三种成像反问题在隐空间中共享同一波方程，仅初值不同。
On a Hidden Property in Computational Imaging
- 将物理量与测量数据映射到隐空间，发现二者遵循相同单向波方程。
- 不同成像任务中，速度图与波形数据的初始条件呈线性相关。
- 该性质揭示了反问题间深层联系，适合研究逆问题建模者参考。
计算成像在地震波反演（FWI）、计算机断层扫描（CT）和电磁反演等科学与医学领域具有重要作用。这些方法通过从测量数据（如地震波形）重建物理属性（如声速分布）来解决反问题，且各类问题由复杂数学方程支配。本文通过实证发现，尽管三类反问题的控制方程不同，但其在隐空间中存在一个隐藏特性：以FWI为例，声速分布与地震波形数据在隐空间中均满足相同的单向波方程，但初始条件不同，且二者呈线性相关。这表明，投影至隐嵌入空间后，两类模态对应于同一方程的不同解，由初始条件关联。实验验证该性质在三种成像问题中均成立，为理解计算成像任务提供了新视角。
原文摘要 · Abstract (English)
Computational imaging plays a vital role in various scientific and medical applications, such as Full Waveform Inversion (FWI), Computed Tomography (CT), and Electromagnetic (EM) inversion. These methods address inverse problems by reconstructing physical properties (e.g., the acoustic velocity map in FWI) from measurement data (e.g., seismic waveform data in FWI), where both modalities are governed by complex mathematical equations. In this paper, we empirically demonstrate that despite their differing governing equations, three inverse problems (FWI, CT, and EM inversion) share a hidden property within their latent spaces. Specifically, using FWI as an example, we show that both modalities (the velocity map and seismic waveform data) follow the same set of one-way wave equations in the latent space, yet have distinct initial conditions that are linearly correlated. This suggests that after projection into the latent embedding space, the two modalities correspond to different solutions of the same equation, connected through their initial conditions. Our experiments confirm that this hidden property is consistent across all three imaging problems, providing a novel perspective for understanding these computational imaging tasks.
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