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Phase retrieval reconstructs a signal from measurements that discard its complex wave phase. A basic diffraction model measures $I(\xi )=|\hat{f}(\xi ){|}^2$, rather than the complex Fourier transform itself. The lost phase cannot generally be recovered simply by taking a square root of the intensity.

There are immediate ambiguities. Multiplication by a constant $e^{i\theta }$ leaves every magnitude unchanged. A translation multiplies the transform by a unit-modulus phase factor, and conjugate reflection $f(x)\mapsto \overline{f(-x)}$ conjugates its transform. These transformations therefore preserve $I$. They need not exhaust the ambiguities of one-dimensional data.^[ref]

By the convolution theorem, magnitude-squared data determine the autocorrelation of a square-integrable function under suitable inversion hypotheses, rather than automatically determining the function itself. Known support, additional masked measurements or other structural assumptions can reduce the ambiguity; their sufficiency must be checked for the actual measurement model. Frequency-resolved optical gating uses a family of delayed nonlinear spectra, giving more information than a single unmodified Fourier magnitude.

Robert Beinert coauthored an account of Fourier phase retrieval with Tamir Bendory and Yonina C. Eldar. Their analysis separates uniqueness questions from reconstruction algorithms and records the symmetries invisible to Fourier-magnitude measurements. This clarifies why an algorithm producing a signal is insufficient unless the measurement design also makes the signal identifiable.^[ref]

Tamir Bendory coauthored a study of Fourier phase retrieval with Robert Beinert and Yonina C. Eldar. It examines reconstruction from Fourier magnitudes and distinguishes unavoidable ambiguities from genuine uniqueness failures. Global phase, translation and conjugate reflection already preserve basic magnitude data, so meaningful recovery claims must account for these symmetries.^[ref]

Yonina C. Eldar coauthored a study of Fourier phase retrieval with Tamir Bendory and Robert Beinert. It compares basic magnitude measurements with richer designs such as masks or short-time measurements. Extra measurement structure can remove ambiguities that persist in the simplest model, linking experimental design with mathematical identifiability.^[ref]