↔ Interactive schematic · inspect the mathematics

Wrapped phase → a continuous surface

The colors repeat, but a smooth surface need not. Integrate neighboring phase differences to recover its shape, then introduce a discontinuity, coarser sampling or an unknown starting depth. Compare this spatial prior with a second-frequency measurement.

Recovery method

Surface depth variation / m

First frequency / MHz

Frequency separation / MHz

Per-phase perturbation / rad

Spatial samples

Surface discontinuity / m

Initial cycle

—single-frequency depth interval λmod / 2 · m

—difference-frequency interval / m

—selected method · m per rad

Continuity supplies the missing cycle count.

The spatial method accumulates principal phase differences. If the underlying unwrapped phase change between neighbors, including measurement error, stays strictly below π in magnitude, the correct local cycle count can be recovered. This condition cannot be verified from wrapped differences alone. A known first depth fixes that global offset.

φ̃ᵢ = φ̃ᵢ₋₁ + principal(φᵢ − φᵢ₋₁)

Try a 6 m step: the plotted curve can remain smooth while representing the wrong depth. More samples help a smooth slope, but cannot resolve an arbitrary discontinuity.

Wavelength sets both range and sensitivity.

Here λmod=c/f is the wavelength of intensity modulation, not the optical wavelength. A single monostatic phase wraps every λmod/2 in depth, and a radian corresponds to λmod/(4π) meters. Higher frequency improves local sensitivity while shortening the unambiguous interval.

z = λmod φ / (4π), Δz per rad = λmod / (4π)

Two close frequencies provide a longer synthetic interval c/(2Δf), with sensitivity c/(4πΔf): the longer interval amplifies phase errors.

Assumptions and derivation

Both modes assume one direct round-trip path and a calibrated reference. The spatial method is one-dimensional nearest-neighbor integration, not a general 2D quality-guided or discontinuity-aware solver. Its sufficient local condition is |Δφ|<π, including measurement perturbations. At equality the direction is ambiguous. A cycle error propagates to all later samples; an unknown absolute anchor leaves a global integer ambiguity. Red lines mark violations using the known synthetic ground truth, which a real sensor would not know.

The two-frequency mode reports depth modulo c/(2|f₂−f₁|). It does not fuse that coarse cycle count into a high-frequency refinement. Neither spatial unwrapping nor a second frequency automatically separates multipath. Optical phase unwrapping uses related mathematics, but its conversion to height depends on the optical geometry, refractive index and reflection/transmission convention.

Computed phase wrapping and nearest-neighbor integration. Antonopoulos et al.: phase unwrapping and the neighbor condition CMU: CW-ToF range ambiguity