Illuminate the same surface with a tilted plane, a delayed sheet, a raster spot or a swept line. Choose a global or synchronized rolling shutter, then inspect the recorded fringes, their spectrum and the field recovered by one Fourier crop.
A delayed planar sheet attaches a phase ramp to X in the scene. A perspective camera sees X proportional to pixel position times depth, so one surface can carry several local fringe frequencies. A calibrated projector or sensor reference can instead attach its ramp directly to image columns. Physical plane-wave tilt changes propagation delay; it does not guarantee a clean, high-frequency image carrier.
Pᵢ = Σⱼ Gᵢⱼ Tᵢⱼ exp[i(2πf Lᵢⱼ/c + βⱼ)]
Hᵢ = Re[Pᵢ exp(iαᵢ)]
For a scan, rolling exposure rejects source columns far from the receiving column. Indirect returns within the same column can survive. A steady flood source has no column-time correspondence, so merely selecting rolling shutter has no such benefit here.
The reconstruction applies a genuine 2D FFT, one circular sideband crop, a frequency shift and an inverse FFT. Known nonlinear phase can be corrected afterward, but that cannot restore frequencies already discarded or overlapped. Four ideal correlation taps recover the reference complex field without requiring sideband separation.
This is a finite synthetic transport operator on a 64×64 perspective-sampled surface. The displayed paths use Euclidean lengths. Direct transport is diagonal; seven explicitly weighted indirect connections include displaced columns, same-column neighbors and an extra 3 m return at the same pixel. These weights make the comparison inspectable; they are not a full visibility/BSDF simulation or a measured room.
For the plane wave, the illumination path term is Z cosθ + X sinθ. For the delayed sheet it is Z, with β=2π d(½+X/13). The projector is collocated with the camera, so its direct angular addressing matches image columns and β=2π d·x/N. A real offset projector requires geometric calibration. The sensor phase is α=2π s·x/N. The displayed signal is a balanced CW correlation cross-term with DC removed, not optical-carrier interference.
Rolling acceptance is Gᵢⱼ=exp[−½((xⱼ−xᵢ−offset)/width)²] for scanned sources and one for steady illumination. Scanning occurs on a slow acquisition clock; the MHz correlation integrates modulation within each exposure. Point and line scans have the same normalized column transport here when a column stays open for its complete scan. Their illumination support and addressed position count differ; this model does not benchmark acquisition speed, noise or optical power. The scan slider inspects illumination position, while the recorded image integrates the complete scan.
The crop is centered at the nominal integer carrier and may overlap a conjugate band or alias at Nyquist: those are meaningful failure cases. The four-tap reference is known from the forward complex sum; it uses additional phase measurements. Removing a world-dependent carrier additionally assumes its phase has been calibrated, potentially requiring known geometry. Arbitrary fringes are not categorically unrecoverable: calibration, resampling or a different inverse model can work when sufficient information remains.
Builds on this manuscript’s illumination/demodulation experiments. Achar et al.: Epipolar Time-of-Flight Imaging Cuche et al.: off-axis Fourier filtering