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Phase unwrapping reconstructs a wave phase field from measurements that determine only its value modulo $2\pi$. If a detector or complex-argument calculation returns ${\theta }_j=\mathrm{Arg}(e^{i{\varphi }_j})$, then all values ${\varphi }_j+2\pi n_j$ with integers $n_j$ give the same wrapped observation. Recovering a continuous wave phase therefore requires information beyond each isolated value. A primary 1982 analysis discusses this problem in wave phase curves obtained from image Fourier transforms before averaging them.^[ref]

Define $W(a)=\mathrm{Arg}(e^{ia})$ with principal interval $(-\pi ,\pi ]$. If adjacent true differences satisfy $|{\varphi }_{j+1}-{\varphi }_j|<\pi$, thenStarting from one reference wave phase, adding these wrapped differences reconstructs the sampled wave phase up to a global multiple of $2\pi$. The proof is direct: wrapping the two endpoint phases changes their difference only by an integer multiple of $2\pi$, and the strict bound selects its unique representative.

For an original example, true phases $0,0.8\pi ,1.6\pi ,2.4\pi$ wrap to $0,0.8\pi ,-0.4\pi ,0.4\pi$. Their wrapped increments are all $0.8\pi$, so reconstruction succeeds. But true increments $1.5\pi$ produce exactly the same wrapped samples as increments $-0.5\pi$. No algorithm can distinguish those two possibilities from the samples alone without a further assumption. Increasing sampling density can restore the bound only when the field's variation and noise allow it.

In two dimensions, different integration paths introduce a consistency test. A nonzero integer multiple of $2\pi$ in the sum of wrapped increments around a closed sampling loop is a phase-unwrapping residue. Noise, undersampling or true wave phase singularities can obstruct a single consistent smooth reconstruction. Choosing an integration path is therefore part of the inference.

For interferometry, low interference visibility can make the wave phase poorly determined even before unwrapping. Unwrapping does not recover information lost where the measured complex amplitude vanishes, nor does it distinguish an actual discontinuity from a smooth but insufficiently sampled change. Report the wave phase reference, principal interval, sampling and any continuity assumptions alongside the reconstructed field.