从空间和时间缩减的数据理论上精确的光声重建
Theoretically exact photoacoustic reconstruction from spatially and temporally reduced data
摘要：
我们调查的波动方程的逆源问题，在光和热声层析成像。在声速为常数和已知的假设下，已有不少理论上精确的反演公式，明确地表达了用实测数据求解这一问题的方法。然而，几乎所有这些公式都需要在无界表面上或完全围绕物体的闭曲面上测量数据。这对于实际应用来说限制太大。另一种方法，我们提出的，在一定的限制下的几何形状，产生理论上准确的重建标准氡投影的来源从有限的开放表面上测量的数据。此外，这种技术减少了应该知道数据的时间间隔。在一般情况下，我们的方法需要预先计算某些单层潜力的密度。然而，在截断的圆形或球形采集表面的情况下，这些密度很容易解析地获得，这导致完全明确的渐近快速算法。我们测试这些算法在一系列的数值模拟。
英文摘要：
We investigate the inverse source problem for the wave equation, arising in photo- and thermoacoustic tomography. There exist quite a few theoretically exact inversion formulas explicitly expressing solution of this problem in terms of the measured data, under the assumption of constant and known speed of sound. However, almost all of these formulas require data to be measured either on an unbounded surface, or on a closed surface completely surrounding the object. This is too restrictive for practical applications. The alternative approach we present, under certain restriction on geometry, yields theoretically exact reconstruction of the standard Radon projections of the source from the data measured on a finite open surface. In addition, this technique reduces the time interval where the data should be known. In general, our method requires a pre-computation of densities of certain single-layer potentials. However, in the case of a truncated circular or spherical acquisition surface, these densities are easily obtained analytically, which leads to fully explicit asymptotically fast algorithms. We test these algorithms in a series of numerical simulations.
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