理想压电传感器光声成像的可解性
Solvability for Photoacoustic Imaging with Idealized Piezoelectric Sensors
摘要：
用于光声成像的大多数重建算法假设压力场由放置在检测表面上的超声传感器测量。然而，这种传感器由于其不均匀的方向和频率响应以及分辨率限制而不能精确地测量压力。这是通常用于光声成像的压电传感器的情况。本文利用匹配渐近展开法和压电材料的基本本构关系，提出了一种简单的压电换能器的数学模型。该方法同时模拟了压力波如何引起压电测量以及传感器的存在如何影响压力波。利用这个模型，我们分析了压电传感器收集的数据是否导致光声成像问题的数学可解性。我们的结论是，这个成像问题是适定在一定的赋范空间和几何假设下。我们还提出了一个迭代重建算法，采用压电测量模型。重建算法的数值实现。
英文摘要：
Most reconstruction algorithms for photoacoustic imaging assume that the pressure field is measured by ultrasound sensors placed on a detection surface. However, such sensors do not measure pressure exactly due to their non-uniform directional and frequency responses, and resolution limitations. This is the case for piezoelectric sensors that are commonly employed for photoacoustic imaging. In this paper, using the method of matched asymptotic expansions and the basic constitutive relations for piezoelectricity, we propose a simple mathematical model for piezoelectric transducers. The approach simultaneously models how the pressure waves induce the piezoelectric measurements and how the presence of the sensors affects the pressure waves. Using this model, we analyze whether the data gathered by piezoelectric sensors leads to the mathematical solvability of the photoacoustic imaging problem. We conclude that this imaging problem is well-posed in certain normed spaces and under a geometric assumption. We also propose an iterative reconstruction algorithm that incorporates the model for piezoelectric measurements. A numerical implementation of the reconstruction algorithm is presented.
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