定向共焦非视距成像模型的重建与范围刻画
Reconstruction and Range Characterization for a Directional Confocal Non-Line-of-Sight Imaging Model
- Yau Mathematical Sciences Center, Tsinghua University（丘成桐数学科学中心，清华大学）
- Yanqi Lake Beijing Institute of Mathematical Sciences and Applications（北京雁栖湖应用数学研究院）
- Qiuzhen College, Tsinghua University（邱士栋学院，清华大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文研究共焦非视距成像中定向反照率场的重建，通过径向预处理和傅里叶-正弦公式唯一确定散度，刻画加权Hilbert空间范围，并提出FFT算法实现高效重建。
AI中文摘要：
我们研究了从共焦非视距测量中重建定向反照率场的问题。对于Sobolev空间中的镜像对称向量场，径向预处理将数据简化为以中继墙上的点为球心的散度的球面均值。完整数据唯一确定该散度，其中无散度场构成全部模糊性。一个显式的傅里叶-正弦公式恢复无旋的Helmholtz分量并重现数据。模型数据的傅里叶-正弦变换构成一个加权Hilbert空间，我们精确刻画了该空间，其范数等于重建场的Sobolev范数。即使预处理数据不属于标准Sobolev空间，该加权范围仍然定义良好。对于模型类中的Schwartz场，当且仅当散度具有零深度积分时，这种Sobolev正则性才成立。对于所有半径$0<r<R_{\max}$，在中继墙的开放子集上的测量唯一确定对应球并集内的散度。一种基于FFT的算法结合Stolt插值在$N^3$网格上以$O(N^3\log N)$次操作实现重建。对显式定义场的测试评估了重建精度以及在无散度扰动下的不变性。我们展示了从模型生成、渲染和测量的瞬态数据中进行势场和向量场重建的成像示例。
英文摘要：
We study the reconstruction of a directional albedo field from confocal non-line-of-sight measurements. For mirror-symmetric vector fields in Sobolev spaces, radial preprocessing reduces the data to spherical means of the divergence with centers on the relay wall. Full data determine this divergence uniquely, with divergence-free fields forming the entire ambiguity. An explicit Fourier--sine formula recovers the irrotational Helmholtz component and reproduces the data. The Fourier--sine transforms of the model data form a weighted Hilbert space that we characterize exactly, with norm equal to the Sobolev norm of the reconstructed field. This weighted range remains well defined even when the preprocessed data fail to belong to standard Sobolev spaces. For Schwartz fields in the model class, such Sobolev regularity holds exactly when the divergence has zero depth integral. Measurements on an open subset of the relay wall, for all radii $0<r<R_{\max}$, uniquely determine the divergence in the union of the corresponding balls. An FFT-based algorithm with Stolt interpolation implements the reconstruction in $O(N^3\log N)$ operations on an $N^3$ grid. Tests on explicitly defined fields assess reconstruction accuracy and invariance under divergence-free perturbations. We present imaging examples of potential and vector-field reconstruction from model-generated, rendered, and measured transients.
