态系统矩阵的显式求逆。迭代DOT算法中计算量最大的部分之一是每次迭代中雅可比矩阵的重新评估，通过使用伴随 - 布罗伊登更新公式为雅可比矩阵提供低秩更新来避免。此外，只要可行，我们还通过沿着包含海森矩阵的扰动方程提供的二次路径进行积分，使算法更高效。然后通过使用模拟和实验数据进行重建，并将PD结果与流行的高斯 - 牛顿方案的结果进行验证，来证明这些算法。这项工作的主要发现如下：（i）PD重建相对无伪影，在定量准确性和对比度恢复方面提供了更好的吸收系数图；（ii）使用雅可比更新公式时，计算时间随测量集维度的缩放比不使用时平缓得多；（iii）数据维度的增加，尽管它使重建问题的不适定性降低，从而提供相对无伪影的重建，但不一定能提供更好的对比度特性恢复。对于后者，还应注意均匀分布测量点，避免靠近源的区域，以使远离源的测量的导数相对强度不会变得微不足道。
