非均匀声速光声层析成像的直接法
A Direct Method for Photoacoustic Tomography with Inhomogeneous Sound Speed
摘要：
用于具有可变声速的光声成像的标准方法是时间反演，其包括在时间上向后求解波动方程的适定最终边值问题。本文研究了迭代Landweber正则化算法，其中收敛性是由标准正则化理论保证的，特别是在捕获声速或测量时间短的情况下。我们制定和解决整个欧几里得空间上的正逆问题，这是常见的标准光声成像，但不是时间反演算法，其中的问题被认为是由测量设备所包围的域。我们制定的直接和伴随的光声运营商的解决方案的内部和外部的微分方程耦合的传输条件。在空间上使用Galerkin方案和在时间上使用有限差分离散化来数值求解先前，而后者包括求解边界积分方程。因此，我们使用边界元-有限元方法的数值解的向前运营商。我们分析了这种方法，证明了收敛性，并提供了数值试验。此外，我们比较了时间反演的方法。
英文摘要：
The standard approach for photoacoustic imaging with variable speed of sound is time reversal, which consists in solving a well-posed final-boundary value problem for the wave equation backwards in time. This paper investigates the iterative Landweber regularization algorithm, where convergence is guaranteed by standard regularization theory, notably also in cases of trapping sound speed or for short measurement times. We formulate and solve the direct and inverse problem on the whole Euclidean space, what is common in standard photoacoustic imaging, but not for time-reversal algorithms, where the problems are considered on a domain enclosed by the measurement devices. We formulate both the direct and adjoint photoacoustic operator as the solution of an interior and an exterior differential equation which are coupled by transmission conditions. The prior is solved numerically using a Galerkin scheme in space and finite difference discretization in time, while the latter consists in solving a boundary integral equation. We therefore use a BEM-FEM approach for numerical solution of the forward operators. We analyze this method, prove convergence, and provide numerical tests. Moreover, we compare the approach to time reversal.
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