Reconstruction and Range Characterization for a Directional Confocal Non-Line-of-Sight Imaging Model

Abstract

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.