SIAM Journal on Mathematical Analysis, Volume 53, Issue 2, Page 2427-2451, January 2021.
We study the problem of recovering the initial data ($f, 0$) of the standard wave equation from the Neumann trace (the normal derivative) of the solution on the boundary of convex domains in arbitrary spatial dimension. Among others, this problem is relevant for tomographic image reconstruction including photoacoustic tomography. We establish explicit inversion formulas of the back-projection type that recover the initial data up to an additive term defined by a smoothing integral operator. In the case that the boundary of the domain is an ellipsoid, the integral operator vanishes, and hence we obtain an analytic formula for recovering the initial data from Neumann traces of the wave equation on ellipsoids.

中文翻译：

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从Neumann迹线中恢复波动方程的初始数据

SIAM数学分析杂志，第53卷，第2期，第2427-2451页，2021年1月。
我们研究了从Neumann迹线（正态导数）中恢复标准波动方程的初始数据（$ f，0 $）的问题。空间上凸域边界上解的解。其中，该问题与包括光声层析成像的层析图像重建有关。我们建立了反投影类型的显式反演公式，可以将初始数据恢复到由平滑积分算符定义的加法项。在畴的边界是椭球的情况下，积分算子消失了，因此我们得到了一个解析公式，用于从椭球上波动方程的诺伊曼迹线中恢复初始数据。